OpenMath Content Dictionary: jacobi

Canonical URL:
http://www.openmath.org/CDs/airy.ocd
CD File:
jacobi.ocd
CD as XML Encoded OpenMath:
jacobi.omcd
Defines:
JacobiAM, JacobiCD, JacobiCN, JacobiCS, JacobiDC, JacobiDN, JacobiDS, JacobiNC, JacobiND, JacobiNS, JacobiP, JacobiSC, JacobiSD, JacobiSN, JacobiTheta1, JacobiTheta2, JacobiTheta3, JacobiTheta4, JacobiZeta
Date:
23/8/2001
Version:
(Revision )
Review Date:
Status:
private

1/1/5000

This content dictionary contains symbols to describe Jacobi related functions. They are defined as in Abromowitz and Stegun (ninth printing on).


JacobiP

This symbol represents the quadnary function which computes the orthogonal Jacobi polynomial.

If the first parameter is a non-negative integer, then the function call JacobiP(n,a,b,x) will compute the nth Jacobi polynomial, with parameters a and b, evaluated at x.

These polynomials are orthogonal on the interval [-1,1] with respect to the weight function w(x) = (1-x)^a*(1+x)^b when a and b are greater than -1. Thus

integral(w(t) JacobiP(m, a, b, t) JacobiP(n, a, b, t),-1 less than or equal to t less than or equal to 1) = 0

if m is not equal to n. They are undefined for negative integer values of a or b.

They satisfy the following recurrence relation:

JacobiP(0,a,b,x) = 1, JacobiP(1,a,b,x) = (a/2-b/2) + (1+a/2+b/2)*x, JacobiP(n,a,b,x) = (2*n+a+b-1-x) * (a^2-b^2+(2*n+a+b-2) * (2*n+a+b)*x) / (2*n*(n+a+b)*(2*n+a+b-2)) * JacobiP(n-1,a,b,x) - 2*(n+a-1)*(n+b-1)*(2*n+a+b) / (2*n*(n+a+b)*(2*n+a+b-2)) * JacobiP(n-2,a,b,x) for n>1.

If the first parameter is not equal to a non-negative integer, then this is the analytic extension of the Jacobi Polynomial and is such that:

JacobiP(mu,alpha,beta,z) = hypergeom([-mu,mu+alpha+beta+1],[alpha+1],(1-z)/2);

It is defined as in chapter 22 of Abromowitz and Stegun Handbook of Mathematical Functions,

Signatures:
sts


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JacobiAM

This symbol represents the binary JacobiAM function, it is the inverse of the trigonometric form of the elliptic integral of the first kind.

Commented Mathematical property (CMP):
JacobiAM(EllipticF(sin(phi),k),k) = phi
Signatures:
sts


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JacobiSN

This symbol represents the binary JacobiSN function, it is defined by: JacobiSN(z,k) = sin(JacobiAM(z,k)) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiCN] [Previous: JacobiAM] [Top]

JacobiCN

This symbol represents the binary JacobiCN function, it is defined by: JacobiCN(z,k) = cos(JacobiAM(z,k)) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiDN] [Previous: JacobiSN] [Top]

JacobiDN

This symbol represents the binary JacobiDN function, it is defined by: JacobiDN(z,k) = sqrt(1-k^2*JacobiSN(z,k)^2) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiNS] [Previous: JacobiCN] [Top]

JacobiNS

This symbol represents the binary JacobiNS function, it is defined by: JacobiNS(z,k) = 1/JacobiSN(z,k) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiNC] [Previous: JacobiDN] [Top]

JacobiNC

This symbol represents the binary JacobiNC function, it is defined by: JacobiNC(z,k) = 1/JacobiCN(z,k) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiND] [Previous: JacobiNS] [Top]

JacobiND

This symbol represents the binary JacobiND function, it is defined by: JacobiND(z,k) = 1/JacobiDN(z,k) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiSC] [Previous: JacobiNC] [Top]

JacobiSC

This symbol represents the binary JacobiSC function, it is defined by: JacobiSC(z,k) = JacobiSN(z,k)/JacobiCN(z,k) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiCS] [Previous: JacobiND] [Top]

JacobiCS

This symbol represents the binary JacobiCS function, it is defined by: JacobiCS(z,k) = JacobiCN(z,k)/JacobiSN(z,k) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiSD] [Previous: JacobiSC] [Top]

JacobiSD

This symbol represents the binary JacobiSD function, it is defined by: JacobiSD(z,k) = JacobiSN(z,k)/JacobiDN(z,k) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiDS] [Previous: JacobiCS] [Top]

JacobiDS

This symbol represents the binary JacobiDS function, it is defined by: JacobiDS(z,k) = JacobiDN(z,k)/JacobiSN(z,k) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiCD] [Previous: JacobiSD] [Top]

JacobiCD

This symbol represents the binary JacobiCD function, it is defined by: JacobiCD(z,k) = JacobiCN(z,k)/JacobiDN(z,k) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiDC] [Previous: JacobiDS] [Top]

JacobiDC

This symbol represents the binary JacobiDC function, it is defined by: JacobiDC(z,k) = JacobiDN(z,k)/JacobiCN(z,k) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 16.

Signatures:
sts


[Next: JacobiTheta1] [Previous: JacobiCD] [Top]

JacobiTheta1

This symbol represents the binary JacobiTheta1 function, it is defined by: JacobiTheta1(z,q) = 2*q^(1/4)*sum((-1)^n*q^(n*(n+1))*sin((2*n+1)*z),n=0..infinity) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions [16.27.1].

Signatures:
sts


[Next: JacobiTheta2] [Previous: JacobiDC] [Top]

JacobiTheta2

This symbol represents the binary JacobiTheta2 function, it is defined by: JacobiTheta2(z,q) = 2*q^(1/4)*sum(q^(n*(n+1))*cos((2*n+1)*z),n=0..infinity) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions [16.27.2].

Signatures:
sts


[Next: JacobiTheta3] [Previous: JacobiTheta1] [Top]

JacobiTheta3

This symbol represents the binary JacobiTheta3 function, it is defined by: JacobiTheta3(z,q) = 1+2*sum(q^(n^2)*cos((2*n)*z),n=1..infinity) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions [16.27.3].

Signatures:
sts


[Next: JacobiTheta4] [Previous: JacobiTheta2] [Top]

JacobiTheta4

This symbol represents the binary JacobiTheta4 function, it is defined by: JacobiTheta4(z,q) = 1+2*sum((-1)^n*q^(n^2)*cos((2*n)*z),n=1..infinity) It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions [16.27.4].

Signatures:
sts


[Next: JacobiZeta] [Previous: JacobiTheta3] [Top]

JacobiZeta

This symbol represents the binary JacobiZeta function, it is defined by: JacobiZeta(z,k) = diff(ln(JacobiTheta4(Pi*z/(2*EllipticK(k)),EllipticNome(k))),z)

which is essentially the logarithmic derivative of JacobiTheta4. JacobiZeta also satisfies the equation EllipticE(z) = JacobiZeta(z,k)+(EllipticE(k)/EllipticK(k))*z It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions [16.34].

Signatures:
sts


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