Anionic Functions for the HP48S/SX/G/GX

Overview
 

-The zip file contains

 >>> A binary file ANION48 ( #17236d / 15926.5 bytes )  which works with the HP48S/SX/G/GX        Both are directories, not libraries.
 >>> A binary file ANION50 ( #59974d / 15849 bytes ) which works with the HP49/50G
>>>  An ASCII file ANION which should work with all these calculators.
 >>> This html page.

-----------------------------------------------------------------------------------------------------------------------------
 

-Like the quaternions may be constructed from the complexes, the octonions may be constructed from the quaternions,
 then the sedenions, 32-ons, 64-ons, 128-ons, ...... , and more generally the "anions".

-The Cayley-Dickinson formula          ( a , b ) ( c , d ) = ( a c - d* b , d a + b c* )       where   * = conjugate

  is used to define the multiplication: a complex number is regarded as a pair of real numbers,
  a quaternion as a pair of complexes and so on...

-This product is already not commutative for the quaternions, not even associative for the octonions,
  and zero-divisors appear with the sedenions.

-However, the formulae that are used to define elementary & special functions involve anions with proportional imaginary parts,
  and in this case, the products are commutative and much easier to compute.
 

Notes:

-When the functions are computed by ascending series, the results are accurate for "small" arguments only.

-The variables are anions ( hypercomplexes ), but the parameters are restricted to real values.
-The anions must be entered as N-dimensional vectors ( N > 1 )
-So, the complexes must be under the form  [ x y ]  but not  ( x , y )

-Set your HP-48 in RECT mode.
 
 

Functions

Description

SOLVA    FXA    peval    tol  Im  AxA0  SPEC     HRMA     GAMA     PSIA     PSINA     ZETAA     HGFA     ALFA       ALF12       ALF13       S       ALF22       ALF23     BESSEL       JNA       YNA       INA       KNA       S1       S2     ELLIPA       JEFA       WEFA       WF2A     ESINT       EIA       ENA       SIA       SHIA       CIA       CHIA       S1       S2     SPHR       SMNA       L       LMN       M       N       C2       ITER     ORTHO       LEGA       LANA       CHB1A       CHB2A       JCPA       USPA       PMN2A       PMN3A       S1       S2    MISC       AIBIA       ANWEB       CATA       CHB1A       CHB2A       DNA       ERFA       GERFA       HMTA       IBFA       IGFA       JCFA       LANA       LOM1A       LOM2A       RCWFA       STRHA       STRLA       TORA       USFA       WHIMA       WHIWA       S1       S2    ASYM       DNA       ENA  AxA  INVA  EXPA  LNA  conj  A­A  A­X  X­A  SHA  CHA  THA  ASHA  ACHA  ATHA  SINA  COSA  TANA  ASINA  ACOSA  ATANA  GDA  AGDA  REC  POL  QxQ  R->P  P->R  SGNI

Directory = Solving an Anionic Equation    Solves f(a)=a    Evaluates a polynomial with real coefficients    tolerance = a small positive number  Imaginary part of an anion  Product of 2 anions whose imaginary parts are proportional  Directory = Special Functions    Harmonic Numbers    Gamma Function    Digamma Function    Polygamma Functions    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 = Elliptic Functions      Jacobian Elliptic Functions      Weierstrass Elliptic Functions      Weierstrass Duplication Formula    Directory = Exponential, Sine & Cosine Integral       Exponential Integral Ei(a)       Exponential Integral En(a)       Sine Integral       Hyperbolic Sine Integral       Cosine Integral       Hyperbolic Cosine Integral       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 = 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 = Miscellaneous Functions        Airy Functions Ai(a) & Bi(a)        Anger & Weber Functions        Catalan Numbers        Chebyshev Functions - 1st kind        Chebyshev Functions - 2nd kind        Parabolic Cylinder Functions        Error Function        Generaized Error Functions        Hermite Functions        Incomplete Beta Functions        Incomplete Gamma Functions        Jacobi Functions        Laguerre's Functions        Lommel Functions - 1st kind        Lommel Functions - 2nd kind        Regular Coulomb Wave Functions        Struve Function H        Struve Function L        Toronto Functions        UltraSpherical Functions        Whittaker's M-Function        Whittaker's W-Function        Subroutine1        Subroutine2     Directory = Asymptotic Expansions        Parabolic Cylinder Functions        Exponential Integral En(a)  Product of 2 anions - general case - Cayley-Dickinson doubl.  Inverse of an anion  Exponential of an anion  Logarithm of an anion  Conjugate of an anion  Raising an anion to an anionic exponent  Raising an anion to a real exponent  Raising a real to an anionic exponent  Hyperbolic Sine  Hyperbolic Cosine  Hyperbolic Tangent  Inverse Hyperbolic Sine  Inverse Hyperbolic Cosine  Inverse Hyperbolic Tangent  Sine of an anion  Cosine of an anion  Tangent of an anion  Arc Sine  Arc Cosine  Arc Tangent  Gudermannian Function  Inverse Gudermannian Function  Polar-Rectangular conversion of an anion  Rectangular-Polar conversion of an anion  Product of 2 Quaternions  Rectangular-Polar Conversion of 2 Real Numbers  Polar-Rectangular Conversion of 2 Real Numbers  "Sign" of Im(a)

 

Imaginary part of an anion
 

Im  returns the imaginary part of  [ a₀ , a₁ , ..... , a_n ] i-e  [ a₁ , ..... , a_n ]

-It's only useful to visualize more easily a quaternion - at least with an HP-48...
 
 

Product of 2 anions 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 ).
-But fortunately, there is a much easier case when the anions have the same imaginary direction,
  it's often the case to compute elementary or special functions of an anion:
 
 

STACK	INPUTS	OUTPUTS
Level 2	A	/
Level 1	A'	A A'

Example:    A = 8 + 3 i + 4 j + 7 k   ;   A' = 5 - 6 i - 8 j - 14 k

  [ 8  3   4   7 ]   ENTER
  [ 5 -6 -8 -14 ]  AxA0      >>>>   [ 188  -33  -44  -77 ]

-So,    A.A' = 188 - 33 i - 44 j - 77 k

Notes:

-Of course, the next program AxA gives the same result but AxA0 is much faster.
-Unlike AxA below, AxA0 also works with N-vectors where N > 1 is not an integer power of 2.
-AxA0 does not check that the imaginary parts are proportional...
 

Product of 2 anions - general case - Cayley-Dickinson doubling
 

-AxA uses the Cayley-Dickinson formula          ( a , b ) ( c , d ) = ( a c - d* b , d a + b c* )       where   * = conjugate

-AxA 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 complexes, it simply calls AxA0
 
 

STACK	INPUTS	OUTPUTS
Level 2	A	/
Level 1	A'	A A'

Example:    Calculate the product of the sedenions:

    A = 2 + 4 e₁ + 9 e₂ + 3 e₃ + 5 e₄ + 7 e₅ + 8 e₆ + 2 e₇ + 6 e₈ + 4 e₉ + 3 e₁₀ + 2 e₁₁ + 9 e₁₂ + 7 e₁₃ + 6 e₁₄ + 8 e₁₅
   A' =  9 + 2 e₁ + 4 e₂ + 7 e₃ + 3 e₄ + 2 e₅ + 6 e₆ + 5 e₇ + 4 e₈ + 2 e₉ + 6 e₁₀ + 9 e₁₁ + 3 e₁₂ + 6 e₁₃ + 5 e₁₄ + 7 e₁₅

    [ 2  4  9  3  5  7  8  2  6  4  3  2  9  7  6  8 ]   ENTER
    [ 9  2  4  7  3  2  6  5  4  2  6  9  3  6  5  7 ]   AxA          >>>>    [  - 357  2  132  20  -53  25  138  101  13 -10  12 -43  41  216  -28  183 ]

  whence

    A.A' = - 357 + 2 e₁ + 132 e₂ + 20 e₃ - 53 e₄ + 25 e₅ + 138 e₆+ 101 e₇ + 13 e₈ - 10 e₉ + 12 e₁₀ - 43 e₁₁ + 41 e₁₂ + 216 e₁₃ - 28 e₁₄ + 183 e₁₅

Notes:

-Since the multiplication is not commutative, place the 1st anion in level2 and the 2nd anion in level1.
-If you enter N-dimensional vectors where N is not an integer power of 2, there will be an error message "somewhere".
-The product of two 128-ons is returned in about 9 minutes (!) with an HP48GX.
 

Inverse of an anion
 

-The reciprocal of an anion A # 0  is given by

    A^-1 = A* / | A |²    where A* is the conjugate of  A   and   | A |² = x₀² + x₁² + ................. + x_N-1²
 
 

STACK	INPUT	OUTPUT
Level 1	A	1/A

Example:    Inverse of the octonion  A = 2 + 4 e₁ + 5 e₂ + 10 e₃ + 3 e₄ + 7 e₅ + 6 e₆ + 9 e₇

     [ 2  4  5  10  3  7  6  9 ]   INVA   gives   [ 2  -4  -5  -10  -3  -7  -6  -9 ] / 320    ( approximately )

     whence    1 / A  = 1/160 - 1/80 e₁ - 1/64 e₂ - 1/32 e₃ - 3/320 e₄ - 7/320 e₅ - 3/160 e₆ - 9/320 e₇
 

Exponential of an anion
 

Formula:       exp( x₀ + x₁ e₁ + .... + x_N-1 e_N-1 ) = e^x0 [ cos µ + ( sin µ ). I ]

    where  µ = ( x₁² + ............... + x_N-1² )^1/2    and    I = ( x₁ e₁ + ............. + x_N-1 e_N-1 ) / µ
 
 

STACK	INPUT	OUTPUT
Level 1	A	Exp(A)

Example:        A = 1 + 0.9 e₁ + 0.8 e₂ + 0.7 e₃ + 0.6 e₄ + 0.5 e₅ + 0.4 e₆ + 0.3 e₇

   [ 1  0.9  0.8  0.7  0.6  0.5  0.4  0.3 ]    EXPA   >>>>   [ -0.27820041773    1.45435861016     1.29276320903     1.13116780790
                                                                                           0.969572406774   0.807977005645   0.646381604517  0.484786203388 ]

-So,  exp w = -0.27820041773 + 1.45435861016  e₁ + 1.29276320903 e₂ + 1.1311678079 e₃
                      + 0.969572406774 e₄ + 0.807977005645 e₅ + 0.646381604517 e₆ + 0.484786203388 e₇
 

Logarithm of an anion
 

-"LNA" uses the formula:      Ln( x₀ + x₁ e₁ + .................+ x_N-1 e_N-1 )

   = Ln ( x₀² + x₁² + .................. + x_N-1² )^1/2 + Atan2(µ,x₀). I

    where  µ = ( x₁² + ............... + x_N-1² )^1/2    and    I = ( x₁ e₁ + ............. + x_N-1 e_N-1 ) / µ

-If  µ = 0  ,    I  is replaced by  e₁
 
 

STACK	INPUT	OUTPUT
Level 1	A	Ln (A)

Example:        A = 1 + 0.9 e₁ + 0.8 e₂ + 0.7 e₃ + 0.6 e₄ + 0.5 e₅ + 0.4 e₆ + 0.3 e₇

   [ 1  0.9  0.8  0.7  0.6  0.5  0.4  0.3 ]    LNA   >>>>   [  0.667500533365   0.555135625689    0.493453889501    0.431772153314
                                                                                         0.370090417126   0.308408680938    0.246726944751    0.185045208563 ]

-So,  Ln A =   0.667500533365 + 0.555135625689  e₁ + 0.493453889501 e₂ + 0.431772153314 e₃
                  + 0.370090417126 e₄ + 0.308408680938 e₅ + 0.246726944751 e₆ + 0.185045208563 e₇
 

Conjugate of an anion
 

-The real part is unchanged and the imaginary part is multiplied by (-1)
 
 

STACK	INPUT	OUTPUT
Level 1	A	conj(A)

Example:        A = 1 + 0.9 e₁ + 0.8 e₂ + 0.7 e₃ + 0.6 e₄ + 0.5 e₅ + 0.4 e₆ + 0.3 e₇

   [ 1  0.9  0.8  0.7  0.6  0.5  0.4  0.3 ]    conj   gives   [ 1  -0.9  -0.8  -0.7  -0.6  -0.5  -0.4  -0.3 ]
 

Raising an anion to an anionic exponent
 

-This may be defined in several ways.
-'A­A'  employs the formula:   A^A' = exp [ ( Ln A ) A' ]
 
 

STACK	INPUTS	OUTPUTS
Level 2	A	/
Level 1	A'	A^A'

Example:

    A = 1 + 0.9 e₁ + 0.8 e₂ + 0.7 e₃ + 0.6 e₄ + 0.5 e₅ + 0.4 e₆ + 0.3 e₇
    A' = 0.3 + 0.4 e₁ + 0.5 e₂ + 0.6 e₃ + 0.7 e₄ + 0.8 e₅ + 0.9 e₆ + e₇

    [ 1  0.9  0.8  0.7  0.6  0.5  0.4  0.3 ]   ENTER
    [ 0.3  0.4  0.5  0.6  0.7  0.8  0.9  1 ]   A­A     >>>>  [ -0.0798554801772   0.0598156552163   0.0747695690201   0.0897234828243
                                                                                         -0.0447059771355  0.0822854669910   0.134585224237     0.074847451159 ]
 

-So,  A^A^' = -0.0798554801772 + 0.0598156552163 e₁ + 0.0747695690201 e₂ +  0.089723482843 e₃
                     - 0.0447059771355 e₄ + 0.082285466991 e₅ + 0.134585224237 e₆ + 0.074847451159 e₇

Note:

-Another definition is possible:   w^w' = exp [ w'  ( Ln w ) ]
 

Raising an anion to a real exponent
 

  A­X   calculates  A^X = exp ( X Ln A )   where  X is a real number and A  is an anion
 
 

STACK	INPUTS	OUTPUTS
Level 2	A	/
Level 1	X	A^X

Example:         A = 1 + 0.9 e₁ + 0.8 e₂ + 0.7 e₃ + 0.6 e₄ + 0.5 e₅ + 0.4 e₆ + 0.3 e₇    and    X = PI

  [ 1  0.9  0.8  0.7  0.6  0.5  0.4  0.3 ]   ENTER  PI  A­X  >>>>   [ -8.10037865985  -0.441313113853   -0.392278323424  -0.343243532998
                                                                                                        -0.29420874257  -0.245173952140   -0.196139161713  -0.147104371284 ]

-So,  A^PI =  -8.10037865985 - 0.441313113853  e₁ - 0.392278323424 e₂ - 0.343243532998 e₃
                     - 0.29420874257 e₄ - 0.24517395214 e₅ - 0.196139161713 e₆ - 0.147104371284 e₇
 

Raising a real to an anionic exponent
 

  X­A   calculates  X^A = exp ( A Ln X )
 
 

STACK	INPUTS	OUTPUTS
Level 2	X	/
Level 1	A	X^A

Example:         X = PI  and  A = 1 + 0.9 e₁ + 0.8 e₂ + 0.7 e₃ + 0.6 e₄ + 0.5 e₅ + 0.4 e₆ + 0.3 e₇

     PI   ENTER   [ 1  0.9  0.8  0.7  0.6  0.5  0.4  0.3 ]     X­A   >>>>   [ -1.06159850987  1.59031853661  1.41361647698  1.23691441735
                                                                                                                 1.06021235773  0.883510298111  0.706808238488  0.530106178866 ]

 and   PI^A = -1.06159850987 + 1.59031853661 e₁ + 1.41361647698 e₂ + 1.23691441735 e₃
                     + 1.06021235773 e₄ + 0.883510298111 e₅ + 0.706808238488 e₆ + 0.530106178866 e₇
 

Hyperbolic Sine
 

-We have:  Sinh ( x₀ + x₁ e₁ + .... + x_N-1 e_N-1 )  =  Sinh x₀ Cos µ  + I ( Cosh x₀ ) ( Sin µ )

    where  µ = ( x₁² + ............... + x_N-1² )^1/2    and    I = ( x₁ e₁ + ............. + x_N-1 e_N-1 ) / µ
 
 

STACK	INPUT	OUTPUT
Level 1	A	Sinh A

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  SHA  >>>>  [ 2.55819839514   1.13145919486   1.50861225981   1.88576532476 ]
 

Hyperbolic Cosine
 

 Cosh ( x₀ + x₁ e₁ + .... + x_N-1 e_N-1 )  =  Cosh x₀ Cos µ  + I ( Sinh x₀ ) ( Sin µ )

    where  µ = ( x₁² + ............... + x_N-1² )^1/2    and    I = ( x₁ e₁ + ............. + x_N-1 e_N-1 ) / µ
 
 

STACK	INPUT	OUTPUT
Level 1	A	Cosh A

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  CHA  >>>>  [ 2.56365605381   1.09075786958   1.4543438261   1.81792978263 ]
 

Hyperbolic Tangent
 

 Tanh A  is defined by  Tanh A = ( Sinh A ) ( Cosh A ) ^-1
 
 

STACK	INPUT	OUTPUT
Level 1	A	Tanh A

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  THA  >>>>  [ 0.999511158203   0.015538758623   0.020718344831   0.025897931037 ]
 

Inverse Hyperbolic Sine
 

 Formula:          ArcSinh A = Ln [ A + ( A² + 1 )^1/2 ]         with a slight modification if  Re(a) = 0
 
 

STACK	INPUT	OUTPUT
Level 1	A	Asinh A

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  ASHA  >>>>  [ 2.68368096566   0.54844598531   0.731261313745   0.914076642184 ]
 

Inverse Hyperbolic Cosine
 

 Formula:          ArcCosh A = Ln [ A + ( A + 1 )^1/2 ( A - 1 )^1/2 ]
 
 

STACK	INPUT	OUTPUT
Level 1	A	Acosh A

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  ACHA  >>>>  [ 2.69156842782   0.550503878188   0.734005170918   0.91750646365 ]
 

Inverse Hyperbolic Tangent
 

 Formula:         ArcTanh A = (1/2) [ Ln ( 1 + A ) - Ln ( 1 - A ) ]
 
 

STACK	INPUT	OUTPUT
Level 1	A	Atanh A

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  ATHA  >>>>  [ 0.036427952795   0.61111743447   0.81482324596   1.01852905745 ]
 

Sine of an anion
 

Formula:     Sin ( x₀ + x₁ e₁ + .... + x_N-1 e_N-1 )  =  Sin x₀ Cosh µ  + I ( Cos x₀ ) ( Sinh µ )

    where  µ = ( x₁² + ............... + x_N-1² )^1/2    and    I = ( x₁ e₁ + ............. + x_N-1 e_N-1 ) / µ
 
 

STACK	INPUT	OUTPUT
Level 1	A	Sin A

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  SINA  >>>>  [ 535.305877182   -103.93893751   -138.585250013   -173.231562516 ]
 

Cosine of an anion
 

Formula:     Cos ( x₀ + x₁ e₁ + .... + x_N-1 e_N-1 )  =  Cos x₀ Cosh µ  + I ( Sin x₀ ) ( Sinh µ )

    where  µ = ( x₁² + ............... + x_N-1² )^1/2    and    I = ( x₁ e₁ + ............. + x_N-1 e_N-1 ) / µ
 
 

STACK	INPUT	OUTPUT
Level 1	A	Cos A

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  COSA  >>>>  [ -244.986778585   -227.110721804   -302.814295738   -378.517869673 ]
 

Tangent of an anion
 

  Formula:    Tan A = ( Sin A ) ( Cos A ) ^-1
 
 

STACK	INPUT	OUTPUT
Level 1	A	Tan A

Example:        A =  4 + 3 i + 2 j + k

    [ 4  3  2  1 ]  TANA  >>>>  [ 0.0011129924136   0.801914456275   0.534609637516   0.267304818758 ]
 

Arc Sine
 

Formula:     If   A  =  x₀ + x₁ e₁ + .... + x_N-1 e_N-1       ,       Arc Sin A     = - I  Arc Sinh ( A  I )         with a slight modification if  Re(a) = 0

    where  µ = ( x₁² + ............... + x_N-1² )^1/2    and    I = ( x₁ e₁ + ............. + x_N-1 e_N-1 ) / µ
 
 

STACK	INPUT	OUTPUT
Level 1	A	Asin A

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  ASINA  >>>>  [ 0.27324624209   1.14193577233   1.52258102977   1.90322628722 ]
 

Arc Cosine
 

      ArcCos q = PI/2 - ArcSin q
 
 

STACK	INPUT	OUTPUT
Level 1	A	Acos A

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  ACOSA  >>>>  [ 1.29755008471   -1.14193577233   -1.52258102977   -1.90322628722 ]
 

Arc Tangent
 

Formula:     If   A  =  x₀ + x₁ e₁ + .... + x_N-1 e_N-1        ,           Arc Tan A     = - I  Arc Tanh ( A  I )         with a slight modification if  Re(a) = 0

    where  µ = ( x₁² + ............... + x_N-1² )^1/2    and    I = ( x₁ e₁ + ............. + x_N-1 e_N-1 ) / µ
 
 

STACK	INPUT	OUTPUT
Level 1	A	Atan A

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  ATANA  >>>>  [ 1.53313188138   0.0557976194604   0.0743968259471   0.0929960324339 ]
 

Gudermannian Function
 

Definition:      Gd(A) = 2 ArcTan [ Tanh (A/2) ]
 
 

STACK	INPUT	OUTPUT
Level 1	A	Gd (A)

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  GDA  >>>>  [ 1.37871508722   0.0809051645136   0.107873552684   0.134841940856 ]
 

Inverse Gudermannian Function
 

Definition:      Agd(A) = 2 ArcTanh [ Tan (A/2) ]
 
 

STACK	INPUT	OUTPUT
Level 1	A	Agd (A)

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  AGDA  >>>>  [ 0.00154457947   0.666732348242   0.888976464322   1.11122058041 ]
 

Rectangular-Polar conversion of an Anion
 
 

STACK	INPUT	OUTPUT
Level 1	A	Pol (A)

Example:        A =  2 + 3 i + 4 j + 5 k

    [ 2  3  4  5 ]  POL  >>>>  [ 7.34846922834  74°2068309517  64°8959097498   51°3401917459 ]
 

-Here, we've assumed that your calculator is in DEG mode.
-POL works in all angular modes.
 

Polar-Rectangular conversion of an Anion
 
 

STACK	INPUT	OUTPUT
Level 1	Pol (A)	A

Example:       If we take the result above:

   [ 7.34846922834  74°2068309517  64°8959097498   51°3401917459 ]   REC  >>>>   [ 2  3  4  5 ]   ( with small roudoff-errors )

-Like POL , REC also works in all angular modes.
 

Product of 2 Quaternions
 
 

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 ]
 

Rectangular-Polar Conversion of 2 Real Numbers
 
 

STACK	INPUTS	OUTPUTS
Level 2	x	r
Level 1	y	µ

Example:

    3   ENTER
    4     R->P     >>>>   r = 5  in level 2  and   µ =  53°1301023542

  R->P  works in all angular modes.
 

Polar-Rectangular Conversion of 2 Real Numbers
 

-Like R->P ,  P->R  uses  x = r cos µ  ,   y = r sin µ
 
 

STACK	INPUTS	OUTPUTS
Level 2	r	x
Level 1	µ	y

Example:

     5   ENTER
   53.1301023542   P->R  returns  3  in level 2  and  4 in level 1

 P->R  works in all angular modes.
 

Sign of Im(a)
 

-Simply returns the sign of the 1st component of Im(a)
-Returns 1 if Im(a) = 0
 

Solving an Anionic Equation
 

-The equation must be rewritten in the form:    f ( a ) = a
  and  f   must satisfy a Lipschitz condition   | f(a) - f(a') | < h | a - a' |   with  h < 1  ,   provided a and a' are close to a solution,
  then the sequence  a_n+1 = f ( a_n)  converges to a root.

-Place a small positive number in 'tol'
-In level 1, place a program that takes a in level 1 and returns f(a) in level 1
-Place an approximation in level 1 and press the key FXA

-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	a	Solution

   where  <<  p  >>  calculates f(a)

Example:        Find a solution of the octonionic equation    w²  - Ln w  + 1 + 0.9 e₁ + 0.8 e₂ + 0.7 e₃ + 0.6 e₄ + 0.5 e₅ + 0.4 e₆ + 0.3 e₇ = 0

         near   1 + e₁ + e₂ + e₃ + e₄ + e₅ + e₆ + e₇

1°) First, we re-write this equation:    w  =  ( Ln w  - 1 - 0.9 e₁ - 0.8 e₂ - 0.7 e₃ - 0.6 e₄ - 0.5 e₅ - 0.4 e₆ - 0.3 e₇ ) ^1/2  =  f ( w )

2°) Store a small number - say  2E-11 - in 'tol'

3°) Then,      <<  LNA  [ 1  .9  .8  .7  .6  .5  .4  .3 ]  -  2  INV  A­X  >>   ENTER
                                                                            [ 1  1  1  1  1  1  1  1 ]     FXA     returns eventually the solution

   [ 1.02254775914  -0.673000418404   -0.598222594136  -0.52344476987  -4486669456  -0.373889121331  -0.299111297064  -0.22433347297 ]
 

Anionic Polynomials
 

'peval'  evaluates  p(a) = c_m a^m + c_m-1 a^m-1 + ................ + c₁ a + c₀    for  a given anion a  and  (m+1) real numbers  c_m , .............. , c₁ , c₀
 
 

STACK	INPUT	OUTPUT
Level 2	polynomial p	/
Level 1	A	p (A)

Example:           a = 1 + 2 i + 3 j + 4 k        ,         p(a) = 6 a⁵ + 2 a⁴ + 3 a³ + 7 a² + 4 a + 1

        [ 6  2  3  7  4  1 ]  ENTER
            [ 1  2  3  4 ]      peval         gives    [ 24383  6104  9156  12208 ]

-Whence,              p(a) = 24383 + 6104 i + 9156 j + 12208 k
 

Harmonic Numbers
 

"HRMA" calculates the generalized harmonic numbers defined as  H_m(a) = 1 + 2 ^-a + 3 ^-a + ............ + m ^-a
 
 

STACK	INPUT	OUTPUT
Level 2	m	/
Level 1	A	Hm(a)

   where  m is a positive integer

Example:    Calculate  H₁₂( 1 + 2 e₁ + 3 e₂ + 4 e₃ + 5 e₄ + 6 e₅ + 7 e₆ + 8 e₇ )

 [ 1  2  3  4  5  6  7  8 ]  ENTER
                12                  HRMA  >>>>  [ 0.248285997  0.012575284  0.018862927  0.025150569
                                                                                        0.031438211  0.037725853  0.044013495  0.050301137 ]        ( rounded to 9D )

Whence:

   H₁₂( 1 + 2 e₁ + 3 e₂ + 4 e₃ + 5 e₄ + 6 e₅ + 7 e₆ + 8 e₇ )  =  0.248285998 + 0.012575285 e₁ + 0.018862927 e₂ + 0.025150570 e₃
                                                                                           + 0.031438213 e₄ + 0.037725855 e₅ + 0.044013498 e₆ + 0.050301140 e₇
 

Gamma Function
 

-Here, the Gamma function is computed by a continued fraction:

    Gam(a) =  exp [ (a-1/2) Ln a + Ln (2.PI)^1/2  - a + ( 1/12 )/( a + ( 1/30 )/( a + ( 53/210)/( a + (195/371)/( a + ...  )))) ]
 

-The relation   Gam(a+1) = a Gam(a)  is used recursively if Re(a) < 8  until   Re(a+1+.......+1)  > 8
 
 

STACK	INPUT	OUTPUT
Level 1	A	Gamma (A)

Example:        A = 4 + 3 i + 2 j + k

     [ 4  3  2  1 ]  GAMA  yields  [ 0.541968821216  -0.740334194929  -0.493556129954  -0.246778064977 ]
 

Digamma Function
 

Formula:             Psi(a) ~ Ln a - 1/(2a) -1/(12a²) + 1/(120a⁴) - 1/(252a⁶) + 1/(240a⁸)    is used if  Re(a) > 8

                        Psi(a+1) = Psi(a) + 1/a    is used recursively until  Re(a+1+....+1) > 8
 
 

STACK	INPUT	OUTPUT
Level 1	A	Psi (A)

Example:        A = 1 + 2 i + 3 j + 4 k

     [ 1  2  3  4 ]   PSIA  >>>>  [ 1.68653155606   0.548896351601   0.823344527405    1.09779270321 ]
 

Polygamma Functions
 

Formulae:     PSINA employs the asymptotic expansions:

  •   If m > 0  ,  y^(m) (a)  ~  (-1)^m-1 [ (m-1)! / a^m + m! / (2.a^m+1) + SUM_k=1,2,.... B_2k (2k+m-1)! / (2k)! / a^2k+m       where  B_2k are Bernoulli numbers

  •   If m = 0  ,  y (a)  ~  Ln a - 1/(2a) - SUM_k=1,2,.... B_2k / (2k) / a^2k              ( digamma function )
 

 and the recurrence relation:   y^(m) (a+p) = y^(m) (a) + (-1)^m m! [ 1/a^m+1 + ....... + 1/(a+p-1)^m+1 ]           where  p  is a positive integer
 
 

STACK	INPUT	OUTPUT
Level 2	m	/
Level 1	A	y(m) (a)

   where  m is a non-negative integer

Example:     m = 3   A =  1 + 0.9 i + 0.8 j + 0.7 k

                  3                ENTER
      [ 1  0.9  0.8  0.7 ]   PSINA    gives   [  -0.673649101  0.147195368   0.130840327   0.114485286 ]         ( rounded to 9D )
 

  whence    y⁽³⁾ ( 1 + 0.9 i + 0.8 j + 0.7 k ) =  -0.673649101 + 0.147195368 i + 0.130840327 j + 0.114485286 k
 

Note:

 m = 0  will give the Digamma Function;
 

Riemann Zeta Function
 

 ZETAA  employs the method given by P. Borwein in  "An Efficient Algorithm for the Riemann Zeta Function"  if  Re(a) >= 1/2
-If  Re(a) < 1/2, it uses:     Zeta(a) = Zeta(1-a)  Pi  ^a-1/2  Gamma((1-a)/2) / Gamma(a/2)
 
 

STACK	INPUT	OUTPUT
Level 1	A	Zeta (A)

Example:        A = -7.6 + 7.7 i + 7.8 j + 7.9 k

     [ -7.6   7.7   7.8   7.9 ]   ZETAA  >>>>  [ 762.985293227  -14.26823184   -14.453533553   -14.638835265 ]
 

-Likewise,  Zeta( 1.1 + 7 i + 8 j + 9 k ) = 0.389296030446 - 0.027737070994 i - 0.0316995097084 j - 0.0356619484218 k

Note:

-The results are faster and more accurate if  Re(a) > 1/2
-Large execution times are to be expected for large imaginary parts.
 

Generalized Hypergeometric Functions
 

 HGFA computes   _pF_q( a₁,a₂,....,a_p ; b₁,b₂,....,b_q ; A ) =  SUM_{k=0,1,2,.....}    [(a₁)_k(a₂)_k.....(a_p)_k] / [(b₁)_k(b₂)_k.....(b_q)_k] . A^k/k!             if  Level1 > 0

           where (a_i)_k = a_i(a_i+1)(a_i+2) ...... (a_i+k-1)   &  (a_i)₀ = 1    ,    likewise for  (b_j)_k    ( Pochhammer's symbol )

 or the regularized function F tilde:

           _pF^~_q( a₁,a₂,....,a_p ; b₁,b₂,....,b_q ; A ) =  SUM_{k=0,1,2,.....}    [(a₁)_k(a₂)_k.....(a_p)_k] / [Gam(k+b₁) Gam(k+b₂).....Gam(k+b_q)] . a^k/k!      if  Level2 < 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	A	/
Level 2	p	/
Level 1	+/- q	f(a)

  Level 1 =  +q  for the non-regularized HGF ,  Level 1 = -q  for the regularized  HGF

Example1:             a₁ = PI  ,   a₂ = e  ;  b₁ = 1 ,  b₂ = 2 ,   b₃ = 4  ;   A = 1 + 2 i + 3 j + 4 k     the non-regularized function:

         PI
          1   E^X
          1   ENTER
          2   ENTER
          4   ENTER
  [ 1  2  3  4]   ENTER
          2   ENTER
          3   ENTER       HGFA    >>>>    [ -6.69112690456  1.30253058324    1.95379587484    2.60506116646 ]
 

  Whence      ₂F₃( PI , e ; 1 , 2 , 4 ;  1 + 2 i + 3 j + 4 k ) = -6.69112690456 + 1.30253058324 i + 1.95379587484 j + 2.60506116646 k
 

Example2:             a₁ = PI  ,   a₂ = e  ;  b₁ = 1 ,  b₂ = 2 ,   b₃ = 4  ;   A = 1 + 2 i + 3 j + 4 k     the regularized function:

         PI
          1   E^X
          1   ENTER
          2   ENTER
          4   ENTER
  [ 1  2  3  4]   ENTER
          2   ENTER
         -3   ENTER       HGFA    >>>>    [ -1.11518781743    0.217088430537    0.325632645804    0.434176861072 ]

    Whence     ₂F^~₃( PI , e ; 1 , 2 , 4 ;  1 + 2 i + 3 j + 4 k ) = -1.11518781743 + 0.217088430537 i + 0.325632645804 j + 0.434176861072 k

-The first result has been simply divided by  Gam(1) Gam(2) Gam(4) = 6
 

Example3:             a₁ = PI  ,   a₂ = e  ;  b₁ = 1 ,  b₂ = -2 ,   b₃ = -4  ;   A = 1 + 2 i + 3 j + 4 k     the regularized function:

         PI
          1   E^X
          1   ENTER
         -2   ENTER
         -4   ENTER
  [ 1  2  3  4]   ENTER
          2   ENTER
         -3   ENTER       HGFA    >>>>    [ -2910223.71103  192140.227467   288210.341193   384280.454929 ]

    Whence     ₂F^~₃( PI , e ; 1 , -2 , -4 ;  1 + 2 i + 3 j + 4 k ) = -2910223.71103 + 192140.227467 i + 288210.341193 j + 384280.454929 k

Note:

  HGFA  is called as a subroutine by several other programs hereunder.
 

Legendre Functions - 1st kind - Type2
 

Formula:       P_n^m(A) = (A+1)^m/2/(1-A)^m/2  ₂F^~₁(-n , n+1 ; 1-m ; (1-A)/2 )            (  A # 1 )                | A - 1 | < 2
 
 

STACK	INPUTS	OUTPUTS
Level 3	m	/
Level 2	n	/
Level 1	A	Pnm(A)

Example:            m = sqrt(2)    n = sqrt(3)       A = 0.4 + 0.5 i + 0.6 j + 0.7 k

     2   SQRT
     3   SQRT
 [ .4  .5  .6  .7 ]   ALF12   >>>>   [ -0.866899242816  -1.80896047218   -2.17075256661   -2.53254466106 ]

     So,     P^sqrt(2)_sqrt(3) ( 0.4 + 0.5 i + 0.6 j + 0.7 k ) = -0.866899242816 - 1.80896047218 i - 2.17075256661 j - 2.53254466106 k

Note:

-See also  PMN2A
 

Legendre Functions - 1st kind - Type3
 

Formula:       P_n^m(A) = (A+1)^m/2/(A-1)^m/2  ₂F^~₁(-n , n+1 ; 1-m ; (1-A)/2 )            (  A # 1 )                | A - 1 | < 2
 
 

STACK	INPUTS	OUTPUTS
Level 3	m	/
Level 2	n	/
Level 1	A	Pnm(A)

Example:            m = sqrt(2)    n = sqrt(3)       A = 0.4 + 0.5 i + 0.6 j + 0.7 k

     2   SQRT
     3   SQRT
 [ .4  .5  .6  .7 ]   ALF13   >>>>   [ -2.49418306364    1.42452961118    1.70943553342   1.99434145566 ]

 Whence       P^sqrt(2)_sqrt(3) ( 0.4 + 0.5 i + 0.6 j + 0.7 k ) = -2.49418306364 + 1.42452961118 i + 1.70943553342 j + 1.99434145566 k

Note:

-See also  PMN3A
 

Legendre Functions - 2nd kind - Type2
 

Formula:

  Q_n^m(A) = 2^m pi^1/2   (A+1)^-m/2/(1-A)^-m/2 [ -Gam((1+m+n)/2)/(2.Gam((2-m+n)/2)) . sin pi(m+n)/2 .  ₂F₁(-n/2-m/2 ; 1/2+n/2-m/2 ; 1/2 ; A² )       | A | < 1
                                                                + A  Gam((2+n+m)/2) / Gam((1+n-m)/2) . cos pi(m+n)/2 . ₂F₁((1-m-n)/2 ; (2+n-m)/2 ; 3/2 ; A² )  ]
 
 

STACK	INPUTS	OUTPUTS
Level 3	m	/
Level 2	n	/
Level 1	A	Qnm(A)

Example:             m = 0.4    n = 1.3       A = 0.1 + 0.2 i + 0.3 j + 0.4 k

      0.4   ENTER
      1.3   ENTER
     [ .1  .2  .3  .4 ]   ALF22   >>>>   [ -0.944581517232    -0.368022564874   -0.552033847312   -0.736045129748 ]

  Whence,       Q_1.3^0.4( 0.1 + 0.2 i + 0.3 j + 0.4 k ) = -0.944581517232 - 0.368022564874 i - 0.552033847312 j - 0.736045129748 k
 

Legendre Functions - 2nd kind - Type3
 

    Q_n^m(A) =  exp( I (m.PI) ) 2 ^-n-1 sqrt(PI) Gam(m+n+1) A ^-m-n-1  (A+1)^m/2/(A-1)^m/2₂F^~₁( (2+m+n)/2 , (1+m+n)/2 ; n+3/2 ; 1/A² )                 | A | > 1

                     where  I = Im(a) / | Im(a) |  multiplied by the sign of the 1st imaginary component of the anion  (  or  I = i  if  Im(a) = 0 )

-I've modified the 1st term  exp( i (m.PI) ) to generalize the definition to quaternions, octonions, .... but the results remain the same for the complexes.
 
 

STACK	INPUTS	OUTPUTS
Level 3	m	/
Level 2	n	/
Level 1	A	Qnm(A)

Example:               m = sqrt(2)    n = - sqrt(3)       A = 1 + 2 i + 3 j + 4 k

      2   SQRT
      3   SQRT   NEG
 [ 1  2  3  4 ]    ALF23   >>>>   [ -1.92629420143   0.74206199724    1.11309299586    1.48412399448 ]
 

    Whence       Q^sqrt(2)_-sqrt(3) ( 1 + 2 i + 3 j + 4 k ) = -1.92629420143 + 0.74206199724 i + 1.11309299586 j + 1.48412399448 k
 

Bessel Functions of the 1st kind
 

Formula:

       J_n(a) = (a/2)ⁿ [ 1/Gam(n+1)  +  (-a²/4)¹/ (1! Gam(n+2) )  + .... + (-a²/4)^k/ (k! Gam(n+k+1) ) + ....  ]      n # -1 ; -2 ; -3 ; ....
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	/
Level 1	A	Jn(A)

Example:        n = PI     A =  1 + 2 i + 3 j + 4 k

        PI
 [ 1  2  3  4 ]   JNA  >>>>   [ -11.2298751376   -3.57080149748    -5.35620224622    -7.14160299493 ]

Note:

 JNA  does not work if n is a negative integer, but we can employ the relation:      J_n  =  (-1)ⁿ  J_-n   in this case.
 

Bessel Functions of the 2nd kind
 

Formulae:

     Y_n(a) = ( J_n(a) cos(n(pi)) - J_-n(a) ) / sin(n(pi))      ;       if    n # .... -3 ; -2 ; -1 ; 0 ; 1 ; 2 ; 3 ....           and if n is a non-negative integer:

     Y_n(a) = -(1/pi) (a/2)^-n SUM_{k=0,1,....,n-1}  (n-k-1)!/(k!) (a²/4)^k + (2/pi) ln(a/2) J_n(a)
                   - (1/pi) (a/2)ⁿ SUM_k=0,1,.....  ( psi(k+1) + psi(n+k+1) ) (-a²/4)^k / (k!(n+k)!)                         where   psi = the digamma function
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	/
Level 1	A	Yn(A)

Example1:         n = PI     a = 1 + 2 i + 3 j + 4 k

         PI
 [ 1  2  3  4 ]   YNA  >>>>   [ 9.61756409422   -4.17135883145    -6.25703824719    -8.34271766288 ]

Example2:         n = 3     a = 1 + 2 i + 3 j + 4 k

         3          ENTER
 [ 1  2  3  4 ]   YNA  >>>>   [ 7.69002723259   -5.22481266244    -7.83721899364    -10.4496253248 ]

Note:

 YNA  does not work if n is a negative integer, but we can employ the relation:      J_n  =  (-1)ⁿ  J_-n   in this case.
 

Modified Bessel Functions of the 1st kind
 

Formula:

     I_n(a) = (a/2)ⁿ [ 1/Gam(n+1)  +  (a²/4)¹/ (1! Gam(n+2) )  + .... + (a²/4)^k/ (k! Gam(n+k+1) ) + ....  ]          n # -1 ; -2 ; -3 ; ....
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	/
Level 1	A	In(A)

Example:         n = PI     A =  1 + 2 i + 3 j + 4 k

        PI
 [ 1  2  3  4 ]   INA  >>>>   [  0.315197911272   -0.12998865977    -0.194982989655    -0.259977319538 ]

Note:

 INA  does not work if n is a negative integer, but we can employ the relation:      I_n  =  I_-n   in this case.
 

Modified Bessel Functions of the 2nd kind
 

Formulae:            K_n(a) = (pi/2) ( I_-n(a) - I_n(a) ) / sin(n(pi))           n # .... -3 ; -2 ; -1 ; 0 ; 1 ; 2 ; 3 ....             and if n is a non-negative integer:

          K_n(a) = (1/2) (a/2)^-n SUM_k=0,1,..,n-1  (n-k-1)!/(k!) (-a²/4)^k - (-1)ⁿ ln(a/2) I_n(a)                                      where   psi = the digamma function
                    + (1/2) (-1)ⁿ (a/2)ⁿ SUM_k=0,1,...( psi(k+1) + psi(n+k+1) ) (a²/4)^k / (k!(n+k)!)
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	/
Level 1	A	Kn(A)

Example1:         n = PI     A =  1 + 2 i + 3 j + 4 k

        PI
 [ 1  2  3  4 ]   KNA  >>>>   [  0.203067063671   -0.0550308926914    -0.0825463390315    -0.110061785386 ]

Example2:         n = 3     a = 1 + 2 i + 3 j + 4 k

         3          ENTER
 [ 1  2  3  4 ]   KNA  >>>>   [ 0.20876779823   -0.0479831967675    -0.0719747951555    -0.09596639354 ]

Note:

 KNA  does not work if n is a negative integer, but we can employ the relation:      K_n  =  K_-n   in this case.
 

Jacobian Elliptic Functions
 

"JEFA" employs Gauss' transformation to calculate  sn ( a | m )  ,  cn ( a | m )  &  dn ( a | m )

-If   m < 1 , we have:

-With  m' = 1-m ,  let be  µ = [ ( 1-sqrt(m') / ( 1+sqrt(m') ]²   and   v = a / ( 1+sqrt(µ) ]  , then:

   sn ( a | m ) = [ ( 1 + sqrt(µ) ) sn ( v | µ ) ] / [ 1 + sqrt(µ) sn² ( v | µ ) ]
   cn ( a | m ) = [ cn ( v | µ ) dn ( v | µ ) ] / [ 1 + sqrt(µ) sn² ( v | µ ) ]
   dn ( a | 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, the program uses the relations:

      sn ( a | m ) = sn ( a m^1/2 | 1/m ) / m^1/2
      cn ( a | m ) = dn ( a m^1/2 | 1/m )
      dn ( a | m ) = cn ( a m^1/2 | 1/m )

-If m = 1:      sn ( a | m ) = tanh a ; cn ( a | m ) = dn ( a | m ) = sech a
 
 

STACK	INPUTS	OUTPUTS
Level 3	/	sn ( a | m )
Level 2	m	cn ( a | m )
Level 1	a	dn ( a | m )

Example:        m = 1/PI   a = 1 + 2 i + 3 j + 4 k

        PI   INV
     [ 1  2  3  4 ]   JEFA  >>>>  [  1.49826225309     0.205407204568    0.308110806852     0.410814419143 ] = sn ( a | m )        in level 3

                                                [ -0.694940079527   0.442849491896    0.664274237842     0.885698983794 ] = cn ( a | m )        in level 2

                                                [ -0.719248898421   0.136199160986    0.204298741478     0.272398321974 ] = dn ( a | m )        in level 1
 

Weierstrass Elliptic Functions
 

-WEFA calculates the Weierstrass Elliptic Function  P(a;g₂,g₃)  by a Laurent series:

      P(a;g₂;g₃) = a ^-2 + c₂.a² + c₃.a⁴ + ...... + c_k.a^2k-2 + ....

  where   c₂ = g₂/20  ;   c₃ = g₃/28  and   c_k =  3 ( c₂. c_k-2 + c₃. c_k-3 + ....... + c_k-2. c₂ ) / (( 2k+1 )( k-3 ))      ( k > 3 )
 
 

STACK	INPUTS	OUTPUTS
Level 3	g2	/
Level 2	g3	/
Level 1	a	P(a;g2;g3)

Example:        a = 0.2 + 0.3 i + 0.4 j + 0.5 k  ;   g₂ = 0.9  ,   g₃ = 1.4

          0.9   ENTER
          1.4   ENTER
    [ 0.2  0.3  0.4  0.5 ]   WEFA   >>>>  [ -1.59163727533   -0.411614082849    -0.548818776066    -0.686023470081 ]
 

   Whence,     P( 0.2 + 0.3 i + 0.4 j + 0.5 k  ;  0.9 ,  1.4 ) = -1.59163727533 - 0.411614082849 i - 0.548818776066 j - 0.686023470081 k
 

Weierstrass Duplication Formula
 

-If the argument a is too "large", the Laurent series don't converge.
-The duplication formula may be used one or several times in this case:

    P(2a) = -2 P(a) + ( 6 P²(a) - g₂/2 )² / ( 4 ( 4 P³(a) - g₂ P(a) - g₃ ) )
 
 

STACK	INPUTS	OUTPUTS
Level 3	g2	g2
Level 2	g3	g3
Level 1	P(a;g2;g3)	P(2a ; g2;g3)

Example:     if you keep the previous result in level 1 and place 0.9  in level 3  and  1.4 in level 2,   WF2A  returns:
 

      P( 0.4 + 0.6 i + 0.8 j + k  ;  0.9 ,  1.4 ) = -0.37089210249 - 0.169841920212 i - 0.22645589363 j - 0.28306986701 k
 

Note:

-In fact, the Laurent series still converge for 2a   and a direct evaluation of P(2a) with "WEFA"  yields:

     P( 0.4 + 0.6 i + 0.8 j + k  ;  0.9 ,  1.4 ) = -0.370892102485 - 0.169841920211 i - 0.226455893616 j - 0.283069867019 k
 

Exponential Integral Ei(a)
 

Formula:        Ei(a)  = C + Ln(a) + Sum_n=1,2,.....  aⁿ/(n.n!)                where  C = 0.577215664901... = Euler's constant.
 
 

STACK	INPUTS	OUTPUTS
Level 1	a	Ei(a)

Example:             a = 1 + 2 i + 3 j + 4 k

           [ 1  2  3  4 ]   EIA  >>>>   [ -0.381065061199   1.05205547601    1.57808321402    2.10411095203 ]
 

Exponential Integral En(a)
 

Formulae:

     E_n(a) = a^n-1 Gam(1-n) - [1/(1-n)]  ₁F₁ ( 1-n , 2-n ; -a )                    if   n # 1 , 2 , 3 , ................     and otherwise:

     E_n(a) = (-a)^n-1 ( -Ln a - gamma + Sum_k=1,...,n-1 1/k ) / (n-1)!  - Sum_k#n-1 (-a)^k / (k-n+1) / k!      where  gamma = Euler's Constant = 0.577215664901...

     and   E₀(a) = (1/a).exp(-a)
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	/
Level 1	a	En(a)

Example1:          Compute:  E_PI( 1 + i/2 + j/3 + k/4 )

           PI
           1   ENTER
           2   INV
           3   INV
           4   INV    4   ->ARRY    ENA  >>>>  [ 0.06644432973   -0.05991613114    -0.03994408743    -0.02995806555 ]
 

Example2:          Compute:  E₂( 1 + i/2 + j/3 + k/4 )

           2   ENTER
           1   ENTER
           2   INV
           3   INV
           4   INV    4   ->ARRY    ENA  >>>>  [ 0.082262051336   -0.087391297264    -0.058260864845    -0.043695648635 ]

Note:

-For large arguments, see the asymptotic expansion.
 

Sine Integral
 

Formula:           Si(a)  = Sum_{m=0,1,2,.....} (-1)^m a^2m+1/((2m+1).(2m+1)!)
 
 

STACK	INPUTS	OUTPUTS
Level 1	a	Si(a)

Example:             a = 1 + 2 i + 3 j + 4 k

           [ 1  2  3  4 ]   SIA  >>>>   [ 17.9895462386    7.07509628771    10.6126444315    14.1501925754 ]
 

Hyperbolic Sine Integral
 

Formula:           Shi(a) = Sum_{m=0,1,2,.....}  a^2m+1/((2m+1).(2m+1)!)
 
 

STACK	INPUTS	OUTPUTS
Level 1	a	Shi(a)

Example:             a = 1 + 2 i + 3 j + 4 k

           [ 1  2  3  4 ]   SHIA  >>>>   [ -0.16077930696    0.52215386342    0.783230795127    1.04430772684 ]
 

Cosine Integral
 

Formula:           Ci(a)  = C + ln(a) + Sum_m=1,2,..... (-1)^m a^2m/(2m.(2m)!)                where  C = 0.577215664901... = Euler's constant.
 
 

STACK	INPUTS	OUTPUTS
Level 1	a	Ci(a)

Example:             a = 1 + 2 i + 3 j + 4 k

           [ 1  2  3  4 ]   CIA  >>>>   [ 19.049991783   -6.09801677188    -9.1470251578    -12.1960335438 ]
 

Hyperbolic Cosine Integral
 

Formula:           Chi(a)= C + ln(a) + Sum_m=1,2,.....  a^2m/(2m.(2m)!)                        where  C = 0.577215664901... = Euler's constant.
 
 

STACK	INPUTS	OUTPUTS
Level 1	a	Chi(a)

Example:             a = 1 + 2 i + 3 j + 4 k

           [ 1  2  3  4 ]   CHIA  >>>>   [ -0.220285754239    0.529901612604    0.794852418908    1.05980322521 ]
 

Angular Spheroidal Wave Function of the 1st kind
 

-"SMNA" 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  | a | <= 1  and  S_mn(a) is computed by      S_mn(a) = ( 1 - a² ) ^m/2   Sum_k=0,1,.... d_k a^k

     with     (k+1)(k+2) d_k+2 - [ k ( k + 2m + 1 ) - l + m ( m + 1 ) ] d_k  - c²  d_k-2 = 0
 

 Flammer's Scheme:           the coefficients are normalized as follows:

      d₀ = P_n^m(0)  =  2^m sqrt(PI) / [ Gam((1-m-n)/2) Gam((2-m+n)/2 ]
      d₁ = P'_n^m(0) = ( m + n ) 2^m sqrt(PI) / [ Gam((2-m-n)/2) Gam((1-m+n)/2 ]
 
 

STACK	INPUTS	OUTPUTS
Level 1	a	Smn(a)

Example:        m = 0.2  ,  n = 0.6 ,  c² = 1.7 ,  l = 2.24686665135     a = 0.1 + 0.2 i + 0.3 j + 0.4 k

     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 )

    [ 0.1  0.2  0.3  0.4 ]   SMNA  >>>>  [ 0.391049683319    0.161100167928    0.241650251888    0.322200335854 ]
 

-Whence     S( 0.1 + 0.2 i + 0.3 j + 0.4 k ) = 0.391049683319 + 0.161100167928 i + 0.241650251888 j + 0.322200335854 k

Note:

-l may be choosen arbitrarily, but the solution will not be regular for  a = 1
 

Calculating the eigenvalues
 

-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(L_mn) + U₂(L_mn) = 0   where    U₁ & U₂  are 2 continued fractions:

                                                 -b_r^m                      -b_r-2^m
   U₁(L_mn) = g_r^m - L_mn +    ---------------       ---------------   ..............
                                           g_r-2^m - L_mn +         g_r-4^m - L_mn +

                                                                                                                       r = n - m
                                     -b_r+2^m                    -b_r+4^m
   U₂(L_mn) =           ---------------       ----------------   ..............
                               g_r+2^m - L_mn +         g_r+4^m - L_mn +

   with      b_r^m  =  [ r.(r-1).(2m+r).(2m+r-1).c⁴ ] / [ (2m+2r-1)².(2m+2r+1).(2m+2r-3) ]
   and      g_r^m  =  (m+r).(m+r+1) + (c²/2).[ 1 - (4m²-1)/((2m+2r-1).(2m+2r+3)) ]
 
 

STACK	INPUTS	OUTPUTS
Level 1	a	Lmn

Example:        m = 0.2  ,  n = 0.6 ,  c² = 1.7

-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 = 2.24686665135

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...
 

Legendre Polynomials
 

Formulae:      n.P_n(a) = (2n-1).a.P_n-1(a) - (n-1).P_n-2(a)  ,  P₀(a) = 1  ,  P₁(a) = a
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	/
Level 1	a	Pn(a)

   Where  n is a non-negative integer

Example:    n = 7     a = 1 + i/2 + j/3 + k/4

      7   ENTER
      1   ENTER
      2   INV
      3   INV
      4   INV
      4   ->ARRY   LEGA  >>>>  [ 36.2081958787   -51.5833728619    -34.3889152414   -25.7916864311 ]

 Whence        P₇( 1 + i/2 + j/3 + k/4 ) =  36.2081958787 - 51.5833728619 i - 34.3889152414 j - 25.7916864311 k
 

Generalized Laguerre's Polynomials
 

Formulae:      L₀^(a) (A) = 1  ,   L₁^(a) (A) = a+1-A   ,    n L_n^(a) (A) = (2.n+a-1-A) L_n-1^(a) (A) - (n+a-1) L_n-2^(a) (A)
 
 

STACK	INPUTS	OUTPUTS
Level 3	a	/
Level 2	n	/
Level 1	A	Ln(a) (A)

   Where  n is a non-negative integer

Example:         a = sqrt(2)     a = 7     A = 1 + 2 i + 3 j + 4 k

         2             SQRT
         7             ENTER
 [ 1  2  3  4 ]    LANA   >>>>  [ 872.266128619    47.1252744791    70.6879117151     94.2505489557 ]
 

Chebyshev Polynomials - 1st kind
 

Formula:            T_n(a) = 2a.T_n-1(a) - T_n-2(a)   ;  T₀(a) = 1  ;   T₁(a) = a
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	/
Level 1	a	Tn(a)

   Where  n is a non-negative integer

Example:    n = 7     a = 1 + 2 i + 3 j + 4 k

          7           ENTER
 [ 1  2  3  4 ]   CHB1A   >>>>  [ -9524759   -1117678   -1676517   -2235356 ]
 

Chebyshev Polynomials - 2nd kind
 

Formula:     U_n(a) = 2a.U_n-1(a) - U_n-2(a)  ;  U₀(a) = 1  ;  U₁(a) = 2a
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	/
Level 1	a	Un(a)

   Where  n is a non-negative integer

Example:    n = 7     a = 1 + 2 i + 3 j + 4 k

         7            ENTER
 [ 1  2  3  4 ]   CHB2A   >>>>  [ -18921448   -2198096    -3297144    -4396192 ]
 

Jacobi Polynomials
 

Formula:      P₀^(a;b) (A) = 1  ;   P₁^(a;b) (A) = (a-b)/2 + A (a+b+2)/2

        2n(n+a+b)(2n+a+b-2) P_n^(a;b) (A) = [ (2n+a+b-1).(a²-b²) + A (2n+a+b-2)(2n+a+b-1)(2n+a+b) ] P_b-1^(a;b) (A)
                                                              - 2(n+a-1)(n+b-1)(2n+a+b) P_n-2^(a;b) (A)
 
 

STACK	INPUTS	OUTPUTS
Level 4	a	/
Level 3	b	/
Level 2	n	/
Level 1	A	Pn(a;b) (A)

   Where  n is a non-negative integer

Example:         a = sqrt(2)     b = sqrt(3)    n = 7      A = 1 + i/2 + j/3 + k/4

      2   SQRT
      3   SQRT
      7   ENTER
      1   ENTER
      2   INV
      3   INV
      4   INV
      4   ->ARRY   JCPA  >>>>  [ 143.530451353   -310.868160336   -207.245440222   -155.434080168 ]
 

UltraSpherical Polynomials
 

Formulae:      C₀^(a) (A) = 1  ;   C₁^(a) (A) = 2.a.A   ;    (n+1).C_n+1^(a) (A) = 2.(n+a).A.C_n^(a) (A) - (n+2a-1).C_n-1^(a) (A)      if  a # 0

                      C_n⁽⁰⁾ (A) = (2/n).T_n(A)
 
 

STACK	INPUTS	OUTPUTS
Level 3	a	/
Level 2	n	/
Level 1	A	Cn(a) (A)

   Where  n is a non-negative integer

Example:         a = sqrt(2)       n = 7      A = 1 + i/2 + j/3 + k/4

      2   SQRT
      7   ENTER
      1   ENTER
      2   INV
      3   INV
      4   INV
      4   ->ARRY   USPA  >>>>  [ 324.544396559   -689.588362439    -459.725574956    -344.794181217 ]

-Likewise,   C₇⁽⁰⁾ ( 1 + i/2 + j/3 + k/4 ) =  29.2455479617 - 33.2791035909 i - 22.1860690606 j - 16.6395517954 k
 

Associated Legendre Functions - 1st kind - Type2
 

-Though this is not always a polynomial, PMN2A is included here to remind that the indexes must be non-negative integers !
 

Formula:      (n-m) P_n^m(a) = a (2n-1) P_n-1^m(a) - (n+m-1) P_n-2^m(a)

     Type 2                P_m^m(a) = (-1)^m (2m-1)!!   (1+a)^m/2/(1-a)^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	A	Pnm (A)

   Where m & n  are non-negative integers  with  m <= n

Example:      m = 3     n = 7      A = 1 + i/2 + j/3 + k/4

      3   ENTER
      7   ENTER
      1   ENTER
      2   INV
      3   INV
      4   INV
      4   ->ARRY   PMN2A  >>>>  [ -12188.5394   -8670.12758288    -5780.08505523    -4335.06379145 ]

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) P_n^m(a) = a (2n-1) P_n-1^m(a) - (n+m-1) P_n-2^m(a)

    Type 3                P_m^m(a) =   (2m-1)!!  (a+1)^m/2/(a-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	A	Pnm (A)

   Where m & n  are non-negative integers  with  m <= n

Example:      m = 3     n = 7      A = 1 + i/2 + j/3 + k/4

      3   ENTER
      7   ENTER
      1   ENTER
      2   INV
      3   INV
      4   INV
      4   ->ARRY   PMN3A  >>>>  [ 11285.9768571   -9363.4953345    -6242.33022303    -4681.74766728 ]

Note:

-See also  ALF13
 

Airy Functions
 

Formulae:

    Ai(a) =   f(a) - g(a)                             with            f(a) = [ 3 ^-2/3 / Gamma(2/3) ]  ₀F₁( 2/3 ; a³/9 )
    Bi(a) = [ f(a) + g(a) ] sqrt(3)               and            g(a) = [ 3 ^-1/3 / Gamma(1/3) ]  ₀F₁( 4/3 ; a³/9 )   a
 
 

STACK	INPUTS	OUTPUTS
Level 2	/	Ai (a)
Level 1	a	Bi (a)

Example:            a = 1 + i/2 + j/3 + k/4

      1   ENTER
      2   INV
      3   INV
      4   INV
      4   ->ARRY   AIBIA  >>>>  [ 0.105299445258   -0.0784420741534    -0.0522947161026    -0.0392210370767 ]  = Ai(a)  in Level 2
                                                    [ 0.97346397687      0.394832414612        0.263221609742       0.197416207307   ]  = Bi(a) in Level 1
 

Thus,    Ai ( 1 + i/2 + j/3 + k/4 ) = 0.105299445258 - 0.0784420741534 i - 0.0522947161026 j - 0.0392210370767 k

 and     Bi ( 1 + i/2 + j/3 + k/4 ) = 0.97346397687 + 0.394832414612 i + 0.263221609742 j + 0.197416207307 k
 

Anger & Weber Functions
 

Formulae:
 

   J_n(a) = + (a/2) sin ( 90°n ) ₁F^~₂( 1 ; (3-n)/2 , (3+n)/2 ; -a²/4 )           Anger's functions

                  + cos ( 90°n ) ₁F^~₂( 1 ; (2-n)/2 , (2+n)/2 ; -a²/4 )

   E_n(a) = - (a/2) cos ( 90°n )  ₁F^~₂( 1 ; (3-n)/2 , (3+n)/2 ; -a²/4 )         Weber's functions

                    + sin ( 90°n ) ₁F^~₂( 1 ; (2-n)/2 , (2+n)/2 ; -a²/4 )
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	Jn(a)
Level 1	a	En(a)

Example:            n = PI     a = 1 + 2 i + 3 j + 4 k        ( quaternion ->  4  STO 00 )

       PI
 [ 1  2  3  4 ]  ANWEB  >>>>   [ -11.2423489533   -3.56496842306    -5.34745263462    -7.12993684617 ]    =   J_n(a)  in level 2
                                                  [ -9.56681734549    4.17720489889     6.26580734852     8.35440979805 ]  =  E_n(a)  in level 1
 

-So,       J_Pi( 1 + 2 i + 3 j + 4 k ) = -11.2423489533 - 3.56496842306 i - 5.34745263462 j - 7.12993684617 k

 and      E_Pi( 1 + 2 i + 3 j + 4 k ) = -9.56681734549 + 4.17720489889 i + 6.26580734852 j + 8.35440979805 k
 

Catalan Numbers
 

-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 anion a
 
 

STACK	INPUTS	OUTPUTS
Level 1	a	C (a)

Example:                  a = 1 + i/2 + j/3 + k/4

      1   ENTER
      2   INV
      3   INV
      4   INV
      4   ->ARRY   CATA  >>>>  [ 0.844036226146    0.2392994035    0.159532935665    0.119649701749 ]

-Thus,     C ( 1 + i/2 + j/3 + k/4 ) =  0.844036226146 + 0.2392994035 i + 0.159532935665 j + 0.119649701749 k
 

Chebyshev Functions - 1st kind
 

Formula:           T_m(cos A) = cos m.A                           with       cos A = a
 
 

STACK	INPUTS	OUTPUTS
Level 2	m	/
Level 1	a	Tm (a)

Example:            m = PI                a = 0.1 + 0.2 i + 0.3 j + 0.4 k

               PI
  [ 0.1  0.2  0.3  0.4 ]   CHB1A   >>>>  [ -0.143159422166   -0.90509061839    -1.35763592759    -1.81018123679 ]

Note:

-See also Chebyshev polynomials 1st kind
 

Chebyshev Functions - 2nd kind
 

Formula:            U_m(cos A) = [ sin (m+1).A ] / sin A                            with       cos A = a
m.A                           with       cos A = a
 
 

STACK	INPUTS	OUTPUTS
Level 2	m	/
Level 1	a	Um (a)

Example:            m = PI                a = 0.1 + 0.2 i + 0.3 j + 0.4 k

               PI
  [ 0.1  0.2  0.3  0.4 ]   CHB2A   >>>>  [  -0.38619952444   -1.36969594746    -2.05454392121    -2.73939189494 ]

Note:

-See also Chebyshev polynomials 2nd kind
 

Parabolic Cylinder Functions
 

Formula:         D_n(a) = 2^n/2 Pi^1/2 exp(-a²/4) [ 1/Gam((1-n)/2)  ₁F₁( -n/2 , 1/2 , a²/2 ) - 2^1/2 ( a / Gam(-n/2) ) M [ (1-n)/2 , 3/2 , a²/2 ]
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	/
Level 1	a	Dn(a)

Example:    n = 0.4      a = 1 + 1.1 i + 1.2 j + 1.3 k

          0.4                ENTER
[ 1  1.1  1.2  1.3 ]     DNA    >>>>  [ 2.60610510664   -0.969116886103    -1.0572184212    -1.1453199563 ]
 

Note:

-For large arguments, an asymptotic expansion is preferable.
 

Error Function
 

Formula:     erf a  = (2/pi^1/2)  SUM_{n=0,1,2,.....}  (-1)ⁿ a²ⁿ⁺¹ / (n! (2n+1))
 
 

STACK	INPUTS	OUTPUTS
Level 1	a	erf (a)

Example:         a = 1 + 1.1 i + 1.2 j + 1.3 k

    [ 1  1.1  1.2  1.3 ]   ERFA   >>>>  [ -2.35599141278   -3.35826113621   -3.66355760278    -3.96885407044 ]
 

Whence,   Erf ( 1 + 1.1 i + 1.2 j + 1.3 k ) =  -2.35599141278 - 3.35826113621i - 3.66355760278 j - 3.96885407044 k
 

Generalized Error Function
 

GERFA calculates  erf_m (a) = a  exp ( -a^m )  M ( 1 ; 1+1/m ; a^m )          where   M = Kummer's function
 
 

STACK	INPUTS	OUTPUTS
Level 2	m	/
Level 1	a	erfm (a)

Example:      m = sqrt(2)    a = 1.1 + 1.2 i + 1.3 j + 1.4 k

                2                 SQRT
  [ 1.1  1.2  1.3  1.4 ]  GERFA  returns  [ 1.205349651  -0.208378333   -0.225743194   -0.243108055 ]         ( rounded to 9D )

 Whence    erf_m(a) = 1.205349651 - 0.208378333 i - 0.225743194 j - 0.243108055 k
 

Hermite Functions
 

Formula:       H_m(a) = 2^m sqrt(PI) [ (1/Gam((1-m)/2))  M(-m/2,1/2,a²) - ( 2.a / Gam(-m/2) ) M((1-m)/2,3/2,a²) ]

     where  Gam = Gamma function
       and      M  = Kummer's function = ₁F₁
 
 

STACK	INPUTS	OUTPUTS
Level 2	m	/
Level 1	a	Hm(a)

Example:        m = PI                   a = 1 + i/2 + j/3 + k/4

       PI
        1    ENTER
        2    INV
        3    INV
        4    INV
        4    ->ARRY   HMTA   >>>>    [ -17.8176048984    2.84549614163    1.89699742776    1.4227480708 ]
 

Incomplete Beta Functions
 

Formula:    B_a(p,q) = ( a^p / p )  F(p,1-q;p+1;a)   where  F = the hypergeometric function
 
 
 

STACK	INPUTS	OUTPUTS
Level 3	p	/
Level 2	q	/
Level 1	a	Ba(p,q)

Example:         Compute  B_{0.1+0.2i+0.3j+0.4k} ( 0.7 , 1.8 )

                0.7                  ENTER
                1.8                  ENTER
   [ 0.1  0.2  0.3  0.4 ]      IBFA        [ 0.653131938  0.244542459   0.366813688   0.489084918 ]

-Whence,    B_{0.1+0.2i+0.3j+0.4k} ( 0.7 , 1.8 ) = 0.653131938 + 0.244542459 i + 0.366813688 j + 0.489084918 k         ( rounded to 9D )

Note:

-In general, the hypergeometric series converge if  | a | < 1
 

Incomplete Gamma Functions
 

Formula:      g(m,a) = ( a^m / m ) exp(-a)  M(1,m+1;a)   where  M = Kummer's function
 
 

STACK	INPUTS	OUTPUTS
Level 2	m	/
Level 1	a	g(m,a)

Example:        Compute  g( 1.6 , 1 + 2 i + 3 j + 4 k )
 

                1.6            ENTER
         [ 1  2  3  4 ]      IGFA        [ 0.955358738  -0.390239935  -0.585359903  -0.780479870 ]           ( rounded to 9D )

-So,     g( 1.6 , 1 + 2 i + 3 j + 4 k ) = 0.955358738 - 0.390239935 i - 0.585359903 j - 0.780479870 k
 
 

Jacobi Functions
 

Formula:           P_m^(a;b) (a) = [ Gam(a+m+1) / Gam(m+1) ]  ₂F^~₁ ( -m , a+b+m+1 , a+1 , (1-a)/2 )

                            where  ₂F^~₁  is the regularized hypergeometric function
 
 

STACK	INPUTS	OUTPUTS
Level 4	a	/
Level 3	b	/
Level 2	m	/
Level 1	A	Pm(a;b) (A)

Example:        a = sqrt(2)    b = sqrt(3)    m = PI                 a = 1 + i / 2 + j / 3 + k / 4

        2   SQRT
        3   SQRT
        PI
        1    ENTER
        2    INV
        3    INV
        4    INV
        4    ->ARRY   JCFA   >>>>   [ -10.1439752498   11.7413999755    7.82759998364    5.87069998775 ]
 

-Likewise,   P_PI^(-4;+4) ( 1 + i/2 + j/3 + k/4 ) =  0.360213557888 - 0.125970280632 i - 0.0839801870903 j - 0.0629851403167 k
 

Laguerre's Functions
 

Formula:      L_n^(a)(A) = [ Gam(a+n+1) / Gam(n+1) ]  ₁F^~₁ ( -n , a+1 , A )
 
 

STACK	INPUTS	OUTPUTS
Level 3	a	/
Level 2	n	/
Level 1	A	Ln(a)(A)

Example:           a = sqrt(3)     n = PI     A = 1 + 2 i + 3 j + 4 k

       3   SQRT
      PI
 [ 1  2  3  4 ]   LANA  >>>>   [ -59.3766393554   3.13535567816    4.70303351732    6.27071135632 ]

Note:

-See also LANA
 

Lommel Functions - 1st kind
 

Formula:                  s⁽¹⁾_m,p(a)  = a^m+1 / [ (m+1)² - p² ]  ₁F₂ ( 1 ; (m-p+3)/2 , (m+p+3)/2 ; -a²/4 )
 
 

STACK	INPUTS	OUTPUTS
Level 3	m	/
Level 2	p	/
Level 1	a	s(1)m,p(a)

Example:         m = sqrt(2)    ,    p = sqrt(3)    ,         a = 1 + 1.1 i + 1.2 j + 1.3 k

            2   SQRT
            3   SQRT
   [ 1  1.1  1.2  1.3 ]  LOM1A  >>>>  [ -2.55566311832    1.08164201813    1.17997311069    1.27830420324 ]
 

Lommel Functions - 2nd kind
 

Formula:

    s⁽²⁾_m,p(a)  = a^m+1 / [ (m+1)² - p² ]  ₁F₂ ( 1 ; (m-p+3)/2 , (m+p+3)/2 ; a²/4 )
                    + 2^m+p-1 Gam(p) Gam((m+p+1)/2) a ^-p / Gam((-m+p+1)/2) ₀F₁ ( ; 1-p ; -a²/4 )
                    + 2^m-p-1 Gam(-p) Gam((m-p+1)/2) a^p / Gam((-m-p+1)/2) ₀F₁ ( ; 1+p ; -a²/4 )

   where  _pF_q is the generalized hypergeometric function and Gam is the Euler Gamma function.
 
 

STACK	INPUTS	OUTPUTS
Level 3	m	/
Level 2	p	/
Level 1	a	s(2)m,p(a)

Example:         m = sqrt(2)    ,    p = sqrt(3)    ,         a = 1 + 1.1 i + 1.2 j + 1.3 k

            2   SQRT
            3   SQRT
   [ 1  1.1  1.2  1.3 ]  LOM2A  >>>>  [ 1.46174379927   -0.935738233153    -1.02080534526    -1.10587245735 ]
 

Regular Coulomb Wave Functions
 

Formulae:         F_L(h,a) = C_L(h) a ^L+1 Sum _k>L A_k^L (h) a ^k-L-1

    with   C_L(h) = (1/Gam(2L+2)) 2^L e ^-pi.h/2 | Gam(L+1+i.h) |
    and    A_L+1^L = 1 ;  A_L+2^L = h/(L+1)  ;  (k+L)(k-L-1) A_k^L = 2.h A_k-1^L - A_k-2^L   ( k > L+2 )
 

-"RCWFA" also uses   | Gam( 1+i y ) |² = (Pi.y) / Sinh (Pi y)
 
 

STACK	INPUTS	OUTPUTS
Level 3	L	/
Level 2	h	/
Level 1	a	FL(h,a)

  Where  L  is a non-negative integer.

Example:            L = 2  ,   h = 0.75  ;   a = 1 + 1.2 i + 1.4 j + 1.6 k

        2    ENTER
     0.75  ENTER
 [ 1  1.2  1.4  1.6 ]   RCWFA   >>>>  [ -0.478767860363    -0.179694771644    -0.209643900251    -0.239593028859 ]
 

-So,   F₂( 0.75 ;  1 + 1.2 i + 1.4 j + 1.6 k ) = -0.478767860363 - 0.179694771644 i - 0.209643900251 j - 0.239593028859 k
 

Struve Function H
 

Formula:           H_m(a) = (a/2)^m+1 ₁F^~₂( 1 ; 3/2 , m + 3/2 ; - a²/4 )
 
 

STACK	INPUTS	OUTPUTS
Level 2	m	/
Level 1	a	Hm(a)

Example:              m = PI     a = 1 + 2 i + 3 j + 4 k

        PI
[ 1  2  3  4 ]    STRHA   >>>>  [ 8.54663322308   -4.08507993257    -6.12761989891    -8.17015986517 ]
 

Struve Function L
 

Formula:               L_m(a) = (a/2)^m+1 ₁F^~₂( 1 ; 3/2 , m + 3/2 ;  a²/4 )
 
 

STACK	INPUTS	OUTPUTS
Level 2	m	/
Level 1	a	Lm(a)

Example:              m = PI     a = 1 + 2 i + 3 j + 4 k

        PI
[ 1  2  3  4 ]    STRLA   >>>>  [ 1.86458278424   -0.116220198782    -0.174330298178   -0.232440397571 ]
 

Toronto Functions
 

Formula:              T(m,n;a) = exp(-a²) [ Gam((m+1)/2) ] a^2n-m+1₁F^~₁( (m+1)/2 , n+1 , a² )
 
 

STACK	INPUTS	OUTPUTS
Level 3	m	/
Level 2	n	/
Level 1	a	T(m,n;a)

Example:        m = sqrt(2)    n = sqrt(3)                  a = 1 + i / 2 + j / 3 + k / 4

       2   SQRT
       3   SQRT
       1    ENTER
       2    INV
       3    INV
       4    INV
       4    ->ARRY   TORA   >>>>  [ 0.321293450074    0.465223021825    0.310148681216    0.232611510913 ]
 

UltraSpherical Functions
 

Formula:    Assuming  l # 0

          C_m^(l)(a) = [ Gam(m+2l) / Gam(m+1) / Gam(2.l) ]  ₂F₁( -m , m+2l , m+1/2 , (1-a)/2 )

                     where  ₂F₁ is the hypergeometric function and Gam = Gamma function
 
 

STACK	INPUTS	OUTPUTS
Level 3	l	/
Level 2	m	/
Level 1	a	Cm(l)(a)

Example:         l = sqrt(2)    m = sqrt(3)    a = 1 + i / 2 + j / 3 + k / 4

       2   SQRT
       3   SQRT
       1    ENTER
       2    INV
       3    INV
       4    INV
       4    ->ARRY    USFA   >>>>  [ 3.24111551742    4.84827789839    3.23218526561    2.4241389492 ]

Note:

-See also  USPA
 

Whittaker's M-Function
 

Formula:

    M_q,p(a) = exp(-a/2)  a^p+1/2  M( p-q+1/2 , 1+2p , a )               where  M = Kummer's functions
 
 

STACK	INPUTS	OUTPUTS
Level 3	q	/
Level 2	p	/
Level 1	a	Mq,p(a)

Example:            q = sqrt(2)   p = sqrt(3)    a = 0.4 + 0.5 i + 0.6 j + 0.7 k
 

                 2                         SQRT
                 3                         SQRT
  [ 0.4  0.5  0.6  0.7 ]           WHIMA     gives   [ -0.807065422   0.373202939   0.447843527   0.522484114 ]          ( rounded to 9D )
 

Whittaker's W-Function
 

Formula:

  W_q,p(a) = [ Gam(2p) / Gam(p-q+1/2) ]  M_q,-p(a) + [ Gam(-2p) / Gam(-p-q+1/2) ]  M_q,p(a)   assuming  2p is not an integer.
 
 

STACK	INPUTS	OUTPUTS
Level 3	q	/
Level 2	p	/
Level 1	a	Wq,p(a)

Example:            q = sqrt(2)   p = sqrt(3)    a = 0.4 + 0.5 i + 0.6 j + 0.7 k
 

                 2                         SQRT
                 3                         SQRT
  [ 0.4  0.5  0.6  0.7 ]           WHIWA     [ 2.275449625  -1.129250837  -1.355101005  -1.580951172 ]              ( rounded to 9D )

Whence,        W_q,p(a)  = 2.275449625 - 1.129250837 i - 1.355101005 j - 1.580951172 k

Note:

-"WHIWA" does not work if 2.p is an integer.
 

Parabolic Cylinder Functions - Asymptotic Expansion
 

-For large arguments, ascending series give poor accuracy or even meaningless results !
-Asymptotic expansions are preferable:
 

Formula:    D_n(a)  ~  aⁿ exp(-a²/4) [ 1 - n(n-1) / ( 2 a² ) + n(n-1)(n-2)(n-3) / ( 2 ( 4 a⁴ ) ) - ....... ]
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	/
Level 1	a	Dn(a)

Example:          n  = PI       a = 1 + 2 i + 3 j + 4 k

       PI
[ 1  2  3  4 ]   DNA  >>>>     [ -33138.8727246    93318.1512975   139977.226946    186636.302594 ]
 

Notes:

-An infinte loop will occur if a is too small.
-However, this program may also be used if a is relatively small when n is a positive integer.
 

Exponential Integral En(a) - Asymptotic Expansion
 

Formula:     E_n(a) ~  (1/a) exp(-a)  ₂F₀(1,n;;-1/a)
 
 

STACK	INPUTS	OUTPUTS
Level 2	n	/
Level 1	a	En(a)

Example:          n = PI   ,    a = 41 + 42 i + 43 j + 44 k

             PI
 [ 41  42  43  44 ]    ENA   >>>>   [ 1.78948114456 10 ^-20   -1.31507219874 10 ^-21    -1.34638344156 10 ^-21    -1.37769468439 10 ^-21 ]

Notes:

-If the argument a is too "small" , the series will diverge too soon.
-However, it may also work with small arguments if  n is a negative integer.
-For instance, it returns correctly:

         E _-2( 1 + 2 i + 3 j + 4 k ) = 0.047465815362  - 0.0202055335051 i - 0.0303083002577 j - 0.0404110670101 k

Remarks:

-These programs only deal with a few special functions.
-They might be completed by many others.
-Since most of the special functions may be expressed in terms of hypergeometric functions,
  the routine  HGFA  is quite useful !
 
 

References:

[1]   Abramowitz and Stegun - "Handbook of Mathematical Functions" -  Dover Publications -  ISBN  0-486-61272-4
[2]   http://functions.wolfram.com/
[3]   Warren D. Smith - Quaternions, Octonions, 16-ons & 2^n-ons, new kinds of numbers