SpheroidalPSPrime

SpheroidalPSPrime[n,m,γ,z]

gives the derivative with respect to of the angular spheroidal function of the first kind.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • SpheroidalPSPrime[n,m,a,γ,z] uses spheroidal functions of type . The types are specified as for SpheroidalPS.
  • For certain special arguments, SpheroidalPSPrime automatically evaluates to exact values.
  • SpheroidalPSPrime can be evaluated to arbitrary numerical precision.
  • SpheroidalPSPrime automatically threads over lists.

Examples

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Basic Examples  (7)

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:

Series expansion at Infinity:

Series expansion at a singular point:

Scope  (26)

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 SpheroidalPSPrime[4,0,1/2,x]:

Evaluate the SpheroidalPSPrime function for half-integer parameters:

Different SpheroidalPSPrime types give different symbolic forms:

Visualization  (3)

Plot the SpheroidalPSPrime function for various orders:

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

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

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

Function Properties  (8)

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

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

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

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

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

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

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

TemplateBox[{2, 0, 1, x}, SpheroidalPSPrime] 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:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

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)

SpheroidalPSPrime of different types have different branch cut structures:

Applications  (1)

Plot prolate and oblate versions of the same angular function:

Possible Issues  (1)

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

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2021_spheroidalpsprime, author="Wolfram Research", title="{SpheroidalPSPrime}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalPSPrime.html}", note=[Accessed: 07-December-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_spheroidalpsprime, organization={Wolfram Research}, title={SpheroidalPSPrime}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalPSPrime.html}, note=[Accessed: 07-December-2021 ]}