JacobiP

JacobiP[n,a,b,x]

gives the Jacobi polynomial .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Explicit polynomials are given when possible.
  • satisfies the differential equation .
  • The Jacobi polynomials are orthogonal with weight function .
  • For certain special arguments, JacobiP automatically evaluates to exact values.
  • JacobiP can be evaluated to arbitrary numerical precision.
  • JacobiP automatically threads over lists.
  • JacobiP[n,a,b,z] has a branch cut discontinuity in the complex z plane running from to .

Examples

open allclose all

Basic Examples  (7)

Evaluate numerically:

Compute the 2^(nd) Jacobi polynomial:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

Scope  (38)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

Specific Values  (6)

Values of JacobiP at fixed points:

Values at zero:

Find the first positive minimum of JacobiP[10,2,3,x]:

Compute the associated JacobiP polynomial:

Compute the associated JacobiP polynomial for half-integer arguments:

Different JacobiP types give different symbolic forms:

Visualization  (4)

Plot the JacobiP function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

Types 2 and 3 of JacobiP function have different branch cut structures:

Function Properties  (11)

Domain of JacobiP of integer orders:

Domain for noninteger orders:

The range for JacobiP of integer orders:

The range for complex values is the whole plane:

JacobiP has the mirror property for integer , and :

Jacobi polynomials are analytic functions:

However, TemplateBox[{n, a, b, x}, JacobiP] is not an analytic function of for noninteger , and :

Nor is it meromorphic:

TemplateBox[{10, 2, 2, x}, JacobiP] is neither non-decreasing nor non-increasing:

TemplateBox[{{1, /, 2}, {/, 2}, {/, 2}, x}, JacobiP] is increasing on its real domain:

TemplateBox[{10, 2, 2, x}, JacobiP] is not injective:

TemplateBox[{{1, /, 2}, {/, 2}, {/, 2}, x}, JacobiP] is:

TemplateBox[{10, 2, 2, x}, JacobiP] is not surjective:

TemplateBox[{{1, /, 2}, {/, 2}, {/, 2}, x}, JacobiP] is:

TemplateBox[{10, 2, 2, x}, JacobiP] is neither non-negative nor non-positive:

TemplateBox[{n, a, b, x}, JacobiP] has no singularities or discontinuities for integer , and :

TemplateBox[{10, 2, 2, x}, JacobiP] is neither convex nor concave:

TemplateBox[{{1, /, 2}, {/, 2}, {/, 2}, x}, JacobiP] is concave on its real domain:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to x:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x:

Formula for the ^(th) derivative with respect to x:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Function Identities and Simplifications  (3)

JacobiP is defined through the identity:

Normalization of JacobiP:

Recurrence relations:

Applications  (3)

Expected value of the number of real eigenvalues of a complex matrix:

Solve a Jacobi differential equation:

Solution of the Schrödinger equation with a PöschlTeller potential:

Calculate the energy eigenvalue from the differential equation:

Properties & Relations  (2)

Use FunctionExpand to expand into other functions:

The generating function for JacobiP:

Possible Issues  (1)

Cancellations in the polynomial form may lead to inaccurate numerical results:

Evaluate the function directly:

Wolfram Research (1988), JacobiP, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiP.html.

Text

Wolfram Research (1988), JacobiP, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiP.html.

BibTeX

@misc{reference.wolfram_2021_jacobip, author="Wolfram Research", title="{JacobiP}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiP.html}", note=[Accessed: 27-July-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_jacobip, organization={Wolfram Research}, title={JacobiP}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiP.html}, note=[Accessed: 27-July-2021 ]}

CMS

Wolfram Language. 1988. "JacobiP." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiP.html.

APA

Wolfram Language. (1988). JacobiP. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiP.html