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SpheroidalJoiningFactor[n,m,γ]

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

Details
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Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Applications  
Possible Issues  
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SpheroidalJoiningFactor

SpheroidalJoiningFactor[n,m,γ]

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

Details

Examples

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Basic Examples  (2)

Evaluate numerically:

Wolfram Language code: SpheroidalJoiningFactor[1, 0, 0.5]//Chop

Plot over a subset of the reals:

Wolfram Language code: Plot[SpheroidalJoiningFactor[1, 0, c], {c, 0, 3}]

Scope  (9)

Numerical Evaluation  (4)

Evaluate numerically:

Wolfram Language code: SpheroidalJoiningFactor[5, 1, .1]
Wolfram Language code: SpheroidalJoiningFactor[1, 0.5, -5]

Evaluate to high precision:

Wolfram Language code: N[SpheroidalJoiningFactor[247, 5, 1], 50]
Wolfram Language code: N[SpheroidalJoiningFactor[2, -1, -1], 20]

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

Wolfram Language code: SpheroidalJoiningFactor[0.211111111111111111, 2, 5]
Wolfram Language code: SpheroidalJoiningFactor[2, 0.211111111111111111, 5]

Complex number inputs:

Wolfram Language code: N[SpheroidalJoiningFactor[23, 5 - I, 2]]

Evaluate efficiently at high precision:

Wolfram Language code: SpheroidalJoiningFactor[31, 5, 1`100]//Timing
Wolfram Language code: SpheroidalJoiningFactor[15, 5, 1`1000];//Timing

Specific Values  (3)

Value at zero:

Wolfram Language code: SpheroidalJoiningFactor[0, 0, 0]//N

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

Wolfram Language code: xval = x /. FindRoot[SpheroidalJoiningFactor[0, 1 / 2, x] == 5, {x, 3}]//Chop
Wolfram Language code: Plot[SpheroidalJoiningFactor[0, 1 / 2, x], {x, 0, 5}, Epilog -> Style[Point[{xval, SpheroidalJoiningFactor[0, 1 / 2, xval ]}], PointSize[Large], Red]]//Quiet

SpheroidalJoiningFactor threads elementwise over lists:

Wolfram Language code: SpheroidalJoiningFactor[1, 0, {1., 2., 3.}]

Visualization  (2)

Plot the SpheroidalJoiningFactor function:

Wolfram Language code: Plot[{SpheroidalJoiningFactor[2, 2, x], SpheroidalJoiningFactor[3, 2, x], SpheroidalJoiningFactor[4, 2, x]}, {x, -4, 4}]

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

Wolfram Language code: ContourPlot[Re[SpheroidalJoiningFactor[2, 1, x + I y]], {x, -3, 3}, {y, -6, 6}, Contours -> 24]

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

Wolfram Language code: ContourPlot[Im[SpheroidalJoiningFactor[2, 1, x + I y]], {x, -3, 3}, {y, -6, 6}, Contours -> 24]

Applications  (1)

A relation between radial and angular spheroidal functions:

Wolfram Language code: SpheroidalS1[n, 0, c, z] == SpheroidalJoiningFactor[n, 0, c]SpheroidalPS[n, 0, c, z];

Check numerically:

Wolfram Language code: % /. {n -> 2, c -> 1, z -> RandomComplex[]}

Possible Issues  (1)

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

Wolfram Language code: SpheroidalJoiningFactor[3 / 2, 1, 0.5]
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.

CMS

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

BibTeX

@misc{reference.wolfram_2026_spheroidaljoiningfactor, author="Wolfram Research", title="{SpheroidalJoiningFactor}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalJoiningFactor.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_spheroidaljoiningfactor, organization={Wolfram Research}, title={SpheroidalJoiningFactor}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalJoiningFactor.html}, note=[Accessed: 13-June-2026]}