- Mathematical function, suitable for both symbolic and numerical manipulation.
- Explicit polynomials are given for integer n and for any m.
- satisfies the differential equation .
- The Gegenbauer polynomials are orthogonal on the interval with weight function , corresponding to integration over a unit hypersphere.
- For certain special arguments, GegenbauerC automatically evaluates to exact values.
- GegenbauerC can be evaluated to arbitrary numerical precision.
- GegenbauerC automatically threads over lists.
- GegenbauerC[n,0,x] is always zero.
- GegenbauerC[n,m,z] has a branch cut discontinuity in the complex z plane running from to .
Examplesopen allclose all
Basic Examples (7)
Compute the 10 Gegenbauer polynomial:
Compute the 10 renormalized Gegenbauer 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 (8)
Values of GegenbauerC at fixed points:
Simple cases give exact symbolic results:
GegenbauerC for symbolic n:
Values at zero:
Find the first positive maximum of GegenbauerC[10,x ]:
Compute the associated GegenbauerC[7,x] polynomial:
Compute the associated GegenbauerC[1/2,x] polynomial for half-integer n:
Different GegenbauerC types give different symbolic forms:
Plot the GegenbauerC 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 GegenbauerC function have different branch cut structures:
Function Properties (7)
Domain of GegenbauerC of integer orders:
The range for GegenbauerC of integer orders:
The range for complex values is the whole plane:
Gegenbauer polynomial of an odd order is odd:
Gegenbauer polynomial of an even order is even:
GegenbauerC threads elementwise over lists:
GegenbauerC has the mirror property :
First derivatives with respect to x:
Higher derivatives with respect to x:
Plot the higher derivatives with respect to x when n=10 and m=1/3:
Formula for the derivative with respect to x:
Compute the indefinite integral using Integrate:
Verify the anti-derivative:
Series Expansions (2)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
Taylor expansion at a generic point:
Function Identities and Simplifications (4)
GegenbauerC is a special case of JacobiP:
Derivative identity of GegenbauerC:
Generating function of Gegenbauer polynomials:
Generalizations & Extensions (2)
Eigenfunctions of the angular part of the four-dimensional Laplace operator:
Radial part of the hydrogen atom eigenfunction in momentum representation:
Properties & Relations (4)
Possible Issues (1)
Cancellations in the polynomial form may lead to inaccurate numerical results:
Evaluate the function directly: