Overview
-The zip file contains
>>> A binary file BIONS48 ( #2003d / 20618.5 bytes ) which
works with the HP48S/SX/G/GX
All are directories, not libraries.
>>> An ascii file BIONS which should work with the HP48-HP49-HP-50
>>> This html page.
-----------------------------------------------------------------------------------------------------------------------------
-In "Anionic Functions for the HP-48", we have used the Cayley-Dickson formula
( a , b ) ( c , d ) = ( a c - d* b , d a + b c* ) where * = conjugate
to construct the complexes from the real numbers, regarding a
complex as a pair of real numbers,
and then the quaternions from the complexes, the octonions from
the quaternions, the sedenions from the octonions,
the 32-ons from the sedenions, the 64-ons from the 32-ons ...
and so on ... for the 2^n-ons
>>> But if we start with the complexes, we gradually get the bi-complexes, bi-quaternions, bi-octonions, bi-sedenions ...
-The programs listed in this page calculate elementary functions and
a few special functions of these "bi-ons" = "complexified anions".
Notes:
-When the functions are computed by ascending series, the results are accurate for "small" arguments only.
-The variables are bions ( hypercomplexes ), but the parameters are
restricted to complex values.
-Bions must be entered as N-dimensional complex vectors ( N > 1 )
-For example, the biquaternion b = ( 6 - 9 i ) + ( -4 + 2 i ) e1 + ( -3 + i ) e2 + ( -4 + 7 i ) e3 must be entered [ ( 6 , -9 ) ( -4 , 2 ) ( -3 , 1 ) ( -4 , 7 ) ]
-Set your HP-48 in RECT mode.
| Functions | Description |
| MU
SOLVB FXB peval tol Im BxB0 SPEC LNGZ GAMZ PSIZ GAMB PSIB PSINB ZETAB HGFB ALFB ALF12 ALF13 S ALF22 ALF23 BESSEL JNB YNB INB KNB S1 S2 CWFB RCWFB ICWFB AECWFB DNB DNB AEDNB ELLIPB JEFB WEFB WF2B ESINT EIB ENB AENB SIB SHIB CIB CHIB S1 S2 MISC AIBIB ANWEB CATB CHB1B CHB2B ERFB GERFB HMTB IBFB IGFB JCFB LANB LERCHB LOM1B LOM2B STRHB STRLB TORB USFB WHIMB WHIWB S1 S2 ORTHO LEGB LANB CHB1B CHB2B JCPB USPB PMN2B PMN3B S1 S2 SPHR SMNB L LMN M N C2 ITER FRIND ELEM DEXPB DLNB DBX DSHB DCHB DSINB DCOSB S1 S2 DAIRYB DSB DCB DERFB DHMTB DJNB DINB DYNB DKNB DBS1B DLANB DEIB DSIB DSHIB DCIB DCHIB S1 S2 S3 S4 S5 S6 BxB INVB EXPB LNB BB BX XB SHB CHB THB ASHB ACHB ATHB SINB COSB TANB ASINB ACOSB ATANB GDB AGDB QxQ |
computes a complex number called µ
Directory = Polynomials & Solving a Bionic Equation Solves f(b) = b Evaluates a bionic polynomial with complex coefficients tolerance = a small positive number Imaginary part of a bion Product of 2 bions whose imaginary parts are proportional Directory = Special Functions LogGamma function of a complex Gamma function of a complex Digamma function of a complex Gamma function of a bion Digamma function Polygamma function Riemann Zeta function Generalized Hypergeometric Functions Directory = Associated Legendre Functions Legendre Functions - 1st kind - Type2 Legendre Functions - 1st kind - Type3 Subroutine Legendre Functions - 2nd kind - Type2 Legendre Functions - 2st kind - Type3 Directory = Bessel Functions Bessel functions of the 1st kind Bessel functions of the 2nd kind Modified Bessel functions of the 1st kind Modified Bessel functions of the 2nd kind Subroutine1 Subroutine2 Directory = Coulomb Wave Functions Regular Coulomb Wave function Irregular Coulomb Wave function Asymptotic Expansion Directory = Parabolic Cylinder Functions Parabolic Cylinder function Asymptotic Expansion Directory = Elliptic Functions Jacobian Elliptic functions Weierstrass Elliptic functions Weierstrass Duplication formula Directory = Exponential, Sine & Cosine Integral Exponential Integral Ei(b) Exponential Integral En(b) Asymptotic Expansion for En(b) Sine Integral Hyperbolic Sine Integral Cosine Integral Hyperbolic Cosine Integral Subroutine1 Subroutine2 Directory = Miscellaneous Functions Airy functions Ai(b) & Bi(b) Anger & Weber functions Catalan Numbers Chebyshev functions - 1st kind Chebyshev functions - 2nd kind Error function Generalized Error function Hermite functions Incomplete Beta function Incomplete Gamma function Jacobi functions Laguerre's functions Lerch Transcendent functions Lommel functions - 1st kind Lommel functions - 2nd kind Struve function H Struve function L Toronto functions UltraSpherical functions Whittaker's M-functions Whittaker's W-functions Subroutine1 Subroutine2 Directory = Orthogonal Polynomials Legendre Polynomials Generalized Laguerre's Polynomials Chebyshev Polynomials - 1st kind Chebyshev Polynomials - 2nd kind Jacobi Polynomials UltraSpherical Polynomials Associated Legendre Functions - 1st kind - Type2 Associated Legendre Functions - 1st kind - Type3 Subroutine1 Subroutine2 Directory = Spheroidal Wave Functions Angular Spheroidal Wave Function of the 1st kind Eigenvalue Calculating the eigenvalues Parameter m Parameter n Parameter c2 Number of iterations ( default = 12 ) Directory = Fractional Integro-Differentiation Directory = Elementary Functions Exponential Logarithm Bion raised to a real or complex exponent Hyperbolic Sine Hyperbolic Cosine Sine Cosine Subroutine1 Subroutine2 Airy functions Ai(b) & Bi(b) Fresnel Sine Integral Fresnel Cosine Integral Error function Hermite functions Bessel function of the 1st kind Modified Bessel function of the 1st kind Bessel function of the 2nd kind Modified Bessel function of the 2nd kind Spherical Bessel function of the 1st kind Laguerre's function Exponential Integral Sine Integral Hyperbolic Sine Integral Cosine Integral Hyperbolic Cosine Integral Subroutine1 Subroutine2 Subroutine3 Subroutine4 Subroutine5 Subroutine6 Product of 2 bions - general case - Cayley-Dickson doubl. Inverse of a bion Exponential of a bion Logarithm of a bion Raising a bion to a bionic exponent Raising a bion to a real or complex exponent Raising a real or a complex to a bionic exponent Hyperbolic Sine Hyperbolic Cosine Hyperbolic Tangent Inverse Hyperbolic Sine Inverse Hyperbolic Cosine Inverse Hyperbolic Tangent Sine of a bion Cosine of a bion Tangent of a bion Arc Sine Arc Cosine Arc Tangent Gudermannian Function Inverse Gudermannian Function Product of 2 Quaternions |
Computing the complex number µ
-If B = b0 + b1 e1 + ............... + bn-1 en-1 is a "bi-on",
MU returns µ = ( b12
+ ............... + bN-12 )1/2
| STACK | INPUT | OUTPUT |
| Level 1 | B | µ |
Example: B = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8) ] MU gives ( 0.910380820346 , 1.07647259048 )
-This elementary program is called as a subroutine to compute several
functions.
Im returns the "imaginary part" of [ b0 , b1 , ..... , bn ] i-e [ b1 , ..... , bn ]
-It's useful to visualize more easily a quaternion - at least with an
HP-48, but it's also called as a subroutine.
-The "imaginary part" of a bion could also be the anion defined as
[ Im b0 , Im b1 , ..... , Im bn ]
Product of 2 bions whose imaginary parts are proportional
-The Cayley-Dickinson formula is not very easy to use and it is also
very slow ( see the next paragraph ).
-Fortunately, the formulas are much simpler if the 2 bi-ons have the
same imaginary direction, I mean if, given b & b'
b = b0 + b1 e1
+ ............... + bn-1 en-1
b' = b'0 + b'1 e1
+ ............... + b'n-1 en-1
there exist a complex number c such that, for all i > 0
, b'i = c bi or bi =
c b'i
-It's often the case to compute elementary or special functions of
a bion:
| STACK | INPUTS | OUTPUTS |
| Level 2 | B | / |
| Level 1 | B' | B B' |
Example: Find the product of the biquaternions
B = ( 6 - 9 i ) + ( -4 + 2 i ) e1 + ( -3 + i ) e2
+ ( -4 + 7 i ) e3
here, Im(b') = ( 2 + 3i ) Im(b)
B' = ( 4 + 7 i ) + ( -14 - 8 i ) e1 + ( -9 - 7 i
) e2 + ( -29 + 2 i ) e3
[ ( 6 , -9 ) ( -4 , 2 ) ( -3 , 1 ) ( -4 , 7 ) ]
ENTER
[ ( 4 , 7 ) ( -14 , -8 ) ( -9 , -7 ) ( -29 , 2 ) ] BxB0
>>>> [ ( -121 , 201 ) ( -186 , 58 ) ( -136 , 22 ) ( -221 ,
273 ) ]
-So, B B' = ( -121 + 201 i ) + ( -186 + 58 i ) e1 + ( -136 + 22 i ) e2 + ( -221 + 273 i ) e3
Notes:
-Of course, the next program BxB gives the same result but BxB0 is much
faster.
-Unlike BxB below, BxB0 also works with N-vectors where N > 1 is not
an integer power of 2.
-BxB0 does not check that the imaginary parts are proportional...
Product of 2 bions - general case - Cayley-Dickson doubling
-BxB uses the Cayley-Dickson formula ( a , b ) ( c , d ) = ( a c - d* b , d a + b c* ) where * = conjugate
-BxB is a recursive program: it calls itself as a subroutine until we
reach the multiplication of 2 quaternions which is computed by QxQ
-But for the product of 2 bicomplexes, it simply calls BxB0
| STACK | INPUTS | OUTPUTS |
| Level 2 | B | / |
| Level 1 | B' | B B' |
Example: Find the product of the bisedenions:
B = ( 25 + 108 i ) + ( 105 + 113 i ) e1 + ( 48 + 3
i ) e2 + ( 123 + 65 i ) e3 + ( 45 + 11 i ) e4
+ ( 58 + 20 i ) e5 + ( 34 + 84 i ) e6 + ( 38 + 117
i ) e7
+ ( 81 + 46 i ) e8 + ( 52 + 36 i ) e9
+ ( 35 + 125 i ) e10 + ( 16 + i ) e11 + ( 41 + 109
i ) e12 + ( 15 + 91 i ) e13 + ( 63 + 94 i ) e14
+ ( 55 + 28 i ) e15
B' = ( 100 + 39 i ) + ( 27 + 59 i ) e1 + ( 61 + 12
i ) e2 + ( 99 + 129 i ) e3 + ( 49 + 44 i ) e4
+ ( 101 + 80 i ) e5 + ( 5 + 74 i ) e6 + ( 21 + 75
i ) e7
+ ( 62 + 53 i ) e8 + ( 77
+ 13 i ) e9 + ( 9 + 107 i ) e10 + ( 64 + 4 i ) e11
+ ( 33 + 43 i ) e12 + ( 60 + 102 i ) e13 + ( 121
+ 114 i ) e14 + ( 89 + 112 i ) e15
-You should find:
B.B' = ( 22450 - 93103 i ) + ( -3665 + 12974 i )
e1 + ( 32291 - 31812 i ) e2 + ( 18482 + 29668 i )
e3 + ( -7840 + 8985 i ) e4 + ( 1001 + 48348 i ) e5
+ ( -10494 +
6090 i ) e6 + ( -1657 + 25709 i ) e7 + ( -14243 -
595 i ) e8 + ( 9295 + 1562 i ) e9 + ( -19125
+ 35197 i ) e10 + ( 18449 - 20036 i ) e11
+ ( -3315 +
35478 i ) e12 + ( -22176 + 9676 i ) e13 + ( 11818
+ 23190 i ) e14 + ( -14895 + 36483 i ) e15
Notes:
-Since the multiplication is not commutative, place the 1st bion in
level2 and the 2nd bion in level1.
-If you enter N-dimensional vectors where N is not an integer power
of 2, there will be an error message "somewhere".
B-1 = B* / | B |2 where B* is the conjugate of B and | B |2 = b02 + b12 + ................. + bN-12
-Note that here | B | may equal 0 even if b # 0 since the
components bj are complexes !
-So, b # 0 may not have an inverse !
| STACK | INPUT | OUTPUT |
| Level 1 | B | 1/B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
INVB returns ( 0.2710 - 0.2285 i ) +
( -0.2115 + 0.1835 i ) e1 + ( -0.1521 + 0.1385 i ) e2
+ ( -0.0927 + 0.0936 i ) e3
rounded to 4 decimals
Formula: exp( b0 + b1 e1 + .... + bN-1 bN-1 ) = eb0 [ cos µ + ( sin µ ). I ]
where µ = ( b12
+ ............... + bN-12 )1/2
and I = ( b1 e1 + .............
+ bN-1 eN-1 ) / µ
| STACK | INPUT | OUTPUT |
| Level 1 | B | Exp(B) |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
EXPB gives exp(b) = ( 3.1095 - 0.0905 i ) + ( 0.7810
+ 2.6836 i ) e1 + ( 0.6211 + 1.9573 i ) e2 + ( 0.4613
+ 1.2310 i ) e3
rounded to 4 D
-"LNB" uses the formulae:
Ln( b0 + b1 e1 + .... + bN-1 bN-1 ) = Ln ( b02 + b12 + ................. + bN-12 )1/2 + A(µ,b0). I
where µ = ( b12 + ............... + bN-12 )1/2 and I = ( b1 e1 + ............. + bN-1 eN-1 ) / µ
>>> Here, A(µ,b0) generalizes the ATAN2 function to complexes:
-We have to solve Sin Z = µ / ( µ2 + b02 )1/2 , Cos Z = b0 / ( µ2 + b02 )1/2
it yields Z = A(µ,b0) = - i Ln ( b0 + i µ ) / ( µ2 + b02 )1/2
>>> If µ = 0 and Im(b) # 0 , Ln b = Ln
b0 + ( b1 e1 + ............. + bN-1
eN-1 ) / b0 ( if b0
= 0 , Ln b does not exist )
>>> If µ = 0 and Im(b) = 0 , Ln b = Ln
b0
| STACK | INPUT | OUTPUT |
| Level 1 | B | Ln (B) |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
LNB >>>> Ln (b) = ( 0.6669 + 0.7167
i ) + ( 0.6100 - 0.0013 i ) e1 + ( 0.4480 - 0.0118 i ) e2
+ ( 0.2860 - 0.0222 i ) e3
rounded to 4 D
Raising a bion to a bionic exponent
-This may be defined in several ways.
-'BB' employs
the formula: B^B' = exp [ ( Ln B ) B' ]
| STACK | INPUTS | OUTPUTS |
| Level 2 | B | / |
| Level 1 | B' | B^B' |
Example:
B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1
+ ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
B' = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1
+ ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
BB returns ( 1.0076 - 0.6123 i ) + ( 0.3685 + 0.432 i ) e1 + ( -0.0661 + 0.4207 i ) e2 + ( 0.5126 + 0.745 i ) e3 rounded to 4D
Note:
-Another definition is possible: bb' = exp [
b' ( Ln b ) ]
Raising a bion to a real or complex exponent
BX
calculates B^X = exp ( X Ln B ) where X is a real
or complex number and B is a bion
| STACK | INPUTS | OUTPUTS |
| Level 2 | B | / |
| Level 1 | X | B^X |
Example:
B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2
+ ( 0.4 + 0.3 i ) e3 X =
2 + 3 i
BX
>>>> ( -0.4735 + 2.3593 i ) + ( -1.8006 - 0.3988 i ) e1
+ ( -1.3296 - 0.2611 i ) e2 + ( -0.8587 - 0.1233 i ) e3
rounded to 4 D
Raising a real or a complex to a bionic exponent
XB
calculates X^B = exp ( B Ln X )
| STACK | INPUTS | OUTPUTS |
| Level 2 | X | / |
| Level 1 | B | X^B |
Example: X = 2 + 3 i and B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
XB
>>>> ( -0.6233 + 6.9092 i ) + ( -5.1426 - 0.4513 i ) e1
+ ( -3.7853 - 0.2404 i ) e2 + ( -2.428 - 0.0296 i ) e3
rounded to 4 D
-We have: Sinh ( x0 + x1 e1 + .... + xN-1 eN-1 ) = Sinh x0 Cos µ + I ( Cosh x0 ) ( Sin µ )
where µ = ( x12
+ ............... + xN-12 )1/2
and I = ( x1 e1 + .............
+ xN-1 eN-1 ) / µ
| STACK | INPUT | OUTPUT |
| Level 1 | B | Sinh B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
SHB >>>> ( 1.6085 + 0.1583 i )
+ ( 0.5553 + 1.2491 i ) e1 + ( 0.4300 + 0.9076 i ) e2
+ ( 0.3047 + 0.5662 i ) e3
rounded to 4 D
Cosh ( x0 + x1 e1 + .... + xN-1 eN-1 ) = Cosh x0 Cos µ + I ( Sinh x0 ) ( Sin µ )
where µ = ( x12
+ ............... + xN-12 )1/2
and I = ( x1 e1 + .............
+ xN-1 eN-1 ) / µ
| STACK | INPUT | OUTPUT |
| Level 1 | B | Cosh B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
CHB >>>> ( 1.5010 - 0.2488 i ) + ( 0.2256
+ 1.4345 i ) e1 + ( 0.1911 + 1.0497 i ) e2 + ( 0.1566
+ 0.6648 i ) e3
rounded to 4 D
Tanh B is defined by Tanh B = ( Sinh B ) ( Cosh B
) -1
| STACK | INPUT | OUTPUT |
| Level 1 | B | Tanh B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
THB >>>> ( 0.7548 - 1.1142 i ) + ( -0.8316
+ 0.1404 i ) e1 + ( -0.6083 + 0.1179 i ) e2 + ( -0.3851
+ 0.0953 i ) e3
rounded to 4 D
Formula:
ArcSinh B = Ln [ B + ( B2 + 1 )1/2 ]
| STACK | INPUT | OUTPUT |
| Level 1 | B | Asinh B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
ASHB >>>> ( 1.3553 + 0.7183 i ) + ( 0.6025
+ 0.0482 i ) e1 + ( 0.4434 + 0.0247 i ) e2 + ( 0.2843
+ 0.0013 i ) e3
rounded to 4 D
Formula:
ArcCosh B = Ln [ B + ( B + 1 )1/2 ( B - 1 )1/2 ]
| STACK | INPUT | OUTPUT |
| Level 1 | B | Acosh B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
ACHB >>>> ( 1.3521 + 0.7113 i ) + ( 0.6185
- 0.0512 i ) e1 + ( 0.4534 - 0.0485 i ) e2 + ( 0.2883
- 0.0459 i ) e3
rounded to 4 D
Formula:
ArcTanh B = (1/2) [ Ln ( 1 + B ) - Ln ( 1 - B ) ]
| STACK | INPUT | OUTPUT |
| Level 1 | B | Atanh B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
ATHB >>>> ( 0.2955 - 0.2011 i ) + ( 0.9797
+ 0.2256 i ) e1 + ( 0.7236 + 0.1484 i ) e2 + ( 0.4675
+ 0.0711 i ) e3
rounded to 4 D
Formula: Sin ( x0 + x1 e1 + .... + xN-1 eN-1 ) = Sin x0 Cosh µ + I ( Cos x0 ) ( Sinh µ )
where µ = ( x12
+ ............... + xN-12 )1/2
and I = ( x1 e1 + .............
+ xN-1 eN-1 ) / µ
| STACK | INPUT | OUTPUT |
| Level 1 | B | Sin B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
SINB >>>> ( 0.6434 + 1.7921 i ) + ( 1.3025
+ 0.2589 i ) e1 + ( 0.9613 + 0.1671 i ) e2 + ( 0.6201
+ 0.0753 i ) e3
rounded to 4 D
Formula: Cos ( x0 + x1 e1 + .... + xN-1 eN-1 ) = Cos x0 Cosh µ + I ( Sin x0 ) ( Sinh µ )
where µ = ( x12
+ ............... + xN-12 )1/2
and I = ( x1 e1 + .............
+ xN-1 eN-1 ) / µ
| STACK | INPUT | OUTPUT |
| Level 1 | B | Cos B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
COSB >>>> ( 1.6624 - 0.0748 i ) + ( -0.1571
- 1.5114 i ) e1 + ( -0.1421 - 1.1074 i ) e2 + ( -0.1272
- 0.7033 i ) e3
rounded to 4 D
Formula: Tan B = ( Sin B ) (
Cos B ) -1
| STACK | INPUT | OUTPUT |
| Level 1 | B | Tan B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
TANB >>>> ( -1.0993 - 0.1510 i ) + (
0.8538 - 0.8196 i ) e1 + ( 0.6127 - 0.6171 i ) e2
+ ( 0.3715 - 0.4146 i ) e3
rounded to 4 D
Formula: If B = x0 + x1 e1 + .... + xN-1 eN-1 , Arc Sin B = - I Arc Sinh ( B I )
where µ = ( x12
+ ............... + xN-12 )1/2
and I = ( x1 e1 + .............
+ xN-1 eN-1 ) / µ
| STACK | INPUT | OUTPUT |
| Level 1 | B | Asin B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
ASINB >>>> ( 0.7517 + 0.0817 i ) + (
1.0103 + 0.5534 i ) e1 + ( 0.7519 + 0.3886 i ) e2
+ ( 0.4935 + 0.2238 i ) e3
rounded to 4 D
ArcCos B = PI/2 - ArcSin B
| STACK | INPUT | OUTPUT |
| Level 1 | B | Acos B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
ACOSB >>>> ( 0.8191 - 0.0817 i ) + (
-1.0103 - 0.5534 i ) e1 + ( -0.7519 - 0.3886 i ) e2
+ ( -0.4935 - 0.2238 i ) e3
rounded to 4 D
Formula: If B = x0 + x1 e1 + .... + xN-1 eN-1 , Arc Tan B = - I Arc Tanh ( B I )
where µ = ( x12
+ ............... + xN-12 )1/2
and I = ( x1 e1 + .............
+ xN-1 eN-1 ) / µ
| STACK | INPUT | OUTPUT |
| Level 1 | B | Atan B |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
ATANB >>>> ( -0.2551 + 0.2588 i ) + (
0.2065 + 1.0188 i ) e1 + ( 0.1697 + 0.7447 i ) e2
+ ( 0.1329 + 0.4705 i ) e3
rounded to 4 D
Definition: Gd(B) = 2 ArcTan
[ Tanh (B/2) ]
| STACK | INPUT | OUTPUT |
| Level 1 | B | Gd (B) |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
GDB >>>> ( -1.0420 + 0.7537 i ) + ( 0.5771
+ 1.7617 i ) e1 + ( 0.4551 + 1.2838 i ) e2 + ( 0.3331
+ 0.8058 i ) e3
rounded to 4D
Definition: Agd(B) = 2 ArcTanh
[ Tan (B/2) ]
| STACK | INPUT | OUTPUT |
| Level 1 | B | Agd (B) |
Example: B = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
AGDB >>>> ( 0.8355 - 0.6779 i ) + ( 1.4782
+ 0.6366 i ) e1 + ( 1.0970 + 0.4414 i ) e2 + ( 0.7158
+ 0.2463 i ) e3
rounded to 4D
| STACK | INPUTS | OUTPUTS |
| Level 2 | Q | / |
| Level 1 | Q' | Q Q' |
Example: Calculate the product of the quaternions: q = 2 - 3i + 4j - 7k and q' = 1 - 4i + 2j + 5k
[ 2 -3 4 -7 ]
ENTER
[ 1 -4 2 5 ]
QxQ gives [ 17 23
51 13 ]
Note:
QxQ also works with biquaternions.
Solving a Bionic Equation f(b) =
b
-The equation must be rewritten in the form: f ( b
) = b
and f must satisfy a Lipschitz condition
| f(b) - f(b') | < h | b - b' | with h <
1 , provided b and b' are close to a solution,
then the sequence bn+1 = f ( bn
) converges to a root.
-Place a small positive number in 'tol'
-In level 1, place a program that takes b in level 1 and returns f(b)
in level 1
-Place an approximation in level 1 and press the key FXB
-The HP48 will display the successive approximations and eventually
the ( or a ) solution in level 1
| STACK | INPUTS | OUTPUTS |
| Level 2 | << p >> | << p >> |
| Level 1 | b | Solution |
where << p >> calculates f(b) and b = 1st approximation
Example: Find a solution of the biquaternionic equation
b2 - Ln b + ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3 = 0 near b = 1
1°), we re-write this equation: b = [ Ln b - { ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3 } ] 1/2 = f ( b )
2°) Store a small number - say 2E-11 - in 'tol'
3°) Then, << LNB
[ ( 0.1 0.2 ) ( 0.3 0.4 ) ( 0.5 0.6
) ( 0.7 0.8 ) ] - 2 INV BX
>> ENTER
[ 1 0 0 0 ] FXB
displays the successive approximations
and returns eventually the solution:
b = ( 1.01498310538 + 0.292115635013 i ) +
( -0.307551610372 - 0.118387776545 i ) e1
+ ( -0.489251534305 - 0.160080802581 i ) e2 + ( -0.670951458238
- 0.201773828616 i ) e3
Evaluating a Bionic Polynomial
'peval' evaluates p(b) = cm bm
+ cm-1 bm-1 + ................ + c1 b
+ c0 for a given bion b and
(m+1) complex numbers cm , .............. , c1
, c0
| STACK | INPUTS | OUTPUTS |
| Level 2 | p | p |
| Level 1 | b | p(b) |
where p is a polynomial with complex ( or real ) coefficients and b is a bion.
Example: b = ( 1 + 2 i ) + ( 3 + 4 i ) e1 + ( 5 + 6 i ) e2 + ( 7 + 8 i ) e3
and p(x) = ( 2 + 3 i ) x4
+ ( 4 - 7 i ) x3 + ( 3 - 5 i ) x2 + ( 1 + 4 i ) x
+ ( 6 + 2 i )
[ ( 2 , 3 ) ( 4 , -7 ) ( 3 , -5 ) ( 1
, 4 ) ( 6 , 2 ) ] ENTER
[ ( 1 , 2 ) ( 3 , 4 ) ( 5 , 6 ) ( 7 , 8 ) ]
peval >>>>
[ ( -39087 , -128086 ) ( -863 , 23966 ) ( 571 , 37456 ) ( 2005 , 50946
) ]
whence p(b) = ( -39087 - 128086 i ) + ( -863 + 23966 i
) e1 + ( 571 + 37456 i ) e2 + ( 2005 + 50946 i )
e3
'LNGZ' computes LnGamma(z) = (z-1/2) Ln(z) - z + (1/2) Ln (2.PI) + ( 1/12 )/( z + ( 1/30 )/( z + ( 53/210)/( z + (195/371)/( z + ... ))))
together with the relation LnGamma ( z+1 ) = Ln
z + LnGamma (z)
| STACK | INPUT | OUTPUT |
| Level 1 | z | LnGamma(z) |
where z = complex number
Example: z = 3 + 4 i
( 3 , 4 ) LNGZ gives ( -1.75662678467 , 4.742664438 ) = -1.75662678467 + 4.742664438 i
Notes:
-Remark that Ln ( Gamma ( 3+4.i ) ) = -1.7566... - 1.5405...
i is different.
-So, LnGamma is not always the same as Ln ( Gamma ), though the real
parts are always equal.
-Unlike Ln ( Gamma ) , LnGamma has a single branch cut: the negative
real semi-axis.
'GAMZ' calculates the Gamma function for complex arguments
A continued fraction is used.
However, if z is real number, the built-in factorial function
is employed since Gam x = ( x-1 ) !
| STACK | INPUT | OUTPUT |
| Level 1 | z | Gamma(z) |
where z = complex number
Example: z = 3 + 4 i
( 3 , 4 ) GAMZ >>> ( 0.00522553847158 , - 0.172547079294 )
Notes:
-If you have an HP-50G, store the program << GAMMA
>> in the 'GAMZ' variable
-You'll get a better precision !
-The asymptotic expansion Psi(z) = ln z - 1/(2z) -1/(12z2)
+ 1/(120z4) - 1/(252z6) + 1/(240z8)
is used for Ré(z) > 12
together with the property: Psi(z+1) = Psi(z) + 1/z
| STACK | INPUT | OUTPUT |
| Level 1 | z | Psi (z) |
where z = complex number
Example: z = 3 + 4 i
( 3 , 4 ) PSIZ >>> ( 1.55035981733 , 1.01050220919 )
Notes:
-If you have an HP-50G, store the program << Psi >>
in the 'PSIZ' variable
-You'll probably get a better precision... though in this example,
the results are identical !
-Here, the Gamma function is computed by a continued fraction:
Gam(b) ~ exp [ (b-1/2) Ln b + Ln (2.PI)1/2
- b + ( 1/12 )/( b + ( 1/30 )/( b + ( 53/210)/( b + (195/371)/( b + ...
)))) ]
-The relation Gam(b+1) = b Gam(b) is used recursively
if Re(Re((b)) < 8 until Re(Re(b+1+.......+1))
> 8
| STACK | INPUT | OUTPUT |
| Level 1 | B | Gamma (B) |
Example: b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] GAMB returns
Gam(b) = ( 0.258452256 + 0.166123160 i ) + ( 0.014416807
+ 0.150189863 i ) e1
+ ( 0.034505408 + 0.233142841 i ) e2 + ( 0.054594009
+ 0.316095819 ) e3 rounded to 9D
Formula: Psi(b) ~ Ln b - 1/(2b) -1/(12b2) + 1/(120b4) - 1/(252b6) + 1/(240b8) is used if Re(b) > 8
Psi(b+1) = Psi(b) + 1/b is used recursively until
Re(b+1+....+1) > 8
| STACK | INPUT | OUTPUT |
| Level 1 | B | Psi (B) |
Example: b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] PSIB gives
Psi(b) = ( 0.300919281 + 0.872730291 i ) + ( 0.587618165
- 0.077794679 i ) e1
+ ( 0.910460763 - 0.168369152 i ) e2 + ( 1.233303362
- 0.258943626 ) e3 rounded
to 9D
Formulae:
• If m > 0 , y(m) (b) ~ (-1)m-1 [ (m-1)! / bm + m! / (2.bm+1) + SUMk=1,2,.... B2k (2k+m-1)! / (2k)! / b2k+m where B2k are Bernoulli numbers
• If m = 0 , y (b) ~ Ln b - 1/(2b) - SUMk=1,2,.... B2k / (2k) / b2k ( digamma function )
-So, the digamma function may be computed with "PSINB" too.
and the recurrence relation: y(m)
(a+p) = y(m) (a) + (-1)m
m! [ 1/am+1 + ....... + 1/(a+p-1)m+1 ]
where p is a positive integer
| STACK | INPUT | OUTPUT |
| Level 2 | m | / |
| Level 1 | B | y(m) (B) |
Example: b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3 , m = 3
3 ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6
) ( 0.7 , 0.8 ) ] PSINB gives
Psi(3) (b) = ( 0.349772973
+ 0.790426127 i ) + ( -0.281836208 - 0.432414490 i
) e1
+ ( -0.474257644 - 0.652019707 i ) e2 + ( -0.666679080
- 0.871624925 ) e3
( 9D )
Note:
-With m = 0 , PSINB is another way to calculate the digamma
function.
ZETAB employs the method given by P. Borwein in "An
Efficient Algorithm for the Riemann Zeta Function" if Re(Re(b))
>= 1/2
-If Re(Re(b)) < 1/2, it uses: Zeta(b)
= Zeta(1-b) Pi b-1/2 Gamma((1-b)/2) / Gamma(b/2)
| STACK | INPUT | OUTPUT |
| Level 1 | B | Zeta (B) |
Example: b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) (
0.5 , 0.6 ) ( 0.7 , 0.8 ) ] ZETAB returns
Zeta(b) = ( 0.478160114 + 0.577429237
i ) + ( -0.256274161 + 0.142614011 i ) e1
+ ( -0.388378571 + 0.242979790 i ) e2 + ( -0.520482981
+ 0.343345568 i ) e3
( 9D )
Notes:
-The results are faster and more accurate if Re(Re(b)) > 1/2
-Large execution times are to be expected for large imaginary parts.
Generalized Hypergeometric Functions
HGFB computes pFq( a1,a2,....,ap ; b1,b2,....,bq ; b ) = SUMk=0,1,2,..... [(a1)k(a2)k.....(ap)k] / [(b1)k(b2)k.....(bq)k] . bk/k! if Level 1 > 0
where (ai)k = ai(ai+1)(ai+2) ...... (ai+k-1) & (ai)0 = 1 , likewise for (bj)k ( Pochhammer's symbol )
ai & bj are complexes and b is a "bi-on"
>>> or the regularized function F tilde:
pF~q( a1,a2,....,ap ; b1,b2,....,bq ; b ) = SUMk=0,1,2,..... [ (a1)k(a2)k.....(ap)k ] / [Gam(k+b1) Gam(k+b2).....Gam(k+bq)] . bk/k! if Level 2 < 0
where Gam = Euler's Gamma function.
| STACK | INPUTS | OUTPUTS |
| Level(3+p+q) | a1 | / |
| ............ | ...... | / |
| Level (4+q) | ap | / |
| Level (3+q) | b1 | / |
| .......... | ....... | / |
| Level 4 | bq | / |
| Level 3 | B | / |
| Level 2 | p | / |
| Level 1 | +/- q | f(b) |
Level 1 = +q for the non-regularized HGF , Level 1 = -q for the regularized HGF
Example1: a1
= 0.2 + 0.3 i , b1 = 0.6 +
0.7 i , b3 = 1.4 + 1.6 i
-> Calculate 2F3( a1,a2
;
b1,b2,b3
; b )
with
a2 = 0.4 + 0.5 i , b2 =
1.2 + 1.3 i
b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.2 , 0.3 ) ENTER
( 0.4 , 0.5 ) ENTER
( 0.6 , 0.7 ) ENTER
( 1.2 , 1.3 ) ENTER
( 1.4 , 1.6 ) ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8
) ] ENTER
2 ENTER
3 HGFB
returns
2F3(b) = ( 1.003991856
+ 0.005192296 i ) + ( 0.032853131 + 0.009144529 i
) e1
+ ( 0.051982446 + 0.011637214 i ) e2 + ( 0.071111761
+ 0.014129900 i ) e3
( 9D )
Example2: With the same arguments, calculate the regularized function F tilde: 2F~3( a1,a2 ; b1,b2,b3 ; b )
b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.2 , 0.3 ) ENTER
( 0.4 , 0.5 ) ENTER
( 0.6 , 0.7 ) ENTER
( 1.2 , 1.3 ) ENTER
( 1.4 , 1.6 ) ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8
) ] ENTER
2 ENTER
-3 HGFB
returns
2F~3(b) = ( 9.112735956
+ 3.989347207 i ) + ( 0.262953244 + 0.212181889 i
) e1
+ ( 0.427181612 + 0.309967487 i ) e2 + ( 0.591409979
+ 0.407753085 i ) e3
( 9D )
-The result of the 1st example has been simply divided by [ Gam(b1) Gam(b2) Gam(b3) ]
Example3: Calculate again the regularized function F tilde: 2F~3( a1,a2 ; b1,b2,b3 ; b ) but with
a1 = 0.2 + 0.3 i , b1 =
0.6 + 0.7 i , b3 = -4
a2 = 0.4 + 0.5 i , b2 =
-3.14
b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.2 , 0.3 ) ENTER
( 0.4 , 0.5 ) ENTER
( 0.6 , 0.7 ) ENTER
-3.14 ENTER
-4 ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8
) ] ENTER
2 ENTER
-3 HGFB
returns
2F~3(b)
= ( 0.185476240 - 0.060667557 i ) + ( 0.133911078 + 0.071283992 i ) e1
+ ( 0.214604001 + 0.100490141 i ) e2 + ( 0.295296925 + 0.129696290
i ) e3
( 9D )
Note:
-Since most of the special functions may be expressed in terms of hypergeometric
functions, the routine HGFB is extensively used hereunder !
Legendre Functions - 1st kind - Type2
Formula: Pnm(B)
= [ (B+1)/(1-B) ]m/2 2F~1(-n
, n+1 ; 1-m ; (1-B)/2 )
( B # 1 )
| B - 1 | < 2
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | B | Pnm(B) |
Example: m = 1 + 2 i , n = 3 + 4 i , B = ( 1.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( -0.1 - 0.2 i ) e2 + ( -0.3 - 0.4 i ) e3
( 1 , 2 ) ENTER
( 3 , 4 ) ENTER
[ ( 1.1 , 0.2 ) ( 0.3 , 0.4 ) ( -0.1 , -0.2 ) ( -0.3
, -0.4 ) ] ALF12 gives
Pnm(B) = ( 93.103757941
- 105.0475346 i ) + ( 75.56897107 + 91.98871965
i ) e1
+ ( -25.89124970 - 46.52055433 i ) e2 + ( -75.56897107
- 91.98871965 i ) e3
( 10D )
Note:
-See also PMN2B
Legendre Functions - 1st kind - Type3
Formula: Pnm(B)
= [ (B+1)/(B-1) ]m/2 2F~1(-n
, n+1 ; 1-m ; (1-B)/2 )
( B # 1 )
| B - 1 | < 2
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | B | Pnm(B) |
Example: m = 1 + 2 i , n = 3 + 4 i , b = ( 1.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( -0.1 - 0.2 i ) e2 + ( -0.3 - 0.4 i ) e3
( 1 , 2 ) ENTER
( 3 , 4 ) ENTER
[ ( 1.1 , 0.2 ) ( 0.3 , 0.4 ) ( -0.1 , -0.2 ) ( -0.3
, -0.4 ) ] ALF13 gives
Pnm(b) = ( 57.145293120
+ 522.83368998 i ) + ( 346.12038247 - 37.93305295
i ) e1
+ ( -155.3276125 - 10.99908730 i ) e2 + ( -346.12038247
+ 37.93305295 i ) e3
( 10D )
Note:
-See also PMN3B
Legendre Functions - 2nd kind - Type2
Formula: Qrm(b)
= 2m pi1/2 (1-b2)-m/2
[ -Gam((1+m+r)/2)/(2.Gam((2-m+r)/2)) . sin pi(m+r)/2 . 2F1(-r/2-m/2
; 1/2+r/2-m/2 ; 1/2 ; b2 )
+ b Gam((2+r+m)/2) / Gam((1+r-m)/2) . cos pi(m+r)/2 . 2F1((1-m-r)/2
; (2+r-m)/2 ; 3/2 ; b2 ) ]
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | B | Qnm(B) |
Example: m = 0.2 + 0.3 i , n = 1.4 + 1.6 i , b = ( 0.05 + 0.1 i ) + ( 0.15 + 0.2 i ) e1 + ( 0.25 + 0.3 i ) e2 + ( 0.35 + 0.4 i ) e3
( 0.2 , 0.3 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.05 , 0.1 ) ( 0.15 , 0.2 ) ( 0.25 , 0.3 ) ( 0.35
, 0.4 ) ] ALF22
Qnm(b) = ( -0.5373501978
- 1.483228607 i ) + ( 1.318605387 - 1.639974388
i ) e1
+ ( 1.925826453 - 2.663848476 i ) e2 + ( 2.533047518
- 3.687722564 i ) e3
( 10D )
Legendre Functions - 2nd kind - Type3
Qrm(b) =
exp( i (m.PI) ) 2 -r-1 sqrt(PI) Gam(m+r+1) b -m-r-1
(b2-1)m/2 2F~1(
(2+m+r)/2 , (1+m+r)/2 ; r+3/2 ; 1/b2 )
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | B | Qnm(B) |
Example: m = 1.2 + 1.3 i , n = 1.4 + 1.5 , b = ( 1.1 + 1.2 i ) + ( 1.3 + 1.4 i ) e1 + ( 1.5 + 1.6 i ) e2 + ( 1.7 + 1.8 i ) e3
( 1.2 , 1.3 )
ENTER
( 1.4 , 1.5 )
ENTER
[ ( 1.1 , 1.2 ) ( 1.3 , 1.4 ) ( 1.5 , 1.6 ) ( 1.7 , 1.8
) ] ALF23
Qnm(b) = ( 0.094724134
+ 0.042366477 i ) + ( 0.021457752 - 0.047321593
i ) e1
+ ( 0.024373027 - 0.054440173 i ) e2 + ( 0.027288302
- 0.061558752 i ) e3
( 9D )
Bessel Functions of the 1st kind
Formula:
Jm(b) = (b/2)m
[ 1/Gam(m+1) + (-b2/4)1/ (1! Gam(m+2)
) + .... + (-b2/4)k/ (k! Gam(m+k+1) ) + ....
] n # -1 ; -2 ; -3 ; ....
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | B | Jn(B) |
Example: n = 2 + 3 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7
, 0.8 ) ] JNB returns
J2+3.i
(b) = ( 1.761078793 + 2.178620399 i ) + ( -0.807601645
+ 0.577907524 i ) e1
+ ( -1.213625964 + 0.966143878 i ) e2 + ( -1.619650283
+ 1.354380215 ) e3
( 9D )
Note:
JNB does not work if n is a negative integer, but we can
employ the relation: Jn
= (-1)n J-n in this case.
Bessel Functions of the 2nd kind
Formulae:
Ym(b) = ( Jm(b) cos(m(pi)) - J-m(b) ) / sin(m(pi)) ; Km(b) = (pi/2) ( I-m(b) - Im(b) ) / sin(m(pi)) m # .... -3 ; -2 ; -1 ; 0 ; 1 ; 2 ; 3 ....
or, if m is a positive integer:
Ym(b) =
-(1/pi) (b/2)-m SUMk=0,1,....,m-1 (m-k-1)!/(k!)
(b2/4)k + (2/pi) Ln(b/2) Jm(b)
- (1/pi) (b/2)m SUMk=0,1,..... ( psi(k+1) +
psi(m+k+1) ) (-b2/4)k / (k!(m+k)!)
where psi = the digamma function
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | B | Yn(B) |
Example1: n = 2 + 3 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7
, 0.8 ) ] YNB returns
Yn (b) = ( -2.42905364514 + 27.3193594245
i ) + ( 10.7732044826 + 3.04317564521 i ) e1
+ ( 17.04965304444 + 3.8854976479 i ) e2 + ( 23.3261016062 +
4.7278196506 ) e3
Example2: n = 3 b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
3
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7
, 0.8 ) ] YNB returns
Y3 (b) = ( -0.712755224 - 0.475908558
i ) + ( 0.575617773 + 0.146363743 i ) e1
+ ( 0.909672826 + 0.182278018 i ) e2 + ( 1.243727879
+ 0.218192292 ) e3
( 9D )
Note:
YNB does not work if n is a negative integer, but we can
employ the relation: Yn
= (-1)n Y-n in this case.
Modified Bessel Functions of the 1st kind
Formula:
Im(b) = (b/2)m
[ 1/Gam(m+1) + (b2/4)1/ (1! Gam(m+2)
) + .... + (b2/4)k/ (k! Gam(m+k+1) ) + ....
] m # -1 ; -2 ; -3
; ....
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | B | In(B) |
Example: n = 2 + 3 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7
, 0.8 ) ] INB returns
I2+3.i
(b) = ( 2.04184519843 + 1.6457734642 i ) + ( -0.624872872269 + 0.688412145165
i ) e1
+ ( -0.919728709124 + 1.12391277624 i ) e2 + ( -1.21458454598
+ 1.55941340732 ) e3
Note:
INB does not work if n is a negative integer, but we can
employ the relation: In
= I-n in this case.
Modified Bessel Functions of the 2nd kind
Formulae: Km(b) = (pi/2) ( I-m(b) - Im(b) ) / sin(m(pi)) m # .... -3 ; -2 ; -1 ; 0 ; 1 ; 2 ; 3 ....
or, if m is a positive integer:
Km(b) = (1/2) (b/2)-m SUMk=0,1,..,m-1
(m-k-1)!/(k!) (-b2/4)k - (-1)m Ln(b/2) Im(b)
+ (1/2) (-1)m (b/2)m SUMk=0,1,...( psi(k+1)
+ psi(m+k+1) ) (b2/4)k / (k!(m+k)!)
where psi = the digamma function
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | B | Kn(B) |
Example1: n = 2 + 3 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7
, 0.8 ) ] KNB returns
K2+3.i (b) = ( 14.38068951
- 63.26852067 i ) + ( -22.09224561 - 6.403916047 i
) e1
+ ( -34.97621644 - 8.222729385 i ) e2 + ( -47.86018726
- 10.04154272 ) e3
( 9D )
Example2: n = 3 b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
3
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7
, 0.8 ) ] KNB returns
K3 (b) = ( 0.795781386845 + 1.02245357222
i ) + ( -0.54871862033 - 0.555496991545 i ) e1
+ ( -0.900440807045 - 0.82267781719 i ) e2 + ( -1.25216299375
- 1.08985864283 ) e3
Note:
KNB does not work if n is a negative integer, but we can
employ the relation: Kn
= K-n in this case.
Regular Coulomb Wave Functions
Formulae: FL(h,b) = CL(h) b L+1 exp(-i.b) M ( L+1-i.h ; 2L+2 ; 2i.b ) where M = Kummer's function
with CL(h)
= 2L exp [ (1/2) { -PI.h + Lngamma(L+1+i.h)
+ Lngamma(L+1-i.h) } - Lngamma(2L+2) ]
| STACK | INPUTS | OUTPUTS |
| Level 3 | L | / |
| Level 2 | h | / |
| Level 1 | b | FL(h,b) |
Example: L = 2 + 3 i , h = -0.6 - 0.7 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
( -0.6 , -0.7 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 )
( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] RCWFB returns
FL(h,b)
= ( 1.432663946 + 2.724686888 i ) + ( -0.994443184
+ 0.451696564 i ) e1
+ ( -1.515195642 + 0.784202095 i ) e2 + ( -2.035948100
+ 1.116707625 i ) e3
Notes:
"RCWFB" does not work if 2.L = -1 , -2 , -3 , ...............
Irregular Coulomb Wave Functions
Formulae: GL(h,b) = [ FL(h,b) Cos c - F-L-1(h,b) ] / sin c
with c = sL(h)
- s-L-1(h)
- ( 2.L + 1 ) PI / 2
and sL(h)
= [ Lngamma ( 1 + L + i.h ) - Lngamma
( 1 + L - i.h ) ] / ( 2.i )
| STACK | INPUTS | OUTPUTS |
| Level 3 | L | / |
| Level 2 | h | / |
| Level 1 | b | FL(h,b) |
Example: L = 2 + 3 i , h = -0.6 - 0.7 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
( -0.6 , -0.7 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 )
( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] ICWFB returns
GL(h,b)
= ( -4.805817221 - 31.486587007 i ) + ( -13.144403234
- 0.948059274 i ) e1
+ ( -19.021113786 - 0.507420209 i ) e2 + ( -25.897824339
- 0.066781144 i ) e3
Note:
"ICWFB" does not work if 2.L is an integer
Coulomb Wave Functions - Asymptotic Expansion
with q = b - h Ln 2.b - L PI / 2 + [ Lngamma ( 1 + L + i.h ) - Lngamma ( 1 + L - i.h ) ] / ( 2.i )
HL+(h,b)
~ exp ( i.q ) 2F0
( - L + i.h , 1 + L + i.h
, 1 / 2.i.b )
HL-(h,b)
~ exp ( -i.q ) 2F0
( - L - i.h , 1 + L - i.h
, -1 / 2.i.b )
then, FL(h,b)
= [ HL+(h,b) - HL-(h,b)
] / (2.i)
GL(h,b) = [ HL+(h,b)
+ HL-(h,b) ] / 2
| STACK | INPUTS | OUTPUTS |
| Level 3 | L | / |
| Level 2 | h | FL(h,b) |
| Level 1 | b | GL(h,b) |
Example: L = 2 + 3 i , h = -0.6 - 0.7 i , b = ( 9.1 + 9.2 i ) + ( 9.3 + 9.4 i ) e1 + ( 9.5 + 9.6 i ) e2 + ( 9.7 + 9.8 i ) e3
( 2 , 3 )
ENTER
( -0.6 , -0.7 )
ENTER
[ ( 9.1 , 9.2 ) ( 9.3 , 9.4 )
( 9.5 , 9.6 ) ( 9.7 , 9.8 ) ] AECWFB returns
FL(h,b)
= ( 273770923.157 - 3362001483.11 i ) + ( -1900097678.67 - 154948082.613
i ) e1
+ ( -1940758040.57 - 158045045.004i ) e2 + ( -1981418402.49
- 161142007.411 i ) e3
in level 2 and in level 1:
GL(h,b)
= ( -3361983306.77 - 273776916.434 i ) + ( -154944696.517 + 1900107951.9
i ) e1
+ ( -158041585.306 + 1940768533.15 i ) e2 + ( -161138474.11
+ 1981429114.43 i ) e3
Notes:
-As usual with asymptotic expansions, the series will diverge too soon
if b is too "small"
-Press ON to stop the infinite loop in this case.
Formula: Dm(b) = 2m/2 Pi1/2 exp(-b2/4) [ 1/Gam((1-m)/2) M( -m/2 , 1/2 , b2/2 ) - 21/2 ( b / Gam(-m/2) ) M [ (1-m)/2 , 3/2 , b2/2 ]
where M = Kummer's function.
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | b | Dn(b) |
Example: n = 0.4 + 0.7 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.4 , 0.7 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 )
( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] DNB returns
Dn (b) = ( 0.409729789 +
0.317378268 i ) + ( -0.162419933 + 0.271785446 i )
e1
+ ( -0.231632220 + 0.436979671 i ) e2 + ( -0.300844507
+ 0.602173396 i ) e3
Note:
-For large arguments, an asymptotic expansion is preferable.
Parabolic Cylinder Functions - Asymptotic Expansion
-For large arguments, ascending series give poor accuracy or even meaningless
results !
-Asymptotic expansions are preferable:
Formula: Dm(b)
~ bm exp(-b2/4) [ 1 - m(m-1) / ( 2 b2
) + m(m-1)(m-2)(m-3) / ( 2 ( 4 b4 ) ) - ....... ]
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | b | Dn(b) |
Example: n = 3.14 + 2.718 i b = ( 5 + 6 i ) + ( 7 + 8 i ) e1 + ( 9 + 10 i ) e2 + ( 11 + 12 i ) e3
( 3.14 , 2.718 )
ENTER
[ ( 5 , 6 )
( 7 , 8 ) ( 9 , 10 ) ( 11 , 12 ) ]
AEDNB gives
Dn (b) = ( -4.325945 E34
+ 2.1466037 E36 i ) + ( 9.6478575 E35 + 3.4397021
E34 i ) e1
+ ( 1.2215324 E36 + 2.645312 E34 i ) e2 + ( 1.4782790
E36 + 1.8509221 E34 i ) e3
Notes:
-An infinte loop will occur if a is too small.
-However, this program may also be used if b is relatively small when
n is a positive integer.
"JEFB" employs Gauss' transformation to calculate sn ( b | m ) , cn ( b | m ) & dn ( b | m )
-If m # 1 , let m' = 1-m , µ = [ ( 1-sqrt(m') / ( 1+sqrt(m') ]2 and v = b / ( 1+sqrt(µ) ] , then:
sn ( b | m ) = [ ( 1 + sqrt(µ) ) sn ( v | µ
) ] / [ 1 + sqrt(µ) sn2 ( v | µ ) ]
cn ( b | m ) = [ cn ( v | µ ) dn ( v | µ )
] / [ 1 + sqrt(µ) sn2 ( v | µ ) ]
dn ( b | m ) = [ 1 - sqrt(µ) sn2 ( v
| µ ) ] / [ 1 + sqrt(µ) sn2 ( v | µ ) ]
-These formulas are applied recursively until µ is small enough to use
sn ( v | 0 ) = Sin v
cn ( v | 0 ) = Cos v
dn ( v | 0 ) = 1
-If m = 1: sn ( b | m ) = tanh b ; cn
( b | m ) = dn ( b | m ) = sech b
| STACK | INPUTS | OUTPUTS |
| Level 3 | / | sn ( b | m ) |
| Level 2 | m | cn ( b | m ) |
| Level 1 | b | dn ( b | m ) |
Example: m =
2 + 3.i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1
+ ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7
, 0.8 ) ] JEFB gives
in level 3:
( all rounded to 9D )
sn ( b | 2+3i ) = ( -0.119297320
- 0.145921679 i ) + ( 0.001430949 + 0.135563787 i
) e1
+ ( 0.013077383 + 0.211365032 i ) e2 + ( 0.024723818
+ 0.287166277 i ) e3
and in level 2:
cn ( b | 2+3i ) = ( 0.928961221 - 0.011137455
i ) + ( -0.021319014 + 0.017378296 i ) e1
+ ( -0.031867398 + 0.028815663 i ) e2 + ( -0.042415782
+ 0.040253030 i ) e3
and in level 1
dn ( b | 2+3i ) = ( -0.926505039 + 0.207185357
i ) + ( 0.084840287 + 0.047110613 i ) e1
+ ( 0.136119697 + 0.066705333 i ) e2 + ( 0.187399107
+ 0.086300054 i ) e3
Weierstrass Elliptic Functions
-WEFB calculates the Weierstrass Elliptic Function P(b;g2,g3) by a Laurent series:
P(b;g2;g3) = b -2 + c2.b2 + c3.b4 + ...... + ck.b2k-2 + ....
where c2 = g2/20 ;
c3 = g3/28 and ck =
3 ( c2. ck-2 + c3. ck-3 + .......
+ ck-2. c2 ) / (( 2k+1 )( k-3 ))
( k > 3 )
| STACK | INPUTS | OUTPUTS |
| Level 3 | g2 | / |
| Level 2 | g3 | / |
| Level 1 | b | P(b;g2;g3) |
Example: g2 = 1.2 + 1.3 i , g3 = 1.6 + 1.7 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 1.2 , 1.3 )
ENTER
( 1.6 , 1.7 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7
, 0.8 ) ] WEFB returns
P(b;g2;g3)
= ( 0.089457020 + 0.121607346 i ) + ( -0.005461551
+ 0.122834455 i ) e1
+ ( 0.001306738 + 0.192058673 i ) e2 + (
0.008075026 + 0.261282892 i ) e3
( 9D )
Weierstrass Duplication Formula
-If the argument b is too "large", the Laurent series don't converge.
-The duplication formula may be used one or several times in this case:
P(2b) = -2 P(b) + ( 6 P2(b)
- g2/2 )2 / ( 4 ( 4 P3(b) - g2
P(b) - g3 ) )
| STACK | INPUTS | OUTPUTS |
| Level 3 | g2 | g2 |
| Level 2 | g3 | g3 |
| Level 1 | P(b;g2;g3) | P(2b ; g2;g3) |
Example: if you keep the previous
result in level 1 and place ( 1.2 , 1.3 ) in level 3
and ( 1.6 , 1.7 ) in level 2, WF2B returns:
P(2b ; g2;g3)
= ( -0.17420917947 - 0.31429376681 i ) + ( -0.01995584363 - 0.25246457542
i ) e1
+ ( -0.05132828218 - 0.39224827015 i ) e2 + ( -0.08270072056
- 0.532031964877 i ) e3
Formula: Ei(b)
= C + Ln(b) + Sumn=1,2,..... bn/(n.n!)
where C = 0.577215664901... = Euler's constant.
| STACK | INPUTS | OUTPUTS |
| Level 1 | b | Ei(b) |
Example: b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] EIB returns
Ei(b) = ( 1.140687091 + 0.557945079
i ) + ( 0.829134746 + 0.443634358 i ) e1
+ ( 1.328940953 + 0.625738818 i ) e2 + ( 1.828747159
+ 0.808843279 ) e3
( 9D )
Formulae:
En(b) = bn-1 Gam(1-n) - [1/(1-n)] 1F1 ( 1-n , 2-n ; -b ) if n # 1 , 2 , 3 , ................ and otherwise:
En(b) = (-b)n-1 ( -Ln b - gamma + Sumk=1,...,n-1 1/k ) / (n-1)! - Sumk#n-1 (-b)k / (k-n+1) / k! where gamma = Euler's Constant = 0.577215664901...
and E0(b) = (1/b).exp(-b)
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | b | En(b) |
Example1: n = 2 + 3 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 )
] ENB gives
E2+3.i (b) = ( -0.214869069 - 0.150734076
i ) + ( -0.062956984 + 0.108669920 i ) e1
+ ( -0.089519302 + 0.174561633 i ) e2 + ( -0.116081620
+ 0.240453347 ) e3
Example2: n = 3 b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
3
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 )
] ENB returns
E3 (b) = ( -0.2411685505 - 0.522115506
i ) + ( -0.214772639 + 0.090964781 i ) e1
+ ( -0.327768135 + 0.159086869 i ) e2 + ( -0.440763630
+ 0.227208957 ) e3
Note:
-For large arguments, the program below is better:
Exponential Integral En(b) - Asymptotic Expansion
Formula: En(b) ~
(1/b) exp(-b) 2F0(1,n;;-1/b)
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | b | En(b) |
Example: n = 3.14 + 2.718 i b = ( 30 + 31 i ) + ( 32 + 33 i ) e1 + ( 34 + 35 i ) e2 + ( 36 + 37 i ) e3
( 3.14 , 2.718 )
ENTER
[ ( 30 , 31 ) ( 32 , 33 ) ( 34 , 35 ) ( 36 , 37 ) ] AENB
produces
En (b) = ( -8.3244597 E10 + 7.1723931
E10 i ) + ( 3.8924375 E10 + 4.5261117 E10 i ) e1
+ ( 4.1361995 E10 + 4.8008915 E10 i ) e2 + ( 4.3799615
E10 + 5.0756713 E10 ) e3
Notes:
-If the argument a is too "small" , the series will diverge too soon.
-However, it may also be used with small arguments if n is a
negative
integer.
-For instance, it returns correctly:
E -2[ ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3 ] = ( -0.050340126 + 0.442798110 i )
+ ( -0.089358126
- 0.047200638 i ) e1 + ( -0.143174727 - 0.066484345
i ) e2 + ( -0.196991329 - 0.085768052 ) e3
Formula:
Si(b) = Summ=0,1,2,..... (-1)m b2m+1/((2m+1).(2m+1)!)
| STACK | INPUTS | OUTPUTS |
| Level 1 | b | Si(b) |
Example: b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] SIB returns
Si(b) = ( 0.029007647 + 0.214825822
i ) + ( 0.254033031 + 0.422973668 i ) e1
+ ( 0.430129422 + 0.639516280 i ) e2 + ( 0.606225812
+ 0.856058892 ) e3
Formula:
Shi(b) = Summ=0,1,2,..... b2m+1/((2m+1).(2m+1)!)
| STACK | INPUTS | OUTPUTS |
| Level 1 | b | Shi(b) |
Example: b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] SHIB returns
Shi(b) = ( 0.168831684 + 0.171286637
i ) + ( 0.343698222 + 0.372373658 i ) e1
+ ( 0.565959119 + 0.553407049 i ) e2 + ( 0.788220016
+ 0.734440439 ) e3
Formula:
Ci(b) = C + ln(b) + Summ=1,2,..... (-1)m b2m/(2m.(2m)!)
where C = 0.577215664901... = Euler's constant.
| STACK | INPUTS | OUTPUTS |
| Level 1 | b | Ci(b) |
Example: b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] CIB returns
Ci(b) = ( 0.822497127 + 1.344232914
i ) + ( 0.534656969 - 0.027912750 i ) e1
+ ( 0.831831852 - 0.086316448 i ) e2 + ( 1.129006735
- 0.144720146 ) e3
Formula:
Chi(b)= C + ln(b) + Summ=1,2,..... b2m/(2m.(2m)!)
where C = 0.577215664901... = Euler's constant.
| STACK | INPUTS | OUTPUTS |
| Level 1 | b | Chi(b) |
Example: b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] CHIB returns
Chi(b) = ( 0.971855407 + 0.386658442
i ) + ( 0.485436524 + 0.071260700 i ) e1
+ ( 0.762981834 + 0.072331769 i ) e2 + ( 1.040527143
+ 0.073402839 ) e3
Angular Spheroidal Wave Function of the 1st kind
-"SMNB" computes the angular spheroidal wave function of the
first
kind.
-Given m , n and c2
, the corresponding eigenvalue l
may be calculated by LMN ( see the next program below
)
-We assume that | b | <= 1 and Smn(b) is computed by Smn(b) = ( 1 - b2 ) m/2 Sumk=0,1,.... dk bk
with (k+1)(k+2) dk+2
- [ k ( k + 2m + 1 ) - l
+ m ( m + 1 ) ] dk
- c2 dk-2 = 0
Flammer's Scheme: the coefficients are normalized as follows:
d0 = Pnm(0)
= 2m sqrt(PI) / [ Gam((1-m-n)/2)
Gam((2-m+n)/2 ]
d1 = P'nm(0)
= ( m + n ) 2m
sqrt(PI) / [ Gam((2-m-n)/2)
Gam((1-m+n)/2 ]
| STACK | INPUT | OUTPUT |
| Level 1 | b | Smn(b) |
Example: m = 0.2 + 0.3 i , n = 0.6 + 0.7 i , c2 = 1.7 + 1.8 i , l = 1.46245588721 + 2.19744971776 i
b = ( 0.05 + 0.1 i ) + ( 0.15 + 0.2 i ) e1 + ( 0.25 + 0.3 i ) e2 + ( 0.35 + 0.4 i ) e3
m , n , c2 and l are to be stored in the variables 'L' 'M' 'N' 'C2' ( they are already stored if you've used LMN below )
[ ( 0.05 , 0.1 ) ( 0.15 , 0.2 ) ( 0.25 , 0.3 ) (
0.35 , 0.4 ) ] SMNB returns
Smn(b)
= ( 0.628023288 - 0.491662142 i ) + ( -0.021846567
+ 0.346286108 i ) e1
+ ( -0.006377755 + 0.541954055 i ) e2 + ( 0.009091056
+ 0.737622001 i ) e3
Note:
-l may be choosen arbitrarily, but the solution
will not be regular for b = 1
-Given m , n and
c2 , store these real numbers into 'M'
'N' 'C2' then press LMN
-Your HP-48 will display the successive approximations and will finally
store the eigenvalue l in the variable 'L'
-LMN solves the transcendental equation U1(Lmn) + U2(Lmn) = 0 where U1 & U2 are 2 continued fractions:
-brm
-br-2m
U1(Lmn) = grm
- Lmn + ---------------
--------------- ..............
gr-2m - Lmn +
gr-4m - Lmn +
r = n - m
-br+2m
-br+4m
U2(Lmn) =
--------------- ----------------
..............
gr+2m - Lmn +
gr+4m - Lmn +
with brm
= [ r.(r-1).(2m+r).(2m+r-1).c4 ] / [ (2m+2r-1)2.(2m+2r+1).(2m+2r-3)
]
and grm
= (m+r).(m+r+1) + (c2/2).[ 1 - (4m2-1)/((2m+2r-1).(2m+2r+3))
]
| STACK | INPUTS | OUTPUTS |
| Level 1 | / | Lmn |
Example: m = 0.2 + 0.3 i , n = 0.6 + 0.7 i , c2 = 1.7 + 1.8 i
-Store these 3 numbers in the variables 'M' 'N'
'C2' and press the LMN key.
-The successive approximations are displayed and finally, we get in
level 1 and in 'L' the eigenvalue l
= 1.46245588721 + 2.19744971776 i
Notes:
LMN uses an iteration method to find the eigenvalue.
It's only a first order method - therefore very slow.
It involves 2 continued fractions that are computed "from right
to left" with 12 terms
( 12 is stored in 'ITER' , modify this value to check if there
is no significant change in the result )
Of course, this is not the best way to calculate continued fractions...
