SpheroidalS2Prime[n,m,γ,z]
gives the derivative with respect to 
of the radial spheroidal function 
of the second kind.
SpheroidalS2Prime
SpheroidalS2Prime[n,m,γ,z]
gives the derivative with respect to 
of the radial spheroidal function 
of the second kind.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, SpheroidalS2Prime automatically evaluates to exact values.
- SpheroidalS2Prime can be evaluated to arbitrary numerical precision.
- SpheroidalS2Prime automatically threads over lists. »
Examples
open all close allBasic Examples (4)
SpheroidalS2Prime[3, 0, 10, 1.2]Plot over a subset of the reals:
Plot[SpheroidalS2Prime[1, 0, 2, x], {x, -6, 6}]Plot over a subset of the complexes:
ComplexPlot3D[SpheroidalS2Prime[1, 0, 2, z], {z, -1 - I, 1 + I}, PlotLegends -> Automatic]Series expansion at a singular point:
Series[SpheroidalS2Prime[1, 1, π / 2, x], {x, 1, 2}]//FullSimplifyScope (24)
Numerical Evaluation (5)
SpheroidalS2Prime[5, 1, .1, .3]SpheroidalS2Prime[1, 0.5, -5, 5]N[SpheroidalS2Prime[2, 1, 1 / 3, -11], 20]The precision of the output tracks the precision of the input:
SpheroidalS2Prime[2, 2, 1 / 3, 0.211111111111111111]SpheroidalS2Prime[2, 0.211111111111111000111, 5, 6]N[SpheroidalS2Prime[23, 5 - I, 2, I]]Evaluate efficiently at high precision:
SpheroidalS2Prime[31, 5, 1, 2`100]//TimingSpheroidalS2Prime[12, 0, 2.5, 5`1000];//TimingCompute the elementwise values of an array using automatic threading:
SpheroidalS2Prime[1, 0, .2, {{1.3, .5}, {0, 0.2}}]Or compute the matrix SpheroidalS2Prime function using MatrixFunction:
MatrixFunction[SpheroidalS2Prime[1, 0, .2, #]&, {{1.3, .5}, {0, 0.2}}]Specific Values (5)
Simple exact values are generated automatically:
Table[SpheroidalS2Prime[1, 1, π / 2, x], {x, 2, 4, 1 / 2}]Table[SpheroidalS2Prime[1, 1, π / 2, x], {x, {-1, 1}}]//QuietFind the first positive maximum of SpheroidalS2Prime[2,0,5,x]:
xmax = x /. FindRoot[D[SpheroidalS2Prime[2, 0, 5, x], x] == 0, {x, 2}]//ChopPlot[SpheroidalS2Prime[2, 0, 5, x], {x, 1, 5}, Epilog -> Style[Point[{xmax, SpheroidalS2Prime[2, 0, 5, xmax ]}], PointSize[Large], Red]] SpheroidalS2Prime functions become elementary if 
and 
:
Table[SpheroidalS2Prime[n, 1, n π / 2, x], {n, 1, 3}]//FullSimplifyTraditionalForm typesetting:
SpheroidalS2Prime[n, m, c, η]//TraditionalFormVisualization (3)
Plot the SpheroidalS2Prime function for integer orders:
Plot[{SpheroidalS2Prime[1, 0, 2, x], SpheroidalS2Prime[2, 0, 2, x], SpheroidalS2Prime[3, 0, 2, x]}, {x, -7, 7}]Plot the SpheroidalS2Prime function for non-integer parameters:
Plot[{SpheroidalS2Prime[1, 0, 1 / 2, x], SpheroidalS2Prime[2, 0, 1 / 2, x], SpheroidalS2Prime[3, 0, 1 / 2, x]}, {x, 0, 8}]Plot the real part of SpheroidalS2Prime:
ComplexContourPlot[Re[SpheroidalS2Prime[2, 0, 1, z]], {z, -4 - 4 I, 4 + 4 I}, Contours -> 20]Plot the imaginary part of SpheroidalS2Prime:
ComplexContourPlot[Im[SpheroidalS2Prime[2, 0, 1, z]], {z, -4 - 4 I, 4 + 4 I}, Contours -> 20]Function Properties (5)
SpheroidalS2Prime is not an analytic function:
FunctionAnalytic[SpheroidalS2Prime[1, 1, π / 2, x], x]![TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2Prime]](Files/SpheroidalS2Prime.en/5.png "TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2Prime]"){kind=small}
has both singularities and discontinuities for 
:
FunctionSingularities[SpheroidalS2Prime[1, 2, π / 2, x], x]//QuietFunctionDiscontinuities[SpheroidalS2Prime[1, 2, π / 2, x], x]//Quiet![TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2Prime]](Files/SpheroidalS2Prime.en/7.png "TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2Prime]"){kind=small}
is neither non-decreasing nor non-increasing:
FunctionMonotonicity[SpheroidalS2Prime[1, 1, π / 2, x], x]![TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2Prime]](Files/SpheroidalS2Prime.en/8.png "TemplateBox[{1, 2, {pi, /, 2}, x}, SpheroidalS2Prime]"){kind=small}
FunctionInjective[SpheroidalS2Prime[1, 1, π / 2, x], x]Plot[{SpheroidalS2Prime[1, 1, π / 2, x], .1}, {x, -6, 6}]SpheroidalS2Prime is neither non-negative nor non-positive:
FunctionSign[SpheroidalS2Prime[1, 1, π / 2, x], x]SpheroidalS2Prime is neither convex nor concave:
FunctionConvexity[SpheroidalS2Prime[1, 1, π / 2, x], x]Differentiation (2)
First derivative with respect to 
:
D[SpheroidalS2Prime[n, m, γ, z], z]Higher derivatives with respect to 
:
Table[D[SpheroidalS2Prime[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"}]Integration (2)
Compute the indefinite integral using Integrate:
Integrate[SpheroidalS2Prime[``n``, ``m``, γ, ``z``], z]FullSimplify[D[%, z]]Integrate[SpheroidalS2Prime[``n``, ``m``, γ, ``z``], {z, 0, 4}]Series Expansions (2)
Find the Taylor expansion using Series:
Series[SpheroidalS2Prime[n, m, γ, x], {x, 0, 3}]//Normal//FullSimplifyPlots of the first three approximations around 
:
terms = Normal@Table[Series[SpheroidalS2Prime[2, 0, 1, x], {x, 2, m}], {m, 1, 5, 2}]//N;
Plot[{SpheroidalS2Prime[2, 0, 1, x], terms}, {x, 0, 10}, PlotRange -> {-100, 100}, MaxRecursion -> 1]Taylor expansion at a generic point:
Series[SpheroidalS2Prime[n, m, γ, x], {x, x0, 2}]// Normal//FullSimplify
Text
Wolfram Research (2007), SpheroidalS2Prime, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalS2Prime.html.
CMS
Wolfram Language. 2007. "SpheroidalS2Prime." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SpheroidalS2Prime.html.
APA
Wolfram Language. (2007). SpheroidalS2Prime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SpheroidalS2Prime.html