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:

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:

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:

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:

TraditionalForm formatting:

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:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

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:

JacobiP is defined through the identity:

Normalization of JacobiP:

Recurrence relations: