SpheroidalPS
SpheroidalPS[n,m,γ,z]
gives the angular spheroidal function of the first kind.
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- The angular spheroidal functions satisfy the differential equation
with the spheroidal eigenvalue
given by SpheroidalEigenvalue[n,m,γ].
- SpheroidalPS[n,m,0,z] is equivalent to LegendreP[n,m,z].
- SpheroidalPS[n,m,a,γ,z] gives spheroidal functions of type
. The types are specified as for LegendreP.
- For certain special arguments, SpheroidalPS automatically evaluates to exact values.
- SpheroidalPS can be evaluated to arbitrary numerical precision.
- SpheroidalPS automatically threads over lists.
Examples
open allclose allBasic Examples (6)
Expansion about the spherical case:
Plot over a subset of the reals:
Series expansion at the origin:
Series expansion at Infinity:
Scope (23)
Numerical Evaluation (4)
Specific Values (4)
Find the first positive minimum of SpheroidalPS[4,0,1/2,x]:
Evaluate the SpheroidalPS function for half-integer parameters:
Different SpheroidalPS types give different symbolic forms:
Visualization (3)
Plot the SpheroidalPS function for various orders:
Types 2 and 3 of SpheroidalPS functions have different branch cut structures:
Function Properties (8)
is an even function with respect to
:
has no singularities or discontinuities:
is neither non-decreasing nor non-increasing:
is neither non-negative nor non-positive:
TraditionalForm formatting:
Differentiation (2)
Series Expansions (2)
Find the Taylor expansion using Series:
Generalizations & Extensions (1)
The different types of SpheroidalPS have different branch cut structures:
Applications (3)
Solve the spheroidal differential equation in terms of SpheroidalPS:
Plot prolate and oblate versions of the same angular function:
SpheroidalPS is a band-limited function with bandwidth proportional to :
Properties & Relations (1)
Spheroidal angular harmonics are eigenfunctions of the Sinc transform on the interval :
Possible Issues (2)
Text
Wolfram Research (2007), SpheroidalPS, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalPS.html.
CMS
Wolfram Language. 2007. "SpheroidalPS." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SpheroidalPS.html.
APA
Wolfram Language. (2007). SpheroidalPS. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SpheroidalPS.html