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GegenbauerC

GegenbauerC
gives the Gegenbauer polynomial .
GegenbauerC
gives the renormalized form .
  • 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 has a branch cut discontinuity in the complex z plane running from to .
Compute the 10^(th) Gegenbauer polynomial:
Compute the 10^(th) renormalized Gegenbauer polynomial:
Plot :
Compute the 10^(th) Gegenbauer polynomial:
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Click for copyable input
Out[1]=
Compute the 10^(th) renormalized Gegenbauer polynomial:
In[2]:=
Click for copyable input
Out[2]=
 
Plot :
In[1]:=
Click for copyable input
Out[1]=
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:
Apply GegenbauerC to a power series:
GegenbauerC can deal with real-valued intervals:
Eigenfunctions of the angular part of the four-dimensional Laplace operator:
Radial part of the hydrogen atom eigenfunction in momentum representation:
Use FunctionExpand to expand GegenbauerC into other functions:
Cancellations in the polynomial form may lead to inaccurate numerical results:
Evaluate the function directly:
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