gives the Jacobi polynomial .
- 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 .
Examplesopen allclose all
Basic Examples (7)
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:
Numerical Evaluation (4)
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:
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 (4)
Domain of JacobiP of integer orders:
The range for JacobiP of integer orders:
The range for complex values is the whole plane:
JacobiP has the mirror property :
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:
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:
Expected value of the number of real eigenvalues of a complex matrix:
Solve a Jacobi differential equation:
Solution of the Schrödinger equation with a Pöschl–Teller potential:
Calculate the energy eigenvalue from the differential equation:
Properties & Relations (2)
Possible Issues (1)
Cancellations in the polynomial form may lead to inaccurate numerical results:
Evaluate the function directly: