Fractional Integro-Differentiation for the HP-41

Overview
 

 1°)  A few Elementary Functions
 2°)  A few Special Functions
 

-If a function f is defined by a power series:  f(x) = SUM_{k=0,1,2,.....}   c_k x^k  ,  its fractional integro-differentiation may be computed by

   d^µ f / dx^µ = D^µ f(x) = SUM_{k=0,1,2,.....}   c_k [ Gam(k+1) / Gam(k+1-µ) ] x^k-µ    where  µ is a real number ( integer or fractional )

-If the function may be expressed in terms of hypergeometric functions _pF_q , the following relation is very useful too:

   D^µ _pF_q ( a₁ , .......... , a_p ; b₁ , .......... b_q ; x ) = x ^-µ Gam(b₁).........Gam(b_q)   _p+1F^~_q+1 ( 1 , a₁ , .......... , a_p ; 1-µ , b₁ , .......... b_q ; x )

   where   Gam = Euler's Gamma function  and _pF^~_q  is the regularized generalized hypergeometric function  ( cf "Hypergeometric Functions for the HP-41" )

-We have   D⁰ f(x) = f(x)
-If  µ = 1 , 2 , 3 , ...  we get the 1st , 2nd , 3rd , ...  derivatives
 and if  µ = -1 , -2 , -3 , ....  the results are the repeated integrals of the function f  - usually those that vanish for x = 0.
 

1°)  A few Elementary Functions
 

Formulae:
 

  •  Hyperbolic Sine                   D^µ Sinh x = 2^µ-1 sqrt(PI) x^1-µ₁F^~₂ ( 1 ; (2-µ)/2 , (3-µ)/2 ; x²/4 )

  •  Hyperbolic Cosine               D^µ Cosh x = (2/x)^µ sqrt(PI)  ₁F^~₂ ( 1 ; (1-µ)/2 , (2-µ)/2 ; x²/4 )

  •  Sine                     D^µ Sin x = 2^µ-1 sqrt(PI) x^1-µ₁F^~₂ ( 1 ; (2-µ)/2 , (3-µ)/2 ; -x²/4 )

  •  Cosine                 D^µ Cos x = (2/x)^µ sqrt(PI)  ₁F^~₂ ( 1 ; (1-µ)/2 , (2-µ)/2 ; -x²/4 )

  •  Logarithm            D^µ Ln x = x ^-µ FC^(µ)_log (x)

        where     FC^(µ)_log (x) = (-1)^µ-1 (µ-1) !                                               if  µ is a positive integer
          and       FC^(µ)_log (x) = [ Ln x - Psi(1-µ) - gamma ] / Gam(1-µ)        otherwise

      Psi = Digamma Function , gamma = Euler's constant = 0.5772156649...  and  Gam = Gamma Function.

  •  Exponential          D^µ Exp x = x ^-µ ₁F^~₁ ( 1 ; 1-µ ; x )
 

Data Registers:  R00 to R04: temp
Flags:  F09 F10
Subroutines:  "1/G+"  ( or "GAM+" ... )  "PSI"  ( cf "Gamma Function for the HP-41" )

-The M-Code function  HGF+  may be replaced by  XEQ "HGF+"  ( cf "Hypergeometric Functions" )
  but in this case, register R00 must be replaced by another - unused - data register.
 
 

 
    ( 197 bytes / SIZE 005 )
 
 

 
Examples:

  •  Hyperbolic Sine

     3.14   ENTER^
     1.28   XEQ "DSH"  >>>>   D^3.14 Sinh ( 1.28 ) = 1.999005451

  •  Hyperbolic Cosine

     3.14   ENTER^
     1.28   XEQ "DCH"  >>>>  D^3.14 Cosh ( 1.28 ) = 1.502958219

  •  Sine

     3.14   ENTER^
     1.28   XEQ "DSIN"  >>>>   D^3.14 Sin ( 1.28 ) = -0.019142092

  •  Cosine

     3.14   ENTER^
     1.28   XEQ "DCOS"  >>>>   D^3.14 Cos ( 1.28 ) = 0.888787267

  •  Logarithm

     3.14   ENTER^
     1.28   XEQ "DLN"  >>>>   D^3.14 Ln ( 1.28 ) = 1.138569850

  •  Exponential

     3.14   ENTER^
     1.28   XEQ "DEXP"  >>>>   D^3.14 Exp ( 1.28 ) = 3.501963669
 

2°)  A few Special Functions
 

Formulae:
 

  •  Sine Integral             D^µ Si x = 2^µ-2  PI  x^1-µ₂F^~₃ ( 1/2 , 1 ; 3/2 , (2-µ)/2 , (3-µ)/2 ; -x²/4 )
 

  •  Hyperbolic Sine Integral         D^µ Shi x = 2^µ-2  PI  x^1-µ₂F^~₃ ( 1/2 , 1 ; 3/2 , (2-µ)/2 , (3-µ)/2 ; x²/4 )
 

  •  Cosine Integral           D^µ Ci x = [ FC^(µ)_log (x) + gamma / Gam(1-µ) ] x ^-µ - 2^µ-3  sqrt(PI)  x^2-µ  ₂F^~₃ ( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; -x²/4 )
 

  •  Hyperbolic Cosine Integral     D^µ Chi x = [ FC^(µ)_log (x) + gamma / Gam(1-µ) ] x ^-µ + 2^µ-3  sqrt(PI)  x^2-µ  ₂F^~₃ ( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; x²/4 )
 

  •  Exponential Integral     D^µ Ei x = [ FC^(µ)_log (x) + gamma / Gam(1-µ) ] x ^-µ + x^1-µ  ₂F^~₂ ( 1 , 1 ; 2 , 2-µ ; x )
 

  •  Fresnel Cosine Integral     D^µ C(x) = 2^2µ-3/2 PI^3/2 x^1-µ₃F^~₄ [ 1/4 , 3/4 , 1 ; (2-µ)/4 , (3-µ)/4 , (4-µ)/4 , (5-µ)/4 ; -(PI)² x⁴/16 ]
 

  •  Fresnel Sine Integral    D^µ S(x) = 2^2µ-11/2 PI^5/2 x^3-µ₃F^~₄ [ 3/4 , 1 , 5/4 ; (4-µ)/4 , (5-µ)/4 , (6-µ)/4 , (6-µ)/4 ; -(PI)² x⁴/16 ]
 

  •  Spherical Bessel Function - 1st kind      D^µ j_n (x) = 2^µ-2n-1 PI  x^n-µ  Gam(n+1)  ₂F^~₃ [ (n+1)/2 , (n+2)/2 ; (n+1-µ)/2 , (n+2-µ)/2 , n+3/2 ; -x²/4 ]
 

  •  Modified Bessel Function - 1st kind      D^µ I_n (x) = 2^µ-2n sqrt(PI) x^n-µ  Gam(n+1)  ₂F^~₃ [ (n+1)/2 , (n+2)/2 ; (n+1-µ)/2 , (n+2-µ)/2 , n+1 ; x²/4 ]
 

  •  Bessel Function - 1st kind           D^µ J_n (x) = 2^µ-2n sqrt(PI)  x^n-µ  Gam(n+1)  ₂F^~₃ [ (n+1)/2 , (n+2)/2 ; (n+1-µ)/2 , (n+2-µ)/2 , n+1 ; -x²/4 ]
 

  •  Modified Bessel Function - 2nd kind

               D^µ K_n (x) = 2^µ-2n-1 (PI)^3/2  x^-µ-n csc(n.PI) { 16ⁿ Gam(1-n)  ₂F^~₃ [ (1-n)/2 , (2-n)/2 ; (1-µ-n)/2 , (2-µ-n)/2 , 1-n ; x²/4 ]

                                 - x²ⁿ Gam(n+1)  ₂F^~₃ [ (n+1)/2 , (n+2)/2 ; (n+1-µ)/2 , (n+2-µ)/2 , n+1 ; x²/4 ] }    where n is not an integer.

  •  Bessel Function - 2nd kind

               D^µ Y_n (x) = 2^µ-2n (PI)^1/2  x^-µ-n csc(n.PI) { -16ⁿ Gam(1-n)  ₂F^~₃ [ (1-n)/2 , (2-n)/2 ; (1-µ-n)/2 , (2-µ-n)/2 , 1-n ; -x²/4 ]

                               + x²ⁿ Cos(n.PI) Gam(n+1)  ₂F^~₃ [ (n+1)/2 , (n+2)/2 ; (n+1-µ)/2 , (n+2-µ)/2 , n+1 ; -x²/4 ] }    where n is not an integer.

  •  Generalized Laguerre's Functions   D^µ L^a_n (x) = [ Gam(n+a+1)/Gam(n+1) ] x ^-µ₂F^~₂ ( 1 , -n ; a+1 , 1-µ ; x )
 

  •  Airy Functions

       D^µ Ai(x) = 3^µ-4/3 x ^-µ  { 3^2/3 Gam(1/3) ₂F^~₃ [ 1/3 , 1 ; (1-µ)/3 , (2-µ)/3 , (3-µ)/3 ; x³/9 ] - x Gam(2/3) ₂F^~₃ [ 2/3 , 1 ; (4-µ)/3 , (2-µ)/3 , (3-µ)/3 ; x³/9 ] }

       D^µ Bi(x) = 3^µ-5/6 x ^-µ  { 3^2/3 Gam(1/3) ₂F^~₃ [ 1/3 , 1 ; (1-µ)/3 , (2-µ)/3 , (3-µ)/3 ; x³/9 ] + x Gam(2/3) ₂F^~₃ [ 2/3 , 1 ; (4-µ)/3 , (2-µ)/3 , (3-µ)/3 ; x³/9 ] }

  •  Error Function     D^µ Erf (x) = 2^µ x^1-µ ₂F^~₂ [ 1/2 , 1 ; (2-µ)/2 , (3-µ)/2 ; -x² ]
 

  •  Hermite Function

       D^µ H_n (x) = [ 2^n+µ (PI) x^-µ / Gam((1-n)/2) ] ₂F^~₂ [ 1 , -n/2 ; (1-µ)/2 , (2-µ)/2 ; x² ] - [ 2^n+µ (PI) x^1-µ / Gam((-n)/2) ] ₂F^~₂ [ 1 , (1-n)/2 ; 1-µ/2 , (3-µ)/2 ; x² ]

