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SpheroidalJoiningFactor
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SpheroidalJoiningFactor

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gives the spheroidal joining factor with degree n and order m.

Details

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0e47wwz6am8dle-lwoon
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
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https://wolfram.com/xid/0e47wwz6am8dle-fcl9v1
Out[1]=1

Scope  (9)Survey of the scope of standard use cases

Numerical Evaluation  (4)

Evaluate numerically:

In[1]:=1
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https://wolfram.com/xid/0e47wwz6am8dle-l274ju
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e47wwz6am8dle-whe1w
Out[2]=2

Evaluate to high precision:

In[1]:=1
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https://wolfram.com/xid/0e47wwz6am8dle-b0wt9
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e47wwz6am8dle-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/0e47wwz6am8dle-y7k4a
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0e47wwz6am8dle-hj5hh1
Out[4]=4

Complex number inputs:

In[1]:=1
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https://wolfram.com/xid/0e47wwz6am8dle-hfml09
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0e47wwz6am8dle-di5gcr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e47wwz6am8dle-bq2c6r
Out[2]=2

Specific Values  (3)

Value at zero:

In[1]:=1
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https://wolfram.com/xid/0e47wwz6am8dle-fc9m8o
Out[1]=1

Find a value of x for which SpheroidalJoiningFactor[0,1/2,x]=5:

In[1]:=1
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https://wolfram.com/xid/0e47wwz6am8dle-f2hrld
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e47wwz6am8dle-eokm8z
Out[2]=2

SpheroidalJoiningFactor threads elementwise over lists:

In[1]:=1
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https://wolfram.com/xid/0e47wwz6am8dle-dh5aia
Out[1]=1

Visualization  (2)

Plot the SpheroidalJoiningFactor function:

In[1]:=1
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https://wolfram.com/xid/0e47wwz6am8dle-ecj8m7
Out[1]=1

Plot the real part of SpheroidalJoiningFactor[2,1,x+i y]:

In[1]:=1
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https://wolfram.com/xid/0e47wwz6am8dle-8gdas
Out[1]=1

Plot the imaginary part of SpheroidalJoiningFactor[2,1,x+i y]:

In[2]:=2
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https://wolfram.com/xid/0e47wwz6am8dle-7ohn
Out[2]=2

Applications  (1)Sample problems that can be solved with this function

A relation between radial and angular spheroidal functions:

In[1]:=1
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https://wolfram.com/xid/0e47wwz6am8dle-jsm3s7

Check numerically:

In[2]:=2
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https://wolfram.com/xid/0e47wwz6am8dle-cskzvr
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

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

In[1]:=1
✖
https://wolfram.com/xid/0e47wwz6am8dle-c6p1oc
Out[1]=1
Wolfram Research (2007), SpheroidalJoiningFactor, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalJoiningFactor.html.
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Wolfram Research (2007), SpheroidalJoiningFactor, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalJoiningFactor.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_spheroidaljoiningfactor, author="Wolfram Research", title="{SpheroidalJoiningFactor}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalJoiningFactor.html}", note=[Accessed: 25-May-2025 ]}

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@misc{reference.wolfram_2025_spheroidaljoiningfactor, author="Wolfram Research", title="{SpheroidalJoiningFactor}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalJoiningFactor.html}", note=[Accessed: 25-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_spheroidaljoiningfactor, organization={Wolfram Research}, title={SpheroidalJoiningFactor}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalJoiningFactor.html}, note=[Accessed: 25-May-2025 ]}

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@online{reference.wolfram_2025_spheroidaljoiningfactor, organization={Wolfram Research}, title={SpheroidalJoiningFactor}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalJoiningFactor.html}, note=[Accessed: 25-May-2025 ]}