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

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Basic Examples  (6)

Evaluate numerically:

Expansion about the spherical case:

Plot over a subset of the reals:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

Scope  (23)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values  (4)

Evaluate symbolically:

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:

Plot the real part of TemplateBox[{3, 0, 1, {x, +, {i, , y}}}, SpheroidalPS]:

Plot the imaginary part of TemplateBox[{3, 0, 1, {x, +, {i, , y}}}, SpheroidalPS]:

Types 2 and 3 of SpheroidalPS functions have different branch cut structures:

Function Properties  (8)

The real domain of TemplateBox[{1, 2, 2, x}, SpheroidalPS]:

The complex domain of TemplateBox[{1, 2, 2, x}, SpheroidalPS]:

TemplateBox[{1, 2, gamma, 3}, SpheroidalPS] is an even function with respect to :

TemplateBox[{1, 2, gamma, 3}, SpheroidalPS] has the mirror property TemplateBox[{1, 2, 3, {z, }}, SpheroidalPS]=TemplateBox[{1, 2, 3, z}, SpheroidalPS]:

TemplateBox[{2, 0, 1, x}, SpheroidalPS] has no singularities or discontinuities:

TemplateBox[{2, 0, 1, x}, SpheroidalPS] is neither non-decreasing nor non-increasing:

TemplateBox[{2, 0, 1, x}, SpheroidalPS] is not injective:

TemplateBox[{2, 0, 1, x}, SpheroidalPS] is neither non-negative nor non-positive:

TraditionalForm formatting:

Differentiation  (2)

The first derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when n=10, m=2 and γ=1/3:

Series Expansions  (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The Taylor expansion at a generic point:

Generalizations & Extensions  (1)

The different types of SpheroidalPS have different branch cut structures:

Applications  (4)

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 :

For spheroidicity parameter :

For spheroidicity parameter , the bandwidth is higher:

Build a near-spherical approximation to :

First few terms of the approximation:

Compare numerically:

Properties & Relations  (1)

Spheroidal angular harmonics are eigenfunctions of the Sinc transform on the interval :

Possible Issues  (2)

Spheroidal functions do not evaluate for half-integer values of n or generic values of m:

Angular spheroidal harmonics in the Wolfram Language adopt the Meixner–Schaefke normalization scheme:

Flammer normalization is also common:

Reconstruct table entries from Abramowitz and Stegun table 21.2:

Wolfram Research (2007), SpheroidalPS, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalPS.html.

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

BibTeX

@misc{reference.wolfram_2023_spheroidalps, author="Wolfram Research", title="{SpheroidalPS}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalPS.html}", note=[Accessed: 08-December-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_spheroidalps, organization={Wolfram Research}, title={SpheroidalPS}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalPS.html}, note=[Accessed: 08-December-2023 ]}