  •  Kummer's Function          D^µ F(a;b;x) = x ^-µ Gam(b) ₂F^~₂ ( 1 , a ; 1-µ , b ; x )
 
 

Data Registers:  R00 to R09: temp
Flags:  F01
Subroutines:  "1/G+"  ( or "GAM+" ... )  "PSI" = digamma function ( cf "Gamma Function for the HP-41" )

-The M-Code function  HGF+  may be replaced by  XEQ "HGF+"  ( cf "Hypergeometric Functions" )
  but in this case, register R00 must be replaced by another - unused - data register.
-LBL 08 ( lines 644 to 675 ) is a subroutine that is called by  "DCI"  "DCHI"  &  "DEI"
 
 

 
    ( 1008 bytes / SIZE 010 )
 
 

     µ is always to be entered first, then the order/index - if any - and finally,  x in register X

Examples:

  •  Sine Integral

     3.14   ENTER^
     1.28   XEQ "DSI"  >>>>   D^3.14 Si ( 1.28 ) = -0.045395644

  •  Hyperbolic Sine Integral

     3.14   ENTER^
     1.28   XEQ "DSHI"  >>>>   D^3.14 Shi ( 1.28 ) = 0.576495211

  •  Cosine Integral

     3.14   ENTER^
     1.28   XEQ "DCI"  >>>>   D^3.14 Ci ( 1.28 ) = 1.367323895

  •  Hyperbolic Cosine Integral

     3.14   ENTER^
     1.28   XEQ "DCHI"  >>>>   D^3.14 Chi ( 1.28 ) = 1.405640394

  •  Exponential Integral

     3.14   ENTER^
     1.28   XEQ "DEI"  >>>>   D^3.14 Ei ( 1.28 ) = 1.982135606

  •  Fresnel Cosine Integral

     3.14   ENTER^
     1.28   XEQ "DCX"  >>>>   D^3.14 C ( 1.28 ) = 16.95612253

  •  Fresnel Sine Integral

     3.14   ENTER^
     1.28   XEQ "DSX"  >>>>   D^3.14 S ( 1.28 ) = -11.20302776

  •  Spherical Bessel Function - 1st kind

     3.14   ENTER^
     2.41   ENTER^
     1.28   XEQ "DSB1"  >>>>   D^3.14 j_2.41 ( 1.28 ) = -0.064451622

  •  Modified Bessel Function - 1st kind ,  n # -1 , -2 , -3 , .................

     3.14   ENTER^
     2.41   ENTER^
     1.28   XEQ "DINX"  >>>>   D^3.14 I_2.41 ( 1.28 ) = 0.352247279

  •  Bessel Function - 1st kind ,  n # -1 , -2 , -3 , .................

     3.14   ENTER^
     2.41   ENTER^
     1.28   XEQ "DJNX"  >>>>   D^3.14 J_2.41 ( 1.28 ) = -0.150524582

  •  Modified Bessel Function - 2nd kind - non-integer order

     3.14   ENTER^
     2.41   ENTER^
     1.28   XEQ "DKNX"  >>>>   D^3.14 K_2.41 ( 1.28 ) = -38.98469314

  •  Bessel Function - 2nd kind - non-integer order

     3.14   ENTER^
     2.41   ENTER^
     1.28   XEQ "DYNX"  >>>>   D^3.14 Y_2.41 ( 1.28 ) = 25.49308580

  •  Generalized Laguerre's Functions

     3.14   ENTER^
     1.76   ENTER^
     2.41   ENTER^
     1.28   XEQ "DLANX"  >>>>   D^3.14 L^1.76_2.41 ( 1.28 ) = -1.767203465

  •  Airy Functions

     3.14   ENTER^
     1.28   XEQ "DAIRY"  >>>>   D^3.14 Ai ( 1.28 ) = -0.162004857    X<>Y   D^3.14 Bi ( 1.28 ) = 3.432592624

  •  Error Function

     3.14   ENTER^
     1.28   XEQ "DERF"  >>>>   D^3.14 Erf ( 1.28 ) = 1.250557023

  •  Hermite Function

     3.14   ENTER^
     2.41   ENTER^
     1.28   XEQ "DHMT"  >>>>   D^3.14 H_2.41 ( 1.28 ) = 3.537707646

  •  Kummer's Functions      With  a = sqrt(2)  &  b = sqrt(3)

      2   SQRT   STO 01     3   SQRT   STO 02

     3.14   ENTER^
     1.28   XEQ "DKUM"  >>>>   D^3.14 F ( 2^1/2 ; 3^1/2 ; 1.28 ) = 2.075891500

Notes:

-For Kummer's Functions,  R01 & R02 must be initialized first.
-R02 is modified during the calculations, but its original content is restored at the end.

-As usual when a function is evaluated by a power series, the results are not very accurate for large arguments,
-They may even be meaningless ... unless all the terms have the same sign !
 

Reference:

[1]   http://functions.wolfram.com/