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JacobiP

JacobiP[n, a, b, x]
gives the Jacobi polynomial .
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Explicit polynomials are given when possible.
  • satisfies the differential equation (1-x^2)y^('')+(b-a-(a+b+2)x)y^'+n(n+a+b+1)y=0.
  • The Jacobi polynomials are orthogonal with weight function (1-x)^a(1+x)^b.
  • 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 -infty to -1.
EXAMPLESCLOSE ALL
Basic Examples   (2)
Compute the 2^(nd) Jacobi polynomial:
Plot :
Compute the 2^(nd) Jacobi polynomial:
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Click for copyable input
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Plot :
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Click for copyable input
Out[1]=
Scope   (7)
Evaluate for fractional orders:
Evaluate for complex orders and arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Simple cases give exact symbolic results:
JacobiP threads element-wise over lists:
TraditionalForm formatting:
Applications   (3)
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   (1)
Use FunctionExpand to expand into other functions:
Possible Issues   (1)
Cancellations in the polynomial form may lead to inaccurate numerical results:
Evaluate the function directly:
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