SpheroidalQS

SpheroidalQS[n,m,γ,z]

gives the angular spheroidal function of the second 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].
  • SpheroidalQS[n,m,0,z] is equivalent to LegendreQ[n,m,z].
  • SpheroidalQS[n,m,a,γ,z] gives spheroidal functions of type . The types are specified as for LegendreP.
  • For certain special arguments, SpheroidalQS automatically evaluates to exact values.
  • SpheroidalQS can be evaluated to arbitrary numerical precision.
  • SpheroidalQS automatically threads over lists.

Examples

open allclose all

Basic Examples  (5)

Evaluate numerically:

Expansion about the spherical case:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope  (19)

Numerical Evaluation  (5)

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)

SpheroidalQS[n,m,0,x] is equivalent to the LegendreQ[n,m,x] function:

Find the first positive maximum of SpheroidalQS[4,0,1/2,x]:

The SpheroidalQS function is equal to zero for half-integer parameters:

Different SpheroidalQS types give different symbolic forms:

Visualization  (3)

Plot the SpheroidalQS function for various orders:

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

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

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

Function Properties  (3)

The complex domain of SpheroidalQS:

SpheroidalQS is an even function:

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=5, m=2 and γ=1:

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 SpheroidalQS have different branch cut structures:

Applications  (2)

Solve the spheroidal differential equation in terms of SpheroidalQS:

Plot prolate and oblate versions of the same angular function:

Possible Issues  (1)

Spheroidal functions do not generically evaluate for half-integer values of n:

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

Text

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

BibTeX

@misc{reference.wolfram_2021_spheroidalqs, author="Wolfram Research", title="{SpheroidalQS}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalQS.html}", note=[Accessed: 19-September-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_spheroidalqs, organization={Wolfram Research}, title={SpheroidalQS}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalQS.html}, note=[Accessed: 19-September-2021 ]}

CMS

Wolfram Language. 2007. "SpheroidalQS." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SpheroidalQS.html.

APA

Wolfram Language. (2007). SpheroidalQS. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SpheroidalQS.html