LommelT2
LommelT2[m,n,z]
gives the modified Lommel function of the second kind 
.
Details
- LommelT2 is also known as a modified 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.
- LommelT2[m,n,z] has a branch cut discontinuity in the complex
plane running from 
to 
. - For certain special arguments, LommelT2 automatically evaluates to exact values.
- LommelT2 can be evaluated to arbitrary numerical precision.
- LommelT2 automatically threads over lists.
Examples
open all close allBasic Examples (4)
LommelT2[-0.25, 2 / 3, 1]
ReImPlot[LommelT2[-1 / 4, 2 / 3, x], {x, -3, 5}]Plot over a subset of the complexes:
ComplexPlot3D[LommelT2[3 / 5, 1 / 2, z], {z, -3 - 3I, 3 + 3I}, Rule[...]]Expand LommelT2 in a Taylor series at the origin:
Series[LommelT2[m, n, z], {z, 0, 1}]Scope (26)
Numerical Evaluation (4)
Evaluate numerically to high precision:
N[LommelT2[2, 3 / 2, 4], 50]The precision of the output tracks the precision of the input:
LommelT2[2, 3 / 2, 4.000000000000000000000000]Evaluate for complex arguments and parameters:
LommelT2[7 / 3 + I, 1.2, 4.5 - I]LommelS2 threads elementwise over lists in the last argument:
LommelT2[1 / 2, 1 / 3, {0.5, 1.0, 1.5}]Compute the matrix LommelT2 function using MatrixFunction:
MatrixFunction[LommelT2[1, 1 / 2, #]&, {{1 / 2, 0}, {0, 1 / 2}}]//FullSimplifySpecific Values (2)
LommelT2 automatically evaluates to simpler functions for certain parameters:
LommelT2[2, 2, z]LommelT2[n, n, z]LommelT2[2, 5, z]The exact value of LommelT2 at unity for a specific set of parameters:
LommelT2[2 / 5, 4 / 3, 1]//SimplifyVisualization (3)
Plot the LommelT2 function for 
and varying 
:
ReImPlot[Table[LommelT2[m, 1 / 2, x], {m, 5}]//Evaluate, {x, -4, 8}]Plot the LommelT2 function for 
and varying 
:
ReImPlot[Table[LommelT2[m, 1 / 3, x], {m, 5}]//Evaluate, {x, 0, 8}]
ComplexContourPlot[Re[LommelT2[4 / 3, 1 / 2, z]], {z, -4 - 4I, 4 + 4I}]
ComplexContourPlot[Im[LommelT2[4 / 3, 1 / 2, z]], {z, -4 - 4I, 4 + 4I}]Function Properties (3)
Real domain of LommelT2:
FunctionDomain[LommelT2[m, n, z], z]//SimplifyComplex domain of LommelT2:
FunctionDomain[LommelT2[m, n, z], z, Complexes]Singularity and discontinuity information of LommelT2:
FunctionSingularities[LommelT2[m, n, z], z]FunctionDiscontinuities[LommelT2[m, n, z], z]
has a branch cut discontinuity for 
:
FunctionDiscontinuities[LommelT2[1 / 2, 1 / 3, z], z]
FunctionSingularities[LommelT2[2, 3, z], z]Differentiation (2)
Integration (4)
Indefinite integral of LommelT2:
Integrate[LommelT2[m, n, x], x]Definite integral of LommelT2:
Integrate[LommelT2[m, n, x], {x, 0, 1}]Integral involving a power function:
Integrate[x^α - 1LommelT2[1 / 2, 1 / 3, x], x]//SimplifyMore integrals with LommelT2:
NIntegrate[x^2 LommelT2[1 / 2, 1 / 3, x], {x, 0, 1}]NIntegrate[Sin[Sqrt[x]]^2 LommelT2[1 / 2, 1 / 3, x], {x, -1 / 2, 1 / 3}]Series Expansions (3)
Taylor expansion for LommelT2:
Series[LommelT2[m, n, x], {x, 0, 1}]
terms = Normal@Table[Series[LommelT2[1 / 2, 3 / 4, x], {x, 0, m}], {m, 1, 3}]//FullSimplifyReImPlot[{LommelT2[1 / 2, 3 / 4, x], terms}, {x, -2, 2}]Expand LommelT2 in a series around 
:
Series[LommelT2[6, 1 / 2, x], {x, ∞, 1}]//FullSimplifySeries expansion at a generic point:
Series[LommelT2[m, n, z], {z, z0, 2}]Integral Transforms (1)
HankelTransform[LommelT2[m, n, r], r, s ]Function Representations (4)
Relation to the HypergeometricPFQ and BesselI functions:
LommelT2[m, n, z] == -((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)]) / ((1 + m - n) (1 + m + n))) - (2^-1 + m π^2 BesselI[n, z] Csc[((1 + m/2) - (n/2)) π] Csc[n π] Sin[(m π/2) - (n π/2)]/Gamma[(1/2) - (m/2) - (n/2)] Gamma[(1/2) - (m/2) + (n/2)]) + (2^-1 + m π^2 BesselI[-n, z] Csc[((1 + m/2) + (n/2)) π] Csc[n π] Sin[(m π/2) + (n π/2)]/Gamma[(1/2) - (m/2) - (n/2)] Gamma[(1/2) - (m/2) + (n/2)])//FullSimplifyRelation to the HypergeometricPFQ and Hypergeometric0F1 functions:
LommelT2[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)) - z^n(2^-1 + m - n π^2 Csc[n π] Tan[(1/2) (m - n) π]/Gamma[(1/2) (1 - m - n)] Gamma[1 + n] Gamma[(1/2) (1 - m + n)])Hypergeometric0F1[n + 1, z^2 / 4] + z^-n(2^-1 + m + n π^2 Csc[n π] Tan[(1/2) (m + n) π]/Gamma[1 - n] Gamma[(1/2) (1 - m - n)] Gamma[(1/2) (1 - m + n)])Hypergeometric0F1[1 - n, z^2 / 4]//FullSimplifyLommelT2 can be represented in terms of MeijerG:
MeijerGReduce[LommelT2[m, n, z], z]Activate[%]//FunctionExpand% == LommelT2[m, n, z]//FullSimplifyFoxH representation of the LommelT2 function:
FoxHReduce[ LommelT2[m, n, z], z]Properties & Relations (2)
Use FunctionExpand to expand LommelT2 into hypergeometric and Bessel functions:
FunctionExpand[LommelT2[m, n, z]]FunctionExpand for a specific set of parameters can generate simpler special functions:
FunctionExpand[LommelT2[2, 1 / 3, z]]//SimplifyNeat Examples (1)
Riemann surface of LommelT2:
With[{reIm = Im, μ = 1 / 3, ν = 1 / 2, ε = 1*^-12},
ParametricPlot3D[...]]Text
Wolfram Research (2025), LommelT2, Wolfram Language function, https://reference.wolfram.com/language/ref/LommelT2.html.
CMS
Wolfram Language. 2025. "LommelT2." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LommelT2.html.
APA
Wolfram Language. (2025). LommelT2. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LommelT2.html