Formulae: n.Pn(b)
= (2n-1).b.Pn-1(b) - (n-1).Pn-2(b) ,
P0(b) = 1 , P1(b) = b
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | b | Pn(b) |
Where n is a non-negative integer
Example: n = 7 , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
7
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 )
] LEGB returns
P7(b) = ( -240.8232644
+ 180.2220152 i ) + ( -8.4974508 + 82.1615056
i ) e1
+ ( -6.6831028 + 128.8517448 i ) e2 + ( -4.8687548
+ 175.541984 i ) e3
Generalized Laguerre's Polynomials
Formulae: L0(a)
(B) = 1 , L1(a) (B) = a+1-B
, n Ln(a) (B) = (2.n+a-1-B) Ln-1(a)
(B) - (n+a-1) Ln-2(a) (B)
| STACK | INPUTS | OUTPUTS |
| Level 3 | a | / |
| Level 2 | n | / |
| Level 1 | b | Ln(a) (b) |
Where n is a non-negative integer
Example: a = 2 + 3 i n = 7 b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
7
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 )
] LANB returns
L72+3i (b) =
( 70.827967066 + 59.563907953 i ) + ( 46.126010715 + 4.093189916
i ) e1
+ ( 72.284031989 + 2.695295412 i ) e2 + ( 98.442053102 + 1.297400908
i ) e3
Chebyshev Polynomials - 1st kind
Formula:
Tn(b) = 2b.Tn-1(b) - Tn-2(b)
; T0(b) = 1 ; T1(b) = b
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | b | Tn(b) |
Where n is a non-negative integer
Example: n = 7 , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
7
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 )
] CHB1B returns
T7(b) = ( -558.0438464
+ 462.2275712 i ) + ( -7.670764800 + 203.3249536
i ) e1
+ ( 4.299603200 + 317.8005888 i ) e2 + ( 16.26997120
+ 432.2762240 i ) e3
Chebyshev Polynomials - 2nd kind
Formula: Un(b) = 2b.Un-1(b)
- Un-2(b) ; U0(b) = 1 ; U1(b)
= 2b
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | b | Un(b) |
Where n is a non-negative integer
Example: n = 7 , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
7
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 )
] CHB2B returns
U7(b) = ( -1176.6665728
+ 804.7309824 i ) + ( -61.19336960 + 378.9647872
i ) e1
+ ( -65.14447360 + 596.0805376 i ) e2 + ( -69.09557760
+ 813.1962880 i ) e3
Formula: P0(a;b) (B) = 1 ; P1(a;b) (B) = (a-b)/2 + B (a+b+2)/2
2n(n+a+b)(2n+a+b-2) Pn(a;b)
(B) = [ (2n+a+b-1).(a2-b2) + B (2n+a+b-2)(2n+a+b-1)(2n+a+b)
] Pb-1(a;b) (B)
- 2(n+a-1)(n+b-1)(2n+a+b) Pn-2(a;b) (B)
| STACK | INPUTS | OUTPUTS |
| Level 4 | a | / |
| Level 3 | b | / |
| Level 2 | n | / |
| Level 1 | B | Pn(a;b) (B) |
Where n is a non-negative integer
Example: a = 2 + 3 i b = ( 4 + 5 i ) n = 7 B = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
( 4 , 5 )
ENTER
7
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 )
] JCPB returns
Pn(a;b) (B) = ( 8797.6726159
+ 1795.6342573 i ) + ( 2742.28422507 - 3357.50546715 i ) e1
+ ( 4009.36295372 - 5457.09126675 i ) e2 + ( 5276.44168239 -
7556.67706636 i ) e3
Formulae:
C0(a) (b) = 1 ; C1(a) (b) = 2.a.b ; (m+1).Cm+1(a) (b) = 2.(m+a).b.Cm(a) (b) - (m+2a-1).Cm-1(a) (b) if a # 0
Cm(0) (b)
= (2/m).Tm(b)
| STACK | INPUTS | OUTPUTS |
| Level 3 | a | / |
| Level 2 | n | / |
| Level 1 | b | Cn(a) (b) |
Where n is a non-negative integer
Example: a = 2 + 3 i n = 7 b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
7
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 )
] USPB returns
C72+3i (b) =
( 38963.6390236 + 23801.2308334 i ) + ( 10363.1939383 - 3209.20241371
i ) e1
+ ( 15909.8463506 - 5835.41128046 i ) e2 + ( 21456.498763 -
8461.6201472 i ) e3
-Likewise, you will find:
C7(0) (b) = ( -159.441098971
+ 132.065020343 i ) + ( -2.19164708571 + 58.0928438857
i ) e1
+ ( 1.22845805714 + 90.8001682286 i ) e2 + ( 4.6485632
+ 123.507492571 i ) e3
Associated Legendre Functions - 1st kind - Type2
-Though this is not always a polynomial, PMN2B is included here to remind
that the indexes must be non-negative integers !
Formula: (n-m) Pnm(b) = b (2n-1) Pn-1m(b) - (n+m-1) Pn-2m(b)
Type 2
Pmm(b) = (-1)m
(2m-1)!! ( 1-b2 )m/2
where (2m-1)!! = (2m-1)(2m-3)(2m-5).......5.3.1
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | b | Pnm (b) |
Where m & n are non-negative integers with m <= n
Example: m = 3 , n = 7 , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
3
ENTER
7
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 )
] PMN2B returns
P37 ( b ) = ( 6071.56627353
+ 53115.3198363 i ) + ( 17513.3557081 - 15006.5407838
i ) e1
+ ( 26120.3116418 - 24811.2720794 i ) e2 + ( 34727.2675758
- 34016.003375 i ) e3
Note:
-See also ALF12
Associated Legendre Functions - 1st kind - Type3
-Though this is not always a polynomial, PMN3A is included here to remind
that the indexes must be non-negative integers !
Formula: (n-m) Pnm(b) = b (2n-1) Pn-1m(b) - (n+m-1) Pn-2m(b)
Type 3
Pmm(b) = (2m-1)!!
( b2-1 )m/2
where (2m-1)!! = (2m-1)(2m-3)(2m-5).......5.3.1
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | b | Pnm (b) |
Where m & n are non-negative integers with m <= n
Example: m = 3 , n = 7 , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
3
ENTER
7
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 )
] PMN3B returns
P37 ( b ) = ( -46823.112327
+ 45127.5481765 i ) + ( 1048.34156612 + 18931.3209529
i ) e1
+ ( -3149.91851935 + 29448.9933613 i ) e2 + (
5251.4954726 + 39966.6657695 i ) e3
Note:
-See also ALF13
Formulae:
Ai(b) = f(b) - g(b)
with
f(b) = [ 3 -2/3 / Gamma(2/3) ] 0F1(
2/3 ; b3/9 )
Bi(b) = [ f(b) + g(b) ] sqrt(3)
and g(b)
= [ 3 -1/3 / Gamma(1/3) ] 0F1( 4/3
; b3/9 ) b
| STACK | INPUTS | OUTPUTS |
| Level 2 | / | Ai (b) |
| Level 1 | b | Bi (b) |
Example: b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] AIBIB returns
Ai(b) = ( 0.478468884 - 0.045373965
i ) + ( -0.040623036 - 0.145492499 i ) e1
+ ( -0.075011336 - 0.223718455 i ) e2 + ( -0.109399636
- 0.301944411 i ) e3
in level 2 and in level 1:
Bi(b) = ( 0.650545600 + 0.036453982
i ) + ( 0.247761441 + 0.146528307 i ) e1
+ ( 0.398230113 + 0.208763244 i ) e2 + ( 0.548698784
+ 0.270998180 i ) e3
Formulae:
Jm(b) = + (b/2) sin ( PI.m/2 ) 1F~2( 1 ; (3-m)/2 , (3+m)/2 ; -b2/4 ) Anger's functions
+ cos ( PI.m/2 ) 1F~2( 1 ; (2-m)/2 , (2+m)/2 ; -b2/4 )
Em(b) = - (b/2) cos ( PI.m/2 ) 1F~2( 1 ; (3-m)/2 , (3+m)/2 ; -b2/4 ) Weber's functions
+ sin ( PI.m/2 ) 1F~2( 1 ; (2-m)/2 , (2+m)/2
; -b2/4 )
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | Jn(b) |
| Level 1 | b | En(b) |
Example: n = 0.4 + 0.7 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.4 , 0.7 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
ANWEB returns
Jm (b) = ( 1.425286892
- 0.337000374 i ) + ( -0.117808153 + 0.381684478 i
) e1
+ ( -0.153245961 + 0.604852438 i ) e2 + ( -0.188683769
+ 0.828020398 i ) e3
in level 2 and in level 1:
Em (b) = ( 0.638513901 +
1.155323362 i ) + ( -0.381725526 - 0.246679412 i )
e1
+ ( -0.615226173 - 0.354281841 i ) e2 + ( -0.848726820
- 0.461884270 i ) e3
-The Catalan numbers may be defined by C(n) = 4n Gam(n+1/2)
/ [ sqrt(PI) Gam(n+2) ]
-This formula is used hereunder after replacing n by the bion b
| STACK | INPUTS | OUTPUTS |
| Level 1 | b | C (b) |
Example: b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] CATB returns
C(b) = ( 0.474992331 - 0.263534256
i ) + ( 0.048669669 + 0.153014311 i ) e1
+ ( 0.088165829 + 0.234808752 i ) e2 + ( 0.127661989
+ 0.316603193 ) e3
Chebyshev Functions - 1st kind
Formula:
Tm(cos B) = cos m.B
with cos B = b
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Tm (b) |
Example: m = 1.2 + 1.3 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 1.2 , 1.3 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 )
( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] CHB1B
returns
Tm
(b) = ( 1.070193726 + 1.688865688 i ) + ( -0.040122707
+ 0.878077601 i ) e1
+ ( 0.007654786 + 1.373010874 i ) e2 + (
0.055432278 + 1.867944147 i ) e3
Note:
-See also Chebyshev polynomials 1st kind
Chebyshev Functions - 2nd kind
Formula:
Um(cos B) = [ sin (m+1).B ] / sin B
with cos B = b
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Um (b) |
Example: m = 1.2 + 1.3 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 1.2 , 1.3 )
ENTER
[ ( 0.1 , 0.2 ) ( 0.3 , 0.4 )
( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ] CHB2B
returns
Um
(b) = ( 3.332558957 + 2.316362965 i ) + ( -0.806753415
+ 1.096755417 i ) e1
+ ( -1.170794895 + 1.775478723 i ) e2 + ( -1.534836374
+ 2.454202030 i ) e3
Note:
-See also Chebyshev polynomials 2nd kind
Formula: erf b = (2/pi1/2)
SUMn=0,1,2,..... (-1)n b2n+1 / (n!
(2n+1))
| STACK | INPUT | OUTPUT |
| Level 1 | b | erf (b) |
Example: b = ( 1 + 0.9 i ) + ( 0.8 + 0.7 i ) e1 + ( 0.6 + 0.5 i ) e2 + ( 0.4 + 0.3 i ) e3
[ ( 1 , 0.9 ) ( 0.8 , 0.7 ) ( 0.6 , 0.5 ) ( 0.4 , 0.3 ) ] ERFB returns
Erf(b) = ( 2.952728869 + 8.149265562
i ) + ( 6.172244023 - 1.372589196 i ) e1
+ ( 4.509301553 - 1.117428242 i ) e2 + ( 2.846359084
- 0.862267288 ) e3
Formula:
Erfm(b) = b exp(-bm) M( 1 ; 1+1/m ; bm
) where M = Kummer's function
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | b | Erfn(b) |
Example: n = 0.4 + 0.7 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.4 , 0.7 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
GERFB returns
Erfn (b) = ( -0.073280236
+ 0.530416042 i ) + ( 0.176820829 + 0.257161065 i
) e1
+ ( 0.296413379 + 0.387025596 i ) e2 + ( 0.416005929
+ 0.516890126 i ) e3
Formula: Hm(b) = 2m sqrt(PI) [ (1/Gam((1-m)/2)) M(-m/2,1/2,b2) - ( 2.b / Gam(-m/2) ) M((1-m)/2,3/2,b2) ]
where Gam = Gamma function
and
M = Kummer's function = 1F1
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Hm(b) |
Example: n = 0.4 + 0.7 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.4 , 0.7 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
HMTB returns
Hn (b) = ( 0.817434193 +
0.870134918 i ) + ( -0.140724907 + 0.379062436 i )
e1
+ ( -0.189205861 + 0.602595392 i ) e2 + ( -0.237686814
+ 0.826128348 i ) e3
Formula:
Bb(p,q) = ( bp / p ) F(p,1-q;p+1;b)
where F = "the" hypergeometric function
| STACK | INPUTS | OUTPUTS |
| Level 3 | p | / |
| Level 2 | q | / |
| Level 1 | b | Bb(p,q) |
Example: p = 0.2 + 0.3 i , q = 1.4 + 1.6 i , b = ( 0.05 + 0.1 i ) + ( 0.15 + 0.2 i ) e1 + ( 0.25 + 0.3 i ) e2 + ( 0.35 + 0.4 i ) e3
( 0.2 , 0.3 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.05 , 0.1 ) ( 0.15 , 0.2 ) ( 0.25
, 0.3 ) ( 0.35 , 0.4 ) ] IBFB returns
Bb(p,q) = ( 0.781441740
- 1.586229591 i ) + ( 0.445532319 - 0.184816869
i ) e1
+ ( 0.680245068 - 0.323956902 i ) e2 + ( 0.914957817
- 0.463096934 i ) e3
Formula: g(m,b)
= ( bm / m ) exp(-b) M(1,m+1;b) where
M = Kummer's function
| STACK | INPUTS | OUTPUTS |
| Level 2 | n | / |
| Level 1 | b | g(n,b) |
Example: n = 2 + 3 i b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
IGFB returns
g(n,b) = (
0.316021539 - 0.294439241 i ) + ( 0.097233572 + 0.118344540
i ) e1
+ ( 0.161151936 + 0.176838797 i ) e2 + ( 0.225070299
+ 0.235333054 i ) e3
Formula: Pm(a;b) (b) = [ Gam(a+m+1) / Gam(m+1) ] 2F~1 ( -m , a+b+m+1 , a+1 , (1-b)/2 )
where 2F1 tilde
is the regularized hypergeometric function
| STACK | INPUTS | OUTPUTS |
| Level 4 | a | / |
| Level 3 | b | / |
| Level 2 | m | / |
| Level 1 | B | Pm(a;b) (B) |
Example: a = 0.2 + 0.3 i , b = 0.6 + 0.7 i , m = 1.4 + 1.6 i , B = ( 1.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.2 , 0.3 )
ENTER
( 0.6 , 0.7 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 1.1 , 0.2
) ( 0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
JCFB returns
Pm(a;b)
(B) = ( -1.594720706 + 3.464906148 i ) + ( -1.081150683
- 0.614094387 i ) e1
+ ( -1.735722616 - 0.871495189 i ) e2 + ( -2.390294550
- 1.128895991 i ) e3
Formula: Lm(a)(b) = [ Gam(m+a+1) / Gam(m+1) ] 1F~1 ( -n , a+1 , b ) m # -1 , -2 , -3 , ....
where Gam = Gamma function
and 1F~1
= Regularized Kummer's function
| STACK | INPUTS | OUTPUTS |
| Level 3 | a | / |
| Level 2 | n | / |
| Level 1 | b | Ln(a)(b) |
Example: a = 0.2 + 0.3 i , m = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.2 , 0.3 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
LANB returns
Lm(a)(b) =
( 2.195071129 + 1.392323348 i ) + ( 0.503589441
- 0.948842403 i ) e1
+ ( 0.709692136 - 1.520481304 i ) e2 + ( 0.915794831
- 2.092120204 i ) e3
Note:
-See also LANB
Formula:
F( b , s , a ) = SUM k=0,1,2,.... bk / ( a + k )s ( F is the greek letter "PHI" )
where s & a are complex numbers
| STACK | INPUTS | OUTPUTS |
| Level 3 | s | / |
| Level 2 | a | / |
| Level 1 | b | F( b , s , a ) |
Example: s = 3.1 + 3.2 i , a = 1.2 + 1.4 i , b = ( 0.05 + 0.1 i ) + ( 0.15 + 0.2 i ) e1 + ( 0.25 + 0.3 i ) e2 + ( 0.31 + 0.4 i ) e3
( 3.1 , 3.2 )
ENTER
( 1.2 , 1.4 )
ENTER
[ ( 0.05 , 0.1 ) ( 0.15 , 0.2 ) ( 0.25 , 0.3 ) ( 0.35
, 0.4 ) ] LERCHB returns
F( b , s ,
a ) = ( -0.186440788 + 2.364004293 i ) + ( -0.058004148
+ 0.054445111 i ) e1
+ ( -0.086130862 + 0.089574705 i ) e2 + ( -0.114257576
+ 0.124704299 i ) e3
Formula:
s(1)m,p(b) = bm+1 / [ (m+1)2
- p2 ] 1F2 ( 1 ; (m-p+3)/2 , (m+p+3)/2
; -b2/4 )
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | p | / |
| Level 1 | b | s(1)m,p(b) |
Example: m = 0.2 + 0.3 i , p = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.2 , 0.3 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
LOM1B returns
s(1)m,p(b)
= ( 0.093585317 + 0.033014799 i ) + ( -0.071915563
+ 0.063783295 i ) e1
+ ( -0.107085614 + 0.105255185 i ) e2 + ( -0.142255666
+ 0.146727075 i ) e3
Formula:
s(2)m,p(b) = bm+1
/ [ (m+1)2 - p2 ] 1F2
( 1 ; (m-p+3)/2 , (m+p+3)/2 ; b2/4 )
+ 2m+p-1 Gam(p) Gam((m+p+1)/2) b -p / Gam((-m+p+1)/2)
0F1
( ; 1-p ; -b2/4 )
+ 2m-p-1 Gam(-p) Gam((m-p+1)/2) bp / Gam((-m-p+1)/2)
0F1
( ; 1+p ; -b2/4 )
where pFq is the generalized
hypergeometric function and Gam is the Euler Gamma function.
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | p | / |
| Level 1 | b | s(2)m,p(b) |
Example: m = 0.2 + 0.3 i , p = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.2 , 0.3 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
LOM2B returns
s(2)m,p(b)
= ( -1.431572990 - 2.691843964 i ) + ( -1.074362126
+ 0.548149519 i ) e1
+ ( -1.632152954 + 0.941062220 i ) e2 + ( -2.189943783
+ 1.333974920 i ) e3
Formula:
Hm(b) = (b/2)m+1 1F~2(
1 ; 3/2 , m + 3/2 ; - b2/4 )
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Hm(b) |
Example: m = 2 + 3.i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
STRHB returns
H2+3.i (b) = ( 1.051041908 - 0.109956743
i ) + ( 0.017273464 + 0.374543488 i ) e1
+ ( 0.056910083 + 0.582905964 i ) e2 + ( 0.096546702
+ 0.791268440 ) e3
Formula:
Lm(b) = (b/2)m+1 1F~2(
1 ; 3/2 , m + 3/2 ; b2/4 )
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Lm(b) |
Example: m = 2 + 3.i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 2 , 3 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
STRLB returns
L2+3.i (b) = ( 0.996410287 - 0.241120843
i ) + ( 0.064842485 + 0.357891440 i ) e1
+ ( 0.129785591 + 0.553123248 i ) e2 + ( 0.194728698
+ 0.748355055 ) e3
Formula:
T(m,n;b) = exp(-b2)
[ Gam((m+1)/2) ] b2n-m+11F~1(
(m+1)/2 , n+1 , b2
)
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | b | T(m,n;b) |
Example: m = 0.2 + 0.3 i , n = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.2 , 0.3 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
TORB returns
T(m,n;b)
= ( 2.358850039 - 5.982167484 i ) + ( 2.067337944
+ 0.962846892 i ) e1
+ ( 3.302074944 + 1.336654116 i ) e2 + ( 4.536811945
+ 1.710461340 i ) e3
Formula: Assuming l # 0
Cm(l)(b) = [ Gam(m+2l) / Gam(m+1) / Gam(2.l) ] 2F1( -m , m+2l , m+1/2 , (1-b)/2 )
where 2F1 is the hypergeometric function and
Gam = Gamma function
| STACK | INPUTS | OUTPUTS |
| Level 3 | l | / |
| Level 2 | m | / |
| Level 1 | b | Cm(l)(b) |
Example: l = 0.2 + 0.3 i , m = 1.4 + 1.6 i , b = ( 1.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.2 , 0.3 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 1.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
USFB returns
Cm(l)(b)
= ( -1.082467434 + 0.708001466 i ) + ( -0.196556330
- 0.294814041 i ) e1
+ ( -0.330212998 - 0.444185397 i ) e2 + ( -0.463869665
- 0.593556753 i ) e3
Note:
-See also USPB
Formula:
Mq,p(b) = exp(-b/2)
bp+1/2 M( p-q+1/2 , 1+2p , b )
where M = Kummer's function
| STACK | INPUTS | OUTPUTS |
| Level 3 | q | / |
| Level 2 | p | / |
| Level 1 | b | Mq,p(b) |
Example: q = 0.2 + 0.3 i , p = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.2 , 0.3 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
WHIMB returns
Mq,p(b) = ( 1.697417276
- 1.038735336 i ) + ( 0.312886026 + 0.617812024
i ) e1
+ ( 0.537527163 + 0.938755876 i ) e2 + ( 0.762168300
+ 1.259699728 i ) e3
Formula:
Wq,p(b) = [ Gam(2p) / Gam(p-q+1/2) ] Mq,-p(b) + [ Gam(-2p) / Gam(-p-q+1/2) ] Mq,p(b) assuming 2p is not an integer.
where Mq,p(b)
= Whittaker's M-function above.
| STACK | INPUTS | OUTPUTS |
| Level 3 | q | / |
| Level 2 | p | / |
| Level 1 | b | Wq,p(b) |
Example: q = 0.2 + 0.3 i , p = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( 0.2 , 0.3 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
WHIWB returns
Wq,p(b) = ( 7.064436821
+ 2.782608807 i ) + ( 1.108472804 - 2.517738277
i ) e1
+ ( 1.527798512 - 4.016349537 i ) e2 + ( 1.947124221
- 5.514960797 i ) e3
Fractional Integro-Differentiation (FID)
Formula:
Dµ Exp b = b -µ 1F~1
( 1 ; 1-µ ; b )
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Exp b |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DEXPB returns
Dµ Exp b = ( -206.0578281 - 113.9610513
i ) + ( -44.71994312 + 70.62720219 i ) e1
+ ( -64.11293509 + 113.7560309 i ) e2 + ( -83.50592706 + 156.8848596
i ) e3
Formulae: Dµ Ln b = b -µ FC(µ)log (b)
where
FC(µ)log (b) = (-1)µ-1 (µ-1)
!
if µ is a positive integer
and
FC(µ)log (b) = [ Ln b - Psi(1-µ) - gamma
] / Gam(1-µ) otherwise
Psi = Digamma Function , gamma = Euler's
constant = 0.577215664901... and Gam = Gamma Function.
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Ln b |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DLNB returns
Dµ Ln b
= ( 859.0935890 - 587.9181288 i ) + ( -190.2120867 - 316.4051619 i ) e1
+ ( -322.0432682 - 478.3750856 i ) e2 + ( -453.8744497 - 640.34500927
i ) e3
Note:
-Likewise, with µ = 3 , D3
Ln b = ( 0.233801703 + 0.211686544 i ) + ( -0.191588820 - 0.110389215 i
) e1
+ ( -0.307709696 - 0.156880070 i ) e2 + ( -0.423830572 - 0.203370925
i ) e3
Formula:
Dµ bz = [ z ! / Gam(z-µ+1) ] bz-µ
assuming z # -1 , -2 , -3 , .....
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | b | / |
| Level 1 | z | Dµ bz |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3 , z = 1.4 + 1.6 i
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
ENTER
( 1.4 , 1.6 )
DBX returns
Dµ
bz = ( 0.497693400 + 0.217073384 i ) + ( 0.084362474 -
0.160789881 i ) e1
+ ( 0.118742269 - 0.257581212 i ) e2 + ( 0.153122063 - 0.354372543
i ) e3
Formula:
Dµ Sinh b = 2µ-1 sqrt(PI) b1-µ1F~2
( 1 ; (2-µ)/2 , (3-µ)/2 ; b2/4 )
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Sinh b |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DSHB returns
Dµ
Sinh b = ( 94.46345493 + 13.00295518 i ) + ( 6.540750036 -
33.16801912 i ) e1
+ ( 7.550128527 - 52.26536984 i ) e2 + ( 8.559507017 - 71.36272055
i ) e3
Formula:
Dµ Cosh b = (2/b)µ sqrt(PI) 1F~2
( 1 ; (1-µ)/2 , (2-µ)/2 ; b2/4 )
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Cosh b |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DCHB returns
Dµ
Cosh b = ( -300.52128302 - 126.9640065 i ) + ( -51.26069332
+ 103.7952213 i ) e1
+ ( -71.66306362 + 166.0214007 i ) e2 + ( -92.06543408 + 228.24758010
i ) e3
Formula:
Dµ Sin b = 2µ-1 sqrt(PI) b1-µ1F~2
( 1 ; (2-µ)/2 , (3-µ)/2 ; -b2/4 )
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Sin b |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DSINB returns
Dµ Sin b
= ( 86.10292846 + 18.96360900 i ) + ( 8.479611306 - 30.08538283 i
) e1
+ ( 10.82136301 - 47.61156612 i ) e2 + ( 13.16311472 - 65.13774940
i ) e3
Formula:
Dµ Cos b = (2/b)µ sqrt(PI) 1F~2
( 1 ; (1-µ)/2 , (2-µ)/2 ; -b2/4 )
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Cos b |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DCOSB returns
Dµ
Cos b = ( -254.7189012 - 135.6609115 i ) + ( -53.38061174 + 87.39494685
i ) e1
+ ( -76.28215856 + 140.6065660 i ) e2 + ( -99.18370538 + 193.8181852
i ) e3
Formulae:
Dµ Ai(b) = 3µ-4/3 b -µ { 32/3 Gam(1/3) 2F~3 [ 1/3 , 1 ; (1-µ)/3 , (2-µ)/3 , (3-µ)/3 ; b3/9 ] - b Gam(2/3) 2F~3 [ 2/3 , 1 ; (4-µ)/3 , (2-µ)/3 , (3-µ)/3 ; b3/9 ] }
Dµ Bi(b) = 3µ-5/6
b -µ { 32/3 Gam(1/3) 2F~3
[ 1/3 , 1 ; (1-µ)/3 , (2-µ)/3 , (3-µ)/3 ; b3/9
] + b Gam(2/3) 2F~3 [ 2/3 , 1 ; (4-µ)/3
, (2-µ)/3 , (3-µ)/3 ; b3/9 ] }
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | Dµ Ai b |
| Level 1 | b | Dµ Bi b |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DAIRYB returns
Dµ Ai(b) = ( -120.0272086
- 52.59871418 i ) + ( -21.13210737 + 41.41275684 i ) e1
+ ( -29.65306695 + 66.29446926 i ) e2 + ( -38.17402653 + 91.17618168
i ) e3
in level 2 and in level 1:
Dµ Bi(b) = ( -127.7699689
- 75.03970333 i ) + ( -29.27191693 + 43.70059981 i ) e1
+ ( -42.16814243 + 70.51468906 i ) e2 + ( -55.06436793 + 97.32877831
i ) e3
Formula:
Dµ S(b) = 22µ-11/2
PI5/2 b3-µ3F~4
[ 3/4 , 1 , 5/4 ; (4-µ)/4 , (5-µ)/4 , (6-µ)/4 , (6-µ)/4
; -(PI)2 b4/16 ]
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ S(b) |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DSB returns
Dµ S(b) = ( 14.88193368
- 8.573304423 i ) + ( -2.729648823 - 5.445395137 i ) e1
+ ( -4.693883775 - 8.276444508 i ) e2 + ( -6.658118727 - 11.10749388
i ) e3
Formula:
Dµ C(b) = 22µ-3/2
PI3/2 b1-µ3F~4
[ 1/4 , 3/4 , 1 ; (2-µ)/4 , (3-µ)/4 , (4-µ)/4 , (5-µ)/4
; -(PI)2 b4/16 ]
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ C(b) |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DCB returns
Dµ C(b) = ( 90.17195800
+ 21.68847860 i ) + ( 9.528403546 - 31.46967562 i ) e1
+ ( 12.34673548 - 49.85496625 i ) e2 + ( 15.16506742 - 68.24025687
i ) e3
Formula:
Dµ Erf (b) = 2µ b1-µ2F~2
[ 1/2 , 1 ; (2-µ)/2 , (3-µ)/2 ; -b2 ]
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Erf(b) |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DERFB returns
Dµ Erf (b) = ( 92.69786384
+ 22.88014402 i ) + ( 10.00128913 - 32.33865652 i ) e1
+ ( 13.01491852 - 51.24840731 i ) e2 + ( 16.02854791 - 70.15815809
i ) e3
Formula:
Dµ
Hm (b) = [ 2m+µ (PI) b-µ /
Gam((1-m)/2) ] 2F~2 [ 1 , -m/2 ; (1-µ)/2
, (2-µ)/2 ; b2 ]
- [ 2m+µ (PI) b1-µ / Gam((-m)/2) ] 2F~2
[ 1 , (1-m)/2 ; 1-µ/2 , (3-µ)/2 ; b2 ]
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | b | Dµ Hn (b) |
Example: m = -3.14 + 2.718 i , n = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DHMTB returns
Dµ Hn (b)
= ( -161.6892429 + 1659.074706 i ) + ( 583.8677499 + 91.94274493 i ) e1
+ ( 918.1891095 + 96.72126210 i ) e2 + ( 1252.510469 + 101.4997793
i ) e3
(FID) Bessel function
of the 1st kind
Formula:
Dµ Jm (b) = 2µ-2m
sqrt(PI) b-m-µ Gam(m+1) 2F~3
[ (m+1)/2 , (m+2)/2 ; (m+1-µ)/2 , (m+2-µ)/2 , m+1 ; -b2/4
]
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | b | Dµ Jn (b) |
Example: m = -3.14 + 2.718 i , n = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DJNB returns
Dµ Jn (b)
= ( -0.081011034 - 0.280385817 i ) + ( -0.095601814 + 0.020153180 i ) e1
+ ( -0.147526575 + 0.039087106 i ) e2 + ( -0.199451336 + 0.058021032
i ) e3
(FID) Modified Bessel
function of the 1st kind
Formulae:
Dµ Im (b) = 2µ-2m
sqrt(PI) bm-µ Gam(m+1) 2F~3
[ (m+1)/2 , (m+2)/2 ; (m+1-µ)/2 , (m+2-µ)/2 , m+1 ; b2/4
]
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | b | Dµ In (b) |
Example: m = -3.14 + 2.718 i , n = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DINB returns
Dµ In (b)
= ( -0.111691408 - 0.292384211 i ) + ( -0.100055670 + 0.030356977 i ) e1
+ ( -0.153658287 + 0.055361338 i ) e2 + ( -0.207260904 + 0.080365698
i ) e3
(FID) Bessel function
of the 2nd kind
Formulae:
Assuming m is not an integer,
• Dµ Ym (b) = 2µ-2m (PI)1/2 b-µ-m csc(m.PI) { -16m Gam(1-m) 2F~3 [ (1-m)/2 , (2-m)/2 ; (1-µ-m)/2 , (2-µ-m)/2 , 1-m ; -b2/4 ]
+ b2m Cos(m.PI) Gam(m+1) 2F~3
[ (m+1)/2 , (m+2)/2 ; (m+1-µ)/2 , (m+2-µ)/2 , m+1 ; -b2/4
] }
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | b | Dµ Yn (b) |
Example: m = -3.14 + 2.718 i , n = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DYNB returns
Dµ Yn (b)
= ( -1892.715601 + 7911.974085 i ) + ( 2761.990235 + 834.1329921 i ) e1
+ ( 4375.435405 + 1080.288249 i ) e2 + ( 5988.880576 + 1326.443506
i ) e3
(FID) Modified Bessel
function of the 2nd kind
Formulae: Assuming m is not an integer,
Dµ Km (b) = 2µ-2m-1 (PI)3/2 b-µ-m csc(m.PI) { 16m Gam(1-m) 2F~3 [ (1-m)/2 , (2-m)/2 ; (1-µ-m)/2 , (2-µ-m)/2 , 1-m ; b2/4 ]
- b2m Gam(m+1) 2F~3 [
(m+1)/2 , (m+2)/2 ; (m+1-µ)/2 , (m+2-µ)/2 , m+1 ; b2/4
] }
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | b | Dµ Kn (b) |
Example: m = -3.14 + 2.718 i , n = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DKNB returns
Dµ Kn (b)
= ( 2353.469156 - 12981.08721 i ) + ( -4547.159627 - 1102.355609 i ) e1
+ ( -7181.757467 - 1355.901980 i ) e2 + ( -9816.355307 - 1609.448351
i ) e3
(FID) Spherical Bessel
function of the 1st kind
Formula:
Dµ jm
(b) = 2µ-2m-1 PI bm-µ Gam(m+1)
2F~3
[ (m+1)/2 , (m+2)/2 ; (m+1-µ)/2 , (m+2-µ)/2 , m+3/2 ; -b2/4
]
| STACK | INPUTS | OUTPUTS |
| Level 3 | m | / |
| Level 2 | n | / |
| Level 1 | b | Dµ jn (b) |
Example: m = -3.14 + 2.718 i , n = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DBS1B returns
Dµ jn (b)
= ( -0.090245463 - 0.129873519 i ) + ( -0.045563143 + 0.026935641 i ) e1
+ ( -0.068923652 + 0.045664652 i ) e2 + ( -0.092284161 + 0.064393662
i ) e3
Formula:
Dµ Lpq (b)
= [ Gam(p+q+1)/Gam(q+1) ] b -µ2F~2
( 1 , -q ; p+1 , 1-µ ; b )
| STACK | INPUTS | OUTPUTS |
| Level 4 | m | / |
| Level 3 | p | / |
| Level 2 | q | / |
| Level 1 | b | Dµ Lpq (b) |
Example: m = -3.14 + 2.718 i , p = 0.2 + 0.3 i , q = 1.4 + 1.6 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
( 0.2 , 0.3 )
ENTER
( 1.4 , 1.6 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DLANB returns
Dµ
Lpq (b) = ( -225.3837161 - 483.3707173 i ) + ( -175.8661355
+ 69.7831884 i ) e1
+ ( -268.7685164 + 122.9310648 i ) e2 + ( -361.6708972 + 176.0789411
i ) e3
Formulae:
Dµ Ei b = [ (-1)µ-1 (µ-1) ! ] b -µ + b1-µ 2F~2 ( 1 , 1 ; 2 , 2-µ ; b ) if µ is a positive integer
Dµ Ei b = [ Ln b - Psi(1-µ) ] b
-µ + b1-µ 2F~2
( 1 , 1 ; 2 , 2-µ ; b ) otherwise
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Ei(b) |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DEIB returns
Dµ Ei (b) = ( 779.2227189
- 646.7105078 i ) + ( -212.7491448 - 289.3230092 i ) e1
+ ( -355.0345066 - 434.3239628 i ) e2 + ( -497.3198685 - 579.3249163
i ) e3
Formula:
Dµ Si b = 2µ-2
PI b1-µ2F~3 ( 1/2
, 1 ; 3/2 , (2-µ)/2 , (3-µ)/2 ; -b2/4 )
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Si(b) |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DSIB returns
Dµ Si (b) = ( 88.89749069
+ 17.11318152 i ) + ( 7.881811921 - 31.11285091 i ) e1
+ ( 9.806598523 - 49.16659237 i ) e2 + ( 11.73138513 - 67.22033384
i ) e3
(FID) Hyperbolic Sine
Integral
Formulae:
Dµ Shi b = 2µ-2
PI b1-µ2F~3 ( 1/2
, 1 ; 3/2 , (2-µ)/2 , (3-µ)/2 ; b2/4 )
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Shi(b) |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DSHIB returns
Dµ Shi (b) = ( 91.68566538
+ 15.12680078 i ) + ( 7.235730882 - 32.14085789 i ) e1
+ ( 8.716471545 - 50.71859678 i ) e2 + ( 10.19721221 - 69.29633567
i ) e3
Formulae:
Dµ Ci b = b -µ [ (-1)µ-1 (µ-1) ! ] - 2µ-3 sqrt(PI) b2-µ2F~3 ( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; -b2/4 ) if µ is a positive integer
Dµ Ci b
= b -µ [ Ln b - Psi(1-µ) ] / Gam( 1 - µ
) - 2µ-3 sqrt(PI) b2-µ 2F~3
( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; -b2/4 )
otherwise
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Ci(b) |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DCIB returns
Dµ Ci (b) = ( 710.4485603
- 666.1713704 i ) + ( -221.0395273 - 265.3856399 i ) e1
+ ( -366.0525137 - 396.3184360 i ) e2 + ( -511.0655002 - 527.2512321
i ) e3
(FID) Hyperbolic Cosine
Integral
Formulae:
Dµ Chi b = b -µ [ (-1)µ-1 (µ-1) ! ] + 2µ-3 sqrt(PI) b2-µ2F~3 ( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; b2/4 ) if µ is a positive integer
Dµ
Chi b = b -µ [ Ln b - Psi(1-µ) ] / Gam( 1 - µ
) + 2µ-3 sqrt(PI) b2-µ 2F~3
( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; b2/4 )
otherwise
| STACK | INPUTS | OUTPUTS |
| Level 2 | m | / |
| Level 1 | b | Dµ Chi(b) |
Example: m = -3.14 + 2.718 i , b = ( 0.1 + 0.2 i ) + ( 0.3 + 0.4 i ) e1 + ( 0.5 + 0.6 i ) e2 + ( 0.7 + 0.8 i ) e3
( -3.14 , 2.718 )
ENTER
[ ( 0.1 , 0.2 ) (
0.3 , 0.4 ) ( 0.5 , 0.6 ) ( 0.7 , 0.8 ) ]
DCHIB returns
Dµ Chi (b) = ( 687.5370535
- 661.8373086 i ) + ( -219.9848757 - 257.1821513 i ) e1
+ ( -363.7509782 - 383.6053660 i ) e2 + ( -507.5170807 - 510.0285807
i ) e3
Remarks:
-The fractional-integro-differentiation of a complex or a bicomplex
function is well defined.
-For biquaternions, bioctonions and so on, this is more doubtful:
-Even the 1st derivative of b2 is not 2.b
because the multiplication is not commutative in general.
-So, these functions may be regarded as other special functions not
always related to differentiation of special functions... except for bicomplex
arguments !
-All these programs only deal with a few special functions, they might
be completed by many others.
- HGFB may be used to calculate most
of the special functions.
Reference:
-Most of the formulae come from: