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LommelS1[m,n,z]

gives the Lommel function of the first kind .

Details
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Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
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Function Properties  
Differentiation  
Integration  
Series Expansions  
Integral Transforms  
Function Representations  
Applications  
Properties & Relations  
Neat Examples  
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LommelS1

LommelS1[m,n,z]

gives the Lommel function of the first kind .

Details

  • LommelS1 is also known as a Lommel function of the first kind.
  • Lommel functions are typically used to represent particular solutions of Bessel-type differential equations.
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • satisfies the differential equation . »
  • LommelS1[m,n,z] has a branch cut discontinuity in the complex plane running from to .
  • For certain special arguments, LommelS1 automatically evaluates to exact values.
  • LommelS1 can be evaluated to arbitrary numerical precision.
  • LommelS1 automatically threads over lists.

Examples

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Basic Examples  (5)

Evaluate numerically:

Wolfram Language code: LommelS1[-0.25, 2 / 3, 1]

Evaluate symbolically:

Wolfram Language code: LommelS1[-1, n, z]

Plot :

Wolfram Language code: ReImPlot[LommelS1[-1 / 4, 2 / 3, x], {x, -25, 25}]

Plot over a subset of the complexes:

Wolfram Language code: ComplexPlot3D[LommelS1[3 / 5, 1 / 2, z], {z, -1 - I, 1 + I}, Rule[...]]

Expand LommelS1 in a Taylor series at the origin:

Wolfram Language code: Series[LommelS1[m, n, z], {z, 0, 3}]

Scope  (30)

Numerical Evaluation  (4)

Evaluate numerically to high precision:

Wolfram Language code: N[LommelS1[2, 0, 4], 50]

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

Wolfram Language code: LommelS1[2, 0, 4.000000000000000000000000]

Evaluate for complex arguments and parameters:

Wolfram Language code: LommelS1[7 / 3 + I, 2I, 4.5 - I]

LommelS1 threads elementwise over lists in the last argument:

Wolfram Language code: LommelS1[1 / 2, 1 / 3, {0.5, 1.0, 1.5}]

Compute the matrix LommelS1 function using MatrixFunction:

Wolfram Language code: MatrixFunction[LommelS1[1, -1, #]&, {{1 / 2, 0}, {0, 1 / 2}}]

Specific Values  (3)

LommelS1 automatically evaluates to simpler functions for certain parameters:

Wolfram Language code: LommelS1[-1, n, z]
Wolfram Language code: LommelS1[-1, 1, z]
Wolfram Language code: LommelS1[0, n, z]
Wolfram Language code: LommelS1[n, n, z]

The exact value of LommelS1 at unity for a specific set of parameters:

Wolfram Language code: LommelS1[2, 6, 1]//FunctionExpand//FullSimplify

LommelS1 is singular whenever or is a non-positive odd integer:

Wolfram Language code: LommelS1[2, 5, z]

Visualization  (3)

Plot the LommelS1 function for and varying :

Wolfram Language code: ReImPlot[Table[LommelS1[m, 1 / 2, x], {m, 5}]//Evaluate, {x, -8, 8}]

Plot the LommelS1 function for and varying :

Wolfram Language code: ReImPlot[Table[LommelS1[1 / 2, n, x], {n, 5}]//Evaluate, {x, 0, 20}]

Plot the real part of :

Wolfram Language code: ComplexContourPlot[Re[LommelS1[4 / 3, 1 / 2, z]], {z, -4 - 4I, 4 + 4I}]

Plot the imaginary part of :

Wolfram Language code: ComplexContourPlot[Im[LommelS1[4 / 3, 1 / 2, z]], {z, -4 - 4I, 4 + 4I}]

Function Properties  (3)

Real domain of LommelS1:

Wolfram Language code: FunctionDomain[LommelS1[m, n, z], z]//Simplify

Complex domain of LommelS1:

Wolfram Language code: FunctionDomain[LommelS1[m, n, z], z, Complexes]

Singularity and discontinuity information of LommelS1:

Wolfram Language code: FunctionSingularities[LommelS1[m, n, z], z]
Wolfram Language code: FunctionDiscontinuities[LommelS1[m, n, z], z]

has a branch cut discontinuity for :

Wolfram Language code: FunctionDiscontinuities[LommelS1[1 / 2, 1 / 3, z], z]

is a singular point for :

Wolfram Language code: FunctionSingularities[LommelS1[2, 1 / 3, z], z]

Differentiation  (2)

First derivative:

Wolfram Language code: D[LommelS1[m, n, z], z]

Higher derivatives:

Wolfram Language code: derivs = Table[D[LommelS1[m, n, x], {x, k}], {k, 1, 3}]

Plot the higher derivatives for , :

Wolfram Language code: ReImPlot[Evaluate[derivs /. {m -> 2, n -> 0}], {x, -10, 10}, Rule[...]]

Integration  (4)

Indefinite integral of LommelS1:

Wolfram Language code: Integrate[LommelS1[m, n, x], x]

Definite integral of LommelS1:

Wolfram Language code: Integrate[LommelS1[m, n, x], {x, 0, 1}]

Integral involving a power function:

Wolfram Language code: Integrate[x^α - 1LommelS1[m, n, x], x]

More integrals with LommelS1:

Wolfram Language code: NIntegrate[x^2 LommelS1[1 / 2, 1 / 3, x], {x, 0, 1}]
Wolfram Language code: NIntegrate[Sin[Sqrt[x]]^2 LommelS1[1 / 2, 1 / 3, x], {x, -1, 1 / 3}]

Series Expansions  (3)

Taylor expansion for LommelS1:

Wolfram Language code: Series[LommelS1[m, n, x], {x, 0, 3}]

Approximations for around :

Wolfram Language code: terms = Normal@Table[Series[LommelS1[1 / 2, 3, x], {x, 0, m}], {m, 2, 6, 2}]
Wolfram Language code: ReImPlot[{LommelS1[1 / 2, 2, x], terms}, {x, -2, 2}]

Expand LommelS1 in a series around :

Wolfram Language code: Series[LommelS1[m, n, x], {x, ∞, 1}]

Series expansion at a generic point:

Wolfram Language code: Series[LommelS1[m, n, z], {z, z0, 2}]

Integral Transforms  (5)

Compute the Laplace transform using LaplaceTransform:

Wolfram Language code: LaplaceTransform[LommelS1[m, n, -t], t, z]

Compute the Fourier transform of LommelS1:

Wolfram Language code: FourierTransform[LommelS1[m, n, -t], t, w, GenerateConditions -> True, FourierParameters -> {a, b}]

FourierCosTransform:

Wolfram Language code: FourierCosTransform[LommelS1[m, n, -t], t, w]

FourierSinTransform:

Wolfram Language code: FourierSinTransform[LommelS1[m, n, -t], t, w]

HankelTransform:

Wolfram Language code: HankelTransform[LommelS1[m, n, r], r, s]

MellinTransform:

Wolfram Language code: MellinTransform[LommelS1[m, n, z], z, s]

Function Representations  (3)

Relation to the HypergeometricPFQ function:

Wolfram Language code: LommelS1[m, n, z] == (z^1 + m HypergeometricPFQ[{1}, {(3/2) + (m/2) - (n/2), (3/2) + (m/2) + (n/2)}, -(z^2/4)]/(m + 1)^2 - n^2)//FullSimplify

LommelS1 can be represented in terms of MeijerG:

Wolfram Language code: MeijerGReduce[LommelS1[m, n, z], z]
Wolfram Language code: Activate[%] == LommelS1[m, n, z]//FullSimplify

FoxH representation of the LommelS1 function:

Wolfram Language code: FoxHReduce[LommelS1[m, n, z], z]

Applications  (1)

Solve the inhomogeneous Bessel equation:

Wolfram Language code: DSolve[z^2 y''[z] + z y'[z] + (z^2 - n^2) y[z] == z^m + 1, y, z]

Properties & Relations  (2)

Use FunctionExpand to expand LommelS1 into hypergeometric functions:

Wolfram Language code: FunctionExpand[LommelS1[m, n, z]]

FunctionExpand for a specific set of parameters can generate simpler special functions:

Wolfram Language code: FunctionExpand[