SpheroidalQSPrime

SpheroidalQSPrime[n,m,γ,z]

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

Details

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

Examples

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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  (18)

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  (5)

Evaluate symbolically:

Find the first positive minimum of SpheroidalQSPrime[4,0,1/2,x]:

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

Different SpheroidalQSPrime types give different symbolic forms:

TraditionalForm formatting:

Visualization  (2)

Plot the SpheroidalQSPrime function for various orders:

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

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

Differentiation  (2)

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:

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 :

Taylor expansion at a generic point:

Applications  (1)

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:

Introduced in 2007
 (6.0)