# 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(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: