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:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix SpheroidalS1Prime function using MatrixFunction:

Simple exact values are generated automatically:

Find the first positive maximum of SpheroidalS1Prime[2,0,5,x]:

SpheroidalS1Prime functions become elementary if m=1 and γ=n π/2 :

TraditionalForm typesetting:

Plot the SpheroidalS1Prime function for integer orders:

Plot the SpheroidalS1Prime function for noninteger parameters:

Plot the real part of :

Plot the imaginary part of :

SpheroidalS1Prime is not an analytic function:

has both singularities and discontinuities for :

is neither non-decreasing nor non-increasing:

is not injective:

SpheroidalS1Prime is neither non-negative nor non-positive:

SpheroidalS1Prime is neither convex nor concave:

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:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point: