LommelS2
LommelS2[m,n,z]
gives the Lommel function of the second kind 
.
Details
- LommelS2 is also known as a Lommel function of the second kind.
- Lommel functions are typically used to represent particular solutions of Bessel-type differential equations.
- Mathematical function, suitable for both symbolic and numerical manipulation.
- LommelS2[m,n,z] has a branch cut discontinuity in the complex
plane running from 
to 
. - For certain special arguments, LommelS2 automatically evaluates to exact values.
- LommelS2 can be evaluated to arbitrary numerical precision.
- LommelS2 automatically threads over lists.
Examples
open all close allBasic Examples (5)
LommelS2[-0.25, 2 / 3, 1]LommelS2[-1, n, z]
ReImPlot[LommelS2[1 / 4, 1 / 3 + I, x], {x, -5, 5}]Plot over a subset of the complexes:
ComplexPlot3D[LommelS2[3 / 5, 1 / 2, z], {z, -10 - 10I, 10 + 10I}, Rule[...]]Expand LommelS2 in a Taylor series at the origin:
Series[LommelS2[m, n, z], {z, 0, 1}]Scope (28)
Numerical Evaluation (4)
Evaluate numerically to high precision:
N[LommelS2[2, 2 / 5, 4], 50]The precision of the output tracks the precision of the input:
LommelS2[2, 2 / 5, 4.000000000000000000000000]Evaluate for complex arguments and parameters:
LommelS2[7 / 3 + I, 1.2, 4.5 - I]LommelS2 threads elementwise over lists in the last argument:
LommelS2[1 / 2, 1 / 3, {0.5, 1.0, 1.5}]Compute the matrix LommelS2 function using MatrixFunction:
MatrixFunction[LommelS2[1, 1 / 2, #]&, {{1 / 2, 0}, {0, 1 / 2}}]//FullSimplifySpecific Values (2)
LommelS2 automatically evaluates to simpler functions for certain parameters:
LommelS2[-1, n, z]LommelS2[0, n, z]LommelS2[n, n, z]LommelS2[2, 5, z]The exact value of LommelS2 at unity for a specific set of parameters:
LommelS2[-1 / 2, 6 / 5, 1]//SimplifyVisualization (3)
Plot the LommelS2 function for 
and varying 
:
ReImPlot[Table[LommelS2[m, 1 / 2, x], {m, 5}]//Evaluate, {x, 0, 8}]Plot the LommelS2 function for 
and varying 
:
ReImPlot[Table[LommelS2[m, 1 / 2, x], {m, 5}]//Evaluate, {x, -8, 4}]
ComplexContourPlot[Re[LommelS2[4 / 3, 1 / 2, z]], {z, -4 - 4I, 4 + 4I}]
ComplexContourPlot[Im[LommelS2[4 / 3, 1 / 2, z]], {z, -4 - 4I, 4 + 4I}]Function Properties (3)
Real domain of LommelS2:
FunctionDomain[LommelS2[m, n, z], z]//SimplifyComplex domain of LommelS2:
FunctionDomain[LommelS2[m, n, z], z, Complexes]Singularity and discontinuity information of LommelS2:
FunctionSingularities[LommelS2[m, n, z], z]FunctionDiscontinuities[LommelS2[m, n, z], z]
has a branch cut discontinuity for 
:
FunctionDiscontinuities[LommelS2[1 / 2, 1 / 3, z], z]
FunctionSingularities[LommelS2[2, 3, z], z]Differentiation (2)
Integration (4)
Indefinite integral of LommelS2:
Integrate[LommelS2[m, n, x], x]Definite integral of LommelS2:
Integrate[LommelS2[m, n, x], {x, 0, 1}]Integral involving a power function:
Integrate[x^α - 1LommelS2[1 / 2, 1 / 3, x], x]//FullSimplifyMore integrals with LommelS2:
NIntegrate[x^2 LommelS2[1 / 2, 1 / 3, x], {x, 0, 1}]NIntegrate[Sin[Sqrt[x]]^2 LommelS2[1 / 2, 1 / 3, x], {x, -1 / 2, 1 / 3}]Series Expansions (3)
Taylor expansion for LommelS2:
Series[LommelS2[m, n, x], {x, 0, 3}]
terms = Normal@Table[Series[LommelS2[7 / 2, 3 / 4, x], {x, 0, m}], {m, 1, 3}]//FullSimplifyReImPlot[{LommelS2[7 / 2, 3 / 4, x], terms}, {x, -2, 2}]Expand LommelS2 in a series around 
:
Series[LommelS2[6, 1 / 3, x], {x, ∞, 1}]//FullSimplifySeries expansion at a generic point:
Series[LommelS2[m, n, z], {z, z0, 2}]Integral Transforms (3)
Compute the Laplace transform using LaplaceTransform:
LaplaceTransform[LommelS2[m, n, -t], t, z]HankelTransform[LommelS2[m, n, r], r, s]MellinTransform[LommelS2[m, n, z], z, s]Function Representations (4)
Relation to the HypergeometricPFQ and BesselJ functions:
LommelS2[m, n, z] == -(2^-1 + m π^2 BesselJ[n, z] Csc[((1 + m/2) - (n/2)) π] Csc[n π]/Gamma[(1/2) - (m/2) - (n/2)] Gamma[(1/2) - (m/2) + (n/2)]) + (2^-1 + m π^2 BesselJ[-n, z] Csc[((1 + m/2) + (n/2)) π] Csc[n π]/Gamma[(1/2) - (m/2) - (n/2)] Gamma[(1/2) - (m/2) + (n/2)]) + (1/(1 + m - n) (1 + m + n))z^1 + m Cos[(m π/2) - (n π/2)] Cos[(m π/2) + (n π/2)] Csc[((1/2) (-1 - m) - (n/2)) π] Csc[((1/2) (-1 - m) + (n/2)) π] HypergeometricPFQ[{1}, {(3/2) + (m/2) - (n/2), (3/2) + (m/2) + (n/2)}, -(z^2/4)]//FullSimplifyRelation to the HypergeometricPFQ and Hypergeometric0F1 functions:
LommelS2[m, n, z] == (z^1 + m HypergeometricPFQ[{1}, {(3/2) + (m/2) - (n/2), (3/2) + (m/2) + (n/2)}, -(z^2/4)]/(1 + m - n) (1 + m + n)) + 2^m + n - 1z^-n(Gamma[n]Gamma[(1/2)(m + n + 1)]/Gamma[(1/2)(-m + n + 1)])Hypergeometric0F1[1 - n, -(z^2/4)] + 2^m - n - 1z^n(Gamma[-n]Gamma[(1/2)(m - n + 1)]/Gamma[(1/2)(-m - n + 1)])Hypergeometric0F1[1 + n, -(z^2/4)]//FullSimplifyLommelS2 can be represented in terms of MeijerG:
MeijerGReduce[LommelS2[m, n, z], z]%//Activate//FunctionExpand% == LommelS2[m, n, z]//FullSimplifyFoxH representation of the LommelS2 function:
FoxHReduce[LommelS2[m, n, z], z]Properties & Relations (2)
Use FunctionExpand to expand LommelS2 into hypergeometric and Bessel functions:
FunctionExpand[LommelS2[m, n, z]]FunctionExpand for a specific set of parameters can generate simpler special functions:
FunctionExpand[LommelS2[2, 1 / 3, z]]//SimplifyNeat Examples (1)
Riemann surface of LommelS2:
With[{reIm = Im, μ = 1 / 3, ν = 1 / 2, ε = 1*^-12},
ParametricPlot3D[...]]Text
Wolfram Research (2025), LommelS2, Wolfram Language function, https://reference.wolfram.com/language/ref/LommelS2.html.
CMS
Wolfram Language. 2025. "LommelS2." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LommelS2.html.
APA
Wolfram Language. (2025). LommelS2. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LommelS2.html