JacobiP

JacobiP[n,a,b,x]
gives the Jacobi polynomial .
Details

- 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
.
- JacobiP can be used with Interval and CenteredInterval objects. »
Examples
open allclose allBasic 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:
Scope (39)
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
JacobiP can be used with Interval and CenteredInterval objects:
Specific Values (6)
Visualization (4)
Function Properties (11)
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 for integer
,
and
:
Jacobi polynomials are analytic functions:
However, is not an analytic function of
for noninteger
,
and
:
is neither non-decreasing nor non-increasing:
is increasing on its real domain:
is neither non-negative nor non-positive:
has no singularities or discontinuities for integer
,
and
:
is neither convex nor concave:
is concave on its real domain:
TraditionalForm formatting:
Differentiation (3)
Integration (3)
Compute the indefinite integral using Integrate:
Series Expansions (4)
Applications (3)
Properties & Relations (2)
Text
Wolfram Research (1988), JacobiP, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiP.html (updated 2022).
CMS
Wolfram Language. 1988. "JacobiP." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/JacobiP.html.
APA
Wolfram Language. (1988). JacobiP. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiP.html