SpheroidalS2Prime

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`SpheroidalS2Prime`

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SpheroidalS2Prime[n,m,γ,z]

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

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
For certain special arguments, SpheroidalS2Prime automatically evaluates to exact values.
SpheroidalS2Prime can be evaluated to arbitrary numerical precision.
SpheroidalS2Prime 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/0fq238t3na248-iw7myr

Out[1]=1

Plot over a subset of the reals:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-io64fy

Out[1]=1

Plot over a subset of the complexes:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-kiedlx

Out[1]=1

Series expansion at the origin:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-fwvmoh

Out[1]=1

Series expansion at a singular point:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-20imb

Out[1]=1

Scope (24)Survey of the scope of standard use cases

Numerical Evaluation (5)

Evaluate numerically:

In[3]:=3

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https://wolfram.com/xid/0fq238t3na248-l274ju

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0fq238t3na248-whe1w

Out[4]=4

Evaluate to high precision:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-zn1q5

Out[1]=1

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

In[2]:=2

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https://wolfram.com/xid/0fq238t3na248-y7k4a

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0fq238t3na248-hj5hh1

Out[3]=3

Complex number inputs:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-hfml09

Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-di5gcr

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0fq238t3na248-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/0fq238t3na248-thgd2

Out[1]=1

Or compute the matrix SpheroidalS2Prime function using MatrixFunction:

In[2]:=2

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https://wolfram.com/xid/0fq238t3na248-o5jpo

Out[2]=2

Specific Values (5)

Simple exact values are generated automatically:

In[2]:=2

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https://wolfram.com/xid/0fq238t3na248-fc9m8o

Out[2]=2

Singular points:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-epvazh

Out[1]=1

Find the first positive maximum of SpheroidalS2Prime[2,0,5,x]:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-f2hrld

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0fq238t3na248-u1fjh

Out[2]=2

SpheroidalS2Prime functions become elementary if and :

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-chhice

Out[1]=1

TraditionalForm typesetting:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-el2iv0

Visualization (3)

Plot the SpheroidalS2Prime function for integer orders:

In[2]:=2

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https://wolfram.com/xid/0fq238t3na248-ecj8m7

Out[2]=2

Plot the SpheroidalS2Prime function for non-integer parameters:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-8jp9f

Out[1]=1

Plot the real part of SpheroidalS2Prime:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-dbvuei

Out[1]=1

Plot the imaginary part of SpheroidalS2Prime:

In[2]:=2

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https://wolfram.com/xid/0fq238t3na248-bfq6yk

Out[2]=2

Function Properties (5)

SpheroidalS2Prime is not an analytic function:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-h5x4l2

Out[1]=1

has both singularities and discontinuities for :

In[2]:=2

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https://wolfram.com/xid/0fq238t3na248-mdtl3h

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0fq238t3na248-mn5jws

Out[3]=3

is neither non-decreasing nor non-increasing:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-nlz7s

Out[1]=1

is not injective:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-poz8g

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0fq238t3na248-ctca0g

Out[2]=2

SpheroidalS2Prime is neither non-negative nor non-positive:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-84dui

Out[1]=1

SpheroidalS2Prime is neither convex nor concave:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-8kku21

Out[1]=1

Differentiation (2)

First derivative with respect to :

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-krpoah

Out[1]=1

Higher derivatives with respect to :

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-z33jv

Out[1]=1

Plot the higher derivatives with respect to when , and :

In[2]:=2

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https://wolfram.com/xid/0fq238t3na248-fxwmfc

Out[2]=2

Integration (2)

Compute the indefinite integral using Integrate:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-bponid

Out[1]=1

Verify the anti-derivative:

In[2]:=2

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https://wolfram.com/xid/0fq238t3na248-op9yly

Out[2]=2

Definite integral:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-bfdh5d

Out[1]=1

Series Expansions (2)

Find the Taylor expansion using Series:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-ewr1h8

Out[1]=1

Plots of the first three approximations around :

In[3]:=3

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https://wolfram.com/xid/0fq238t3na248-binhar

Out[4]=4

Taylor expansion at a generic point:

In[1]:=1

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https://wolfram.com/xid/0fq238t3na248-jwxla7

Out[1]=1

Properties & Relations (1)Properties of the function, and connections to other functions

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/0fq238t3na248-c6p1oc

Out[1]=1

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