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SpheroidalRadialFactor[n,m,c]

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

Details
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Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Function Properties  
Applications  
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SpheroidalRadialFactor

SpheroidalRadialFactor[n,m,c]

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

Details

Examples

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Basic Examples  (2)

Evaluate numerically:

Wolfram Language code: SpheroidalRadialFactor[3, 2, 10.]

Plot over a subset of the reals:

Wolfram Language code: Plot[SpheroidalRadialFactor[3, 2, c], {c, 0, 5}]

Scope  (11)

Numerical Evaluation  (4)

Evaluate numerically:

Wolfram Language code: SpheroidalRadialFactor[5, 1, 7.]
Wolfram Language code: SpheroidalRadialFactor[1, 0.5, -5]

Evaluate to high precision:

Wolfram Language code: N[SpheroidalRadialFactor[7, 0, 1 / 4], 50]
Wolfram Language code: N[SpheroidalRadialFactor[2, 1, 1 / 3], 20]

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

Wolfram Language code: SpheroidalRadialFactor[2, 2, 1.211111111000111111111]

Complex number inputs:

Wolfram Language code: N[SpheroidalRadialFactor[3, 5 - I, 1 / 3 + I]]

Evaluate efficiently at high precision:

Wolfram Language code: SpheroidalRadialFactor[31, 5, 1 / 6`100]//Timing
Wolfram Language code: SpheroidalRadialFactor[31, 50, 1 / 7`100];//Timing//Quiet

Specific Values  (2)

Value at zero:

Wolfram Language code: SpheroidalRadialFactor[0, 0, 0.]

Find the first positive maximum of SpheroidalRadialFactor[3,2,x]:

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

Visualization  (2)

Plot the SpheroidalRadialFactor function:

Wolfram Language code: Plot[{SpheroidalRadialFactor[2, 2, x], SpheroidalRadialFactor[3, 2, x], SpheroidalRadialFactor[4, 2, x]}, {x, -5, 5}]

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

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

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

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

Function Properties  (3)

has no singularities or discontinuities:

Wolfram Language code: FunctionSingularities[SpheroidalRadialFactor[3, 2, x], x]//Quiet
Wolfram Language code: FunctionDiscontinuities[SpheroidalRadialFactor[3, 2, x], x]//Quiet

is neither nondecreasing nor nonincreasing:

Wolfram Language code: FunctionMonotonicity[SpheroidalRadialFactor[3, 2, x], x]

is neither convex nor concave:

Wolfram Language code: FunctionConvexity[SpheroidalRadialFactor[3, 2, x], x]

Applications  (1)

Build a near-spherical approximation to :

Wolfram Language code: nsLimitS1[n_, m_, z_, {c_, ord_Integer}] := (Normal[Series[SpheroidalPS[n1, -m1, c, x], {c, 0, ord}] /. LegendreP[k_, -m1, 2, x] :> SphericalBesselJ[k, z]](1 - c ^ 2 / z ^ 2)^m / 2 / SpheroidalRadialFactor[n, -m, c]) /. {n1 -> n, m1 -> m}

First few terms of the approximation:

Wolfram Language code: nsLimitS1[n, m, z, {c, 3}]//TraditionalForm

Compare numerically:

Wolfram Language code: (Table[nsLimitS1[3, 1, 1`30, {c, k}], {k, 2, 10, 2}] - SpheroidalS1[3, 1, c, 1 / c]) /. {c -> 1 / 100}
Wolfram Research (2007), SpheroidalRadialFactor, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalRadialFactor.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_spheroidalradialfactor, author="Wolfram Research", title="{SpheroidalRadialFactor}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalRadialFactor.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_spheroidalradialfactor, organization={Wolfram Research}, title={SpheroidalRadialFactor}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalRadialFactor.html}, note=[Accessed: 13-June-2026]}