Built-in Mathematica Symbol

GegenbauerC

GegenbauerC[n, m, x]
gives the Gegenbauer polynomial

.

GegenbauerC[n, x]
gives the renormalized form

.

MORE INFORMATION

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 .

EXAMPLES

CLOSE ALL

Basic Examples (2)

Compute the 10

Gegenbauer polynomial:

Compute the 10

renormalized Gegenbauer polynomial:

Plot

:

Compute the 10

Gegenbauer polynomial:

In[1]:=

Out[1]=

Compute the 10

renormalized Gegenbauer polynomial:

In[2]:=

Out[2]=

Plot

:

In[1]:=

Out[1]=

Scope (6)

Evaluate for complex orders and arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

GegenbauerC threads element-wise over lists:

Simple cases give exact symbolic results:

TraditionalForm formatting:

Generalizations & Extensions (2)

Apply GegenbauerC to a power series:

GegenbauerC can deal with real-valued intervals:

Applications (2)

Eigenfunctions of the angular part of the four-dimensional Laplace operator:

Radial part of the hydrogen atom eigenfunction in momentum representation:

Properties & Relations (1)

Use FunctionExpand to expand GegenbauerC into other functions:

Possible Issues (1)

Cancellations in the polynomial form may lead to inaccurate numerical results:

Evaluate the function directly: