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Pipeng Free Online Software : Special Functions Calculators
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Pipeng : Maths Special Functions Calculation Module

Special Functions Calculators

Description : Legendre elliptic integral calculators using Carlsons forms.

Discussion : The Legendre elliptic integrals are calculated using Carlsons symmetric forms:

The Legendre forms can be calculated by

K(k) = RF(0, 1 - k2, 1)
E(k) = RF(0, 1 - k2, 1) - 1/3 k2 RD(0, 1 - k2, 1)
Π(k, n) = RF(0, 1 - k2, 1) + 1/3 n RJ(0, 1 - k2, 1, 1 - n)

where

K = complete elliptical integral of the first kind
E = complete elliptical integral of the second kind
Π = complete elliptical integral of the third kind

References :

Calculator Tools In This Module:

CALC : Math : Elliptic 001 : Complete Elliptic Integral Of The First And Second Kind K And E : Calculator
CALC : Math : Elliptic 002 : Complete Elliptic Integral Of The Third Kind Π : Calculator
CALC : Math : Elliptic 003 : Incomplete Elliptic Integral Of The First And Second Kind F And E : Calculator
CALC : Math : Elliptic 004 : Incomplete Elliptic Integral Of The Third Kind Π : Calculator
CALC : Math : Elliptic 011 : Jacobi Elliptic Functions And Elliptic Amplitude : Calculator
CALC : Math : Function 001 : Gamma Function Γ : Calculator


Link

Module List

CALC : Math : Elliptic 001 : Complete Elliptic Integral Of The First And Second Kind K And E : Calculator

Description : Calculate the complete Legrendre elliptic integrals of the first and second kind (K and E).

Discussion : The function EllipticCK() calculates the complete elliptic integral of the first kind K. The function EllipticCE calculates the complete elliptic integral of the second kind E. The function working is not shown. To check the functions check the following table values for K and E.

  • k = 0.00 : K = 1.570796 : E = 1.570796
  • k = 0.10 : K = 1.574746 : E = 1.566862
  • k = 0.50 : K = 1.685750 : E = 1.467462
  • k = 0.75 : K = 1.910990 : E = 1.318472

Input Variables :

  • k = Elliptical k Modulus

Output Variables :

  • E = Elliptical E Integral
  • K = Elliptical K Integral

Calculation :

K = EllipticCK( k )
E = EllipticCE( k )

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CALC : Math : Elliptic 002 : Complete Elliptic Integral Of The Third Kind Π : Calculator

Description : Calculate the complete Legrendre elliptic integral of the third kind (Π).

Discussion : The function EllipticCP() calculates the complete elliptic integral of the third kind Π. The function working is not shown. To check the function check the following table values for Π.

  • n = 0.7 : k = 0.00 : Π = 2.867869
  • n = 0.7 : k = 0.10 : Π = 2.877191
  • n = 0.7 : k = 0.50 : Π = 3.143395
  • n = 0.7 : k = 0.75 : Π = 3.707671

Input Variables :

  • k = Elliptical k Modulus
  • n = Elliptical n Modulus

Output Variables :

  • π = Elliptical Integral Third Kind

Calculation :

Π = EllipticCP( n , k )

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CALC : Math : Elliptic 003 : Incomplete Elliptic Integral Of The First And Second Kind F And E : Calculator

Description : Calculate the incomplete Legrendre elliptic integrals of the first and second kind (F and E).

Discussion : The function EllipticIF() calculates the incomplete elliptic integral of the first kind F. The function EllipticIE calculates the incomplete elliptic integral of the second kind E. The function working is not shown. To check the functions check the following table values for F and E.

  • φ = 1.119770 : k = 0.00 : F = 1.119770 : E = 1.119770
  • φ = 1.119770 : k = 0.10 : F = 1.121596: E = 1.117949
  • φ = 1.119770 : k = 0.50 : F = 1.170444 : E = 1.072664
  • φ = 1.119770 : k = 0.75 : F = 1.254864 : E = 1.008087

Input Variables :

  • φ = Elliptic Integral Angle
  • k = Elliptical k Modulus

Output Variables :

  • E = Elliptical Integral First Kind
  • K = Elliptical Integral Second Kind

Calculation :

K = EllipticIF( φ , k )
E = EllipticIE( φ , k )

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CALC : Math : Elliptic 004 : Incomplete Elliptic Integral Of The Third Kind Π : Calculator

Description : Calculate the incomplete Legrendre elliptic integral of the third kind (Π).

Discussion : The function EllipticIP() calculates the incomplete elliptic integral of the third kind Π. The function working is not shown. To check the function check the following table values for Π.

  • φ = 1.119770 : n = 0.7 : k = 0.00 : Π = 1.545953
  • φ = 1.119770 : n = 0.7 : k = 0.10 : Π = 1.549011
  • φ = 1.119770 : n = 0.7 : k = 0.50 : Π = 1.631549
  • φ = 1.119770 : n = 0.7 : k = 0.75 : Π = 1.777651

Input Variables :

  • φ = Elliptic Integral Angle
  • k = Elliptical k Modulus
  • n = Elliptical n Modulus

Output Variables :

  • π = Elliptical Integral Third Kind

Calculation :

Π = EllipticIP( φ , n , k )

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CALC : Math : Elliptic 011 : Jacobi Elliptic Functions And Elliptic Amplitude : Calculator

Description : Calculate the Jacobi elliptic functions, sn, cn, dn and am.

Discussion : The function EllipticFunc() calculates the sn, cn, dn and am function values. The function working is not shown. To check the function check the following table values.

  • u = 2 : k = 0.5 : sn = 0.9628982 : cn = -0.2698650 : dn = 0.8764741 : am = 1.844049
  • u = 1 : k = 0.5 : sn = 0.8226356 : cn = 0.5685690 : dn = 0.9114920 : am = 0.9660311
  • u = -1 : k = 0.5 : sn = -0.8226356 : cn = 0.5685690 : dn = 0.9114920 : am = -0.9660311
  • u = 3 : k = 0.5 : sn = 0.3610800 : cn = -0.9325348 : dn = 0.9835676 : am = 2.772167
  • u = 3 : k = 0.7 : sn = 0.6186656 : cn = -0.7856544 : dn = 0.9013622 : am = 2.474549
  • u = 3 : k = 0.9 : sn = 0.9442446 : cn = -0.3292447 : dn = 0.5270728 : am = 1.9063000

The EllipticFunc() function is invalid if the absolute value of k is greater than or equal to one.

Input Variables :

  • k = Elliptical k Modulus
  • u = Elliptical u Modulus

Output Variables :

  • am = Elliptical Function am
  • cn = Elliptical Function cn
  • dn = Elliptical Function dn
  • sn = Elliptical Function sn

Calculation :

list( sn , cn , dn , am ) = EllipticFunc( u , k )

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CALC : Math : Function 001 : Gamma Function Γ : Calculator

Description : Calculate the Gamma function value.

Discussion : The function GammaZ() calculates the Gamma function value Γ. The function working is not shown. To check the function check the following table values for Γ.

  • z = 0.55 : Γ = 1.616124
  • z = 1.55 : Γ = 0.8888683
  • z = 2.55 : Γ = 1.377746
  • z = -0.45 : Γ = -3.591387
  • z = -1.45 : Γ = 2.476819

The GammaZ() function is invalid if z equals zero, or if z equals a negative integer values. The function GammaZ() is not recursive and may give inaccurate results for very large absolute values of z, or for z values very close to zero or a negative integer.

Input Variables :

  • z = Z Input Value

Output Variables :

  • Γ = Gamma Value

Calculation :

Γ = GammaZ( z )

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