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SpheroidalEigenvalue[n,m,γ]

gives the spheroidal eigenvalue with degree and order .

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Applications  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
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SpheroidalEigenvalue

SpheroidalEigenvalue[n,m,γ]

gives the spheroidal eigenvalue with degree and order .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The spheroidal eigenvalues for successive correspond to the successive values of for which there exist normalizable solutions to the differential equation .
  • SpheroidalEigenvalue[n,m,0]is equal to .
  • For certain special arguments, SpheroidalEigenvalue automatically evaluates to exact values.
  • SpheroidalEigenvalue can be evaluated to arbitrary numerical precision.
  • SpheroidalEigenvalue automatically threads over lists. »

Examples

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Basic Examples  (4)

Evaluate numerically:

Wolfram Language code: SpheroidalEigenvalue[2, 1, 0.5]

Plot over a subset of the reals:

Wolfram Language code: Plot[SpheroidalEigenvalue[1, 0, x], {x, -6, 6}]

Series expansion in the spherical limit as γ approaches 0:

Wolfram Language code: Series[SpheroidalEigenvalue[n, m, γ], {γ, 0, 3}]

Series expansion at Infinity:

Wolfram Language code: Series[SpheroidalEigenvalue[n, m, x], {x, ∞, 2}] // Normal//FullSimplify

Scope  (14)

Numerical Evaluation  (5)

Evaluate numerically:

Wolfram Language code: SpheroidalEigenvalue[5, 0, 0]
Wolfram Language code: SpheroidalEigenvalue[2, 0.5, -5]

Evaluate to high precision:

Wolfram Language code: N[SpheroidalEigenvalue[117, 5, 1], 50]
Wolfram Language code: N[SpheroidalEigenvalue[11, 1, 21], 50]

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

Wolfram Language code: SpheroidalEigenvalue[0.211111111111111111, 1, 5]
Wolfram Language code: SpheroidalEigenvalue[1, 0.211111111111111111, 5]

Complex number inputs:

Wolfram Language code: N[SpheroidalEigenvalue[237, 5 - I, 2]]

Evaluate efficiently at high precision:

Wolfram Language code: SpheroidalEigenvalue[21, 5, 1`100]//Timing
Wolfram Language code: SpheroidalEigenvalue[151, 5, 1`10000];//Timing

Compute the elementwise values of an array using automatic threading:

Wolfram Language code: SpheroidalEigenvalue[2, 0.5, {{1, 2}, {3, 4}}]

Or compute the matrix SpheroidalEigenvalue function using MatrixFunction:

Wolfram Language code: MatrixFunction[SpheroidalEigenvalue[2, 0.5, #]&, {{1, 2}, {3, 4}}]

Specific Values  (7)

Simple exact values are generated automatically:

Wolfram Language code: Table[SpheroidalEigenvalue[1, m, π / 2], {m, {1, 3}}]//FullSimplify

Evaluate symbolically for integer parameters:

Wolfram Language code: SpheroidalEigenvalue[1, 2, γ]//FullSimplify

Evaluate symbolically for half-integer parameters:

Wolfram Language code: SpheroidalEigenvalue[3 / 2, 1 / 2, γ]//FullSimplify

Find the maximum of SpheroidalEigenvalue[1,2/3,x]:

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

SpheroidalEigenvalue evaluates exactly if m=1 and γ=n π/2:

Wolfram Language code: Table[SpheroidalEigenvalue[n, 1, n π / 2], {n, 1, 3}]//FullSimplify

SpheroidalEigenvalue threads elementwise over lists:

Wolfram Language code: SpheroidalEigenvalue[5, {0, 1, 2, 3, 4, 5}, 0.5]

TraditionalForm formatting:

Wolfram Language code: SpheroidalEigenvalue[n, m, c]//TraditionalForm

Visualization  (2)

Plot the SpheroidalEigenvalue function for integer orders:

Wolfram Language code: Plot[{SpheroidalEigenvalue[1, 1, x], SpheroidalEigenvalue[2, 1, x], SpheroidalEigenvalue[3, 1, x]}, {x, -10, 10}]

Plot the real part of TemplateBox[{2, 1, {x, +, {i, , y}}}, SpheroidalEigenvalue]:

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

Plot the imaginary part of TemplateBox[{2, 1, {x, +, {i, , y}}}, SpheroidalEigenvalue]:

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

Applications  (3)

Solve the spheroidal differential equation:

Wolfram Language code: DSolve[(1 - x ^ 2)y''[x] - 2x y'[x] + (SpheroidalEigenvalue[n, m, c] + c ^ 2(1 - x ^ 2) - (m ^ 2/1 - x ^ 2))y[x] == 0, y, x]

Solve this spheroidal-type differential equation:

Wolfram Language code: DSolve[(a^2 r^2 z^-2 + 2 r SpheroidalEigenvalue[ν, μ, γ]) w[z] - ((-1 + r + 2 s + a^2 (1 + r - 2 s) z^2 r) Derivative[1][w][z]/z) + (1 - a^2 z^2 r) Derivative[2][w][z] == 0, w[z], z]

Find a branch point of SpheroidalEigenvalue:

Wolfram Language code: FindRoot[SpheroidalEigenvalue[2, 0, c] == SpheroidalEigenvalue[0, 0, c], {c, 2 + 3 I}, WorkingPrecision -> 30, AccuracyGoal -> 6]
Wolfram Language code: SpheroidalEigenvalue[2, 0, c] - SpheroidalEigenvalue[0, 0, c] /. %

Properties & Relations  (1)

For half-integer values, the SpheroidalEigenvalue reduces to the MathieuCharacteristicA function:

Wolfram Language code: -(1/4) - (x^2/2) + MathieuCharacteristicA[1, (x^2/4)] == SpheroidalEigenvalue[1 / 2, 1 / 2, x]

Possible Issues  (1)

SpheroidalEigenvalue does not evaluate for half-integer or for generic :

Wolfram Language code: SpheroidalEigenvalue[(1/2), 0, 0.1]

The half-integer values of are singular for the near-spherical expansion:

Wolfram Language code: SeriesCoefficient[SpheroidalEigenvalue[n, m, c], {c, 0, 6}]
Wolfram Research (2007), SpheroidalEigenvalue, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalEigenvalue.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_spheroidaleigenvalue, author="Wolfram Research", title="{SpheroidalEigenvalue}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SpheroidalEigenvalue.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_spheroidaleigenvalue, organization={Wolfram Research}, title={SpheroidalEigenvalue}, year={2007}, url={https://reference.wolfram.com/language/ref/SpheroidalEigenvalue.html}, note=[Accessed: 13-June-2026]}