SpheroidalS2
SpheroidalS2[n,m,γ,z]
gives the radial spheroidal function 
of the second kind.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The radial spheroidal functions satisfy the differential equation
with the spheroidal eigenvalue 
given by SpheroidalEigenvalue[n,m,γ]. - The
are normalized according to the Meixner–Schäfke scheme. - SpheroidalS2 can be evaluated to arbitrary numerical precision.
- SpheroidalS2 automatically threads over lists. »
Examples
open all close allBasic Examples (4)
SpheroidalS2[2, 0, 2.5, 5]Plot over a subset of the reals:
Plot[SpheroidalS2[1, 0, 2, x], {x, -6, 6}]Plot over a subset of the complexes:
ComplexPlot3D[SpheroidalS2[2, 0, 1, z], {z, -1 - I, 1 + I}, PlotLegends -> Automatic]Series expansion at a singular point:
Series[SpheroidalS2[1, 1, π / 2, x], {x, 1, 3}]//FullSimplifyScope (22)
Numerical Evaluation (5)
SpheroidalS2[1, 0, 1.5, 2]SpheroidalS2[1, 0.5, -5, 5]N[SpheroidalS2[7, 5, 1, 2], 50]//ChopN[SpheroidalS2[2, 1, 1 / 3, -11], 20]The precision of the output tracks the precision of the input:
SpheroidalS2[2, 2, 1 / 3, 0.211111111111111111]SpheroidalS2[2, 0.211111111111111000111, 5, 6]N[SpheroidalS2[23, 5 - I, 2, I]]Evaluate efficiently at high precision:
SpheroidalS2[31, 5, 1, 2`100]//TimingSpheroidalS2[12, 0, 2.5, 5`100000];//TimingCompute the elementwise values of an array using automatic threading:
SpheroidalS2[1, 0, .2, {{1.3, .5}, {0, 0.2}}]Or compute the matrix SpheroidalS2 function using MatrixFunction:
MatrixFunction[SpheroidalS2[1, 0, .2, #]&, {{1.3, .5}, {0, 0.2}}]Specific Values (5)
Simple exact values are generated automatically:
Table[SpheroidalS2[1, 1, π / 2, x], {x, 2, 4, 1 / 2}]Table[SpheroidalS2[1, 1, π / 2, x], {x, {-1, 1}}]//QuietFind the first positive maximum of SpheroidalS2[2,0,5,x]:
xmax = x /. FindRoot[D[SpheroidalS2[2, 0, 5, x], x] == 0, {x, 1.5}]//ChopPlot[SpheroidalS2[2, 0, 5, x], {x, 1, 5}, Epilog -> Style[Point[{xmax, SpheroidalS2[2, 0, 5, xmax ]}], PointSize[Large], Red]] SpheroidalS2 functions become elementary if 
and 
:
Table[SpheroidalS2[n, 1, n π / 2, x], {n, 1, 3}]//FullSimplifyTraditionalForm typesetting:
SpheroidalS2[n, m, c, η]//TraditionalFormVisualization (3)
Plot the SpheroidalS2 function for integer orders:
Plot[{SpheroidalS2[1, 0, 2, x], SpheroidalS2[2, 0, 2, x], SpheroidalS2[3, 0, 2, x]}, {x, -10, 10}]Plot the SpheroidalS2 function for non-integer parameters:
Plot[{SpheroidalS2[1, 0, 1 / 2, x], SpheroidalS1[2, 0, 1 / 2, x], SpheroidalS1[3, 0, 1 / 2, x]}, {x, -10, 10}]
ComplexContourPlot[Re[SpheroidalS2[2, 0, 1, z]], {z, -4 - 4 I, 4 + 4 I}, Contours -> 20]
ComplexContourPlot[Im[SpheroidalS2[2, 0, 1, z]], {z, -4 - 4 I, 4 + 4 I}, Contours -> 20]Function Properties (5)
SpheroidalS2 is not an analytic function:
FunctionAnalytic[SpheroidalS2[1, 1, π / 2, x], x]![TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2]](Files/SpheroidalS2.en/9.png "TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2]"){kind=small}
has both singularities and discontinuities for 
:
FunctionSingularities[SpheroidalS2[1, 2, π / 2, x], x]//QuietFunctionDiscontinuities[SpheroidalS2[1, 2, π / 2, x], x]//Quiet![TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2]](Files/SpheroidalS2.en/11.png "TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2]"){kind=small}
is neither non-decreasing nor non-increasing:
FunctionMonotonicity[SpheroidalS2[1, 1, π / 2, x], x]![TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2]](Files/SpheroidalS2.en/12.png "TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2]"){kind=small}
FunctionInjective[SpheroidalS2[1, 1, π / 2, x], x]Plot[{SpheroidalS2[1, 1, π / 2, x], .1}, {x, -6, 6}]SpheroidalS2 is neither non-negative nor non-positive:
FunctionSign[SpheroidalS2[1, 1, π / 2, x], x]SpheroidalS2 is neither convex nor concave:
FunctionConvexity[SpheroidalS2[1, 1, π / 2, x], x]Differentiation (2)
First derivative with respect to 
:
D[SpheroidalS2[n, m, γ, z], z]Higher derivatives with respect to 
:
Table[D[SpheroidalS2[n, m, γ, z], {z, k}], {k, 1, 3}]//FullSimplifyPlot the higher derivatives with respect to 
when 
, 
and 
:
Plot[Evaluate[% /. { n -> 10, m -> 2, γ -> 1 / 3}], {z, -3, 3}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative"}]Series Expansions (2)
Find the Taylor expansion using Series:
Series[SpheroidalS2[n, m, γ, x], {x, 0, 3}]//Normal//FullSimplifyPlots of the first three approximations around 
:
terms = Normal@Table[Series[SpheroidalS2[2, 0, 1 / 2, x], {x, 2, m}], {m, 1, 5, 2}]//N;
Plot[{SpheroidalS2[2, 0, 1 / 2, x], terms}, {x, -5, 5}, PlotRange -> {-100, 100}, MaxRecursion -> 1]Taylor expansion at a generic point:
Series[SpheroidalS2[n, m, γ, x], {x, x0, 2}]// Normal//FullSimplifyApplications (1)
Related Links
MathWorldText
Wolfram Research (2007), SpheroidalS2, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalS2.html.
CMS
Wolfram Language. 2007. "SpheroidalS2." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SpheroidalS2.html.
APA
Wolfram Language. (2007). SpheroidalS2. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SpheroidalS2.html