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 .
• JacobiP can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples(7)

Evaluate numerically:

Compute the 2 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(39)

Numerical Evaluation(5)

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:

JacobiP can be used with Interval and CenteredInterval objects:

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, is not an analytic function of for noninteger , and :

Nor is it meromorphic:

is neither non-decreasing nor non-increasing:

is increasing on its real domain:

is not injective:

is:

is not surjective:

is:

is neither non-negative nor non-positive:

has no singularities or discontinuities for integer , and :

is neither convex nor concave:

is concave on its real domain:

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 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(4)

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:

In an n-point GaussRadau quadrature rule, the value of one of the two extreme nodes is fixed, and the other n-1 nodes are computed from the roots of a certain Jacobi polynomial. Letting the leftmost node be the fixed node, compute the nodes and weights of an n-point GaussRadau quadrature rule:

Use the n-point GaussRadau quadrature rule to numerically evaluate an integral:

Compare the result of the GaussRadau quadrature with the result from NIntegrate:

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 (updated 2022).

Text

Wolfram Research (1988), JacobiP, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiP.html (updated 2022).

CMS

Wolfram Language. 1988. "JacobiP." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. 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

BibTeX

@misc{reference.wolfram_2024_jacobip, author="Wolfram Research", title="{JacobiP}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiP.html}", note=[Accessed: 26-May-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_jacobip, organization={Wolfram Research}, title={JacobiP}, year={2022}, url={https://reference.wolfram.com/language/ref/JacobiP.html}, note=[Accessed: 26-May-2024 ]}