SpheroidalS1

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`SpheroidalS1`

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SpheroidalS1[n,m,γ,z]

gives the radial spheroidal function of the first kind.

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
The radial spheroidal functions satisfy the differential equation with the spheroidal eigenvalue given by SpheroidalEigenvalue[n,m,γ].
The are normalized according to the Meixner–Schäfke scheme.
SpheroidalS1 can be evaluated to arbitrary numerical precision.
SpheroidalS1 automatically threads over lists. »

Examples

open allclose all

Basic Examples (5)Summary of the most common use cases

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-g6a2f7

Out[1]=1

Plot over a subset of the reals:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-io64fy

Out[1]=1

Plot over a subset of the complexes:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-kiedlx

Out[1]=1

Series expansion at the origin:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-fwvmoh

Out[1]=1

Series expansion at a singular point:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-20imb

Out[1]=1

Scope (21)Survey of the scope of standard use cases

Numerical Evaluation (5)

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-l274ju

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-cksbl4

Out[2]=2

Evaluate to high precision:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-b0wt9

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-zn1q5

Out[2]=2

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

In[3]:=3

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https://wolfram.com/xid/0enwjuj7d-y7k4a

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0enwjuj7d-hj5hh1

Out[4]=4

Complex number inputs:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-hfml09

Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-di5gcr

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-bq2c6r

Out[2]=2

Compute the elementwise values of an array using automatic threading:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-thgd2

Out[1]=1

Or compute the matrix SpheroidalS1 function using MatrixFunction:

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-o5jpo

Out[2]=2

Specific Values (4)

Simple exact values are generated automatically:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-fc9m8o

Out[1]=1

Find the first positive maximum of SpheroidalS1[2,0,5,x]:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-f2hrld

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-h3ihr5

Out[2]=2

SpheroidalS1 functions become elementary if m=1 and γ=n π/2 :

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-chhice

Out[1]=1

TraditionalForm typesetting:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-el2iv0

Visualization (3)

Plot the SpheroidalS1 function for integer orders:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-ecj8m7

Out[1]=1

Plot the SpheroidalS1 function for noninteger parameters:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-8jp9f

Out[1]=1

Plot the real part of :

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-dbvuei

Out[1]=1

Plot the imaginary part of :

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-kbvpn

Out[2]=2

Function Properties (5)

SpheroidalS1 is not an analytic function:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-h5x4l2

Out[1]=1

has both singularities and discontinuities for :

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-mdtl3h

Out[2]=2

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-mn5jws

Out[1]=1

is neither non-decreasing nor non-increasing:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-nlz7s

Out[1]=1

is not injective:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-poz8g

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-ctca0g

Out[2]=2

SpheroidalS1 is neither non-negative nor non-positive:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-84dui

Out[1]=1

SpheroidalS1 is neither convex nor concave:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-8kku21

Out[1]=1

Differentiation (2)

First derivative with respect to z:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-krpoah

Out[1]=1

Higher derivatives with respect to z:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-z33jv

Out[1]=1

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

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-fxwmfc

Out[2]=2

Series Expansions (2)

Find the Taylor expansion using Series:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-ewr1h8

Out[1]=1

Plots of the first three approximations around :

In[5]:=5

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https://wolfram.com/xid/0enwjuj7d-binhar

Out[6]=6

Taylor expansion at a generic point:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-jwxla7

Out[1]=1

Applications (4)Sample problems that can be solved with this function

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

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-dbsamh

Out[1]=1

Plot the eigenvalue:

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-fvwqwj

Out[2]=2

Find resonant frequencies for the Dirichlet problem in the prolate spheroidal cavity:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-cpivcf

Out[1]=1

Determine the first few frequencies:

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-blbz2q

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0enwjuj7d-b8ff34

Out[3]=3

Plot the prolate and oblate functions:

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-jcujiz

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-edag2r

Out[2]=2

Build a near-spherical approximation to :

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-gzcleh

First few terms of the approximation:

In[2]:=2

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https://wolfram.com/xid/0enwjuj7d-b4ghrk

Compare numerically:

In[3]:=3

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https://wolfram.com/xid/0enwjuj7d-cjl5gk

Out[3]=3

Possible Issues (1)Common pitfalls and unexpected behavior

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

In[1]:=1

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https://wolfram.com/xid/0enwjuj7d-c6p1oc

Out[1]=1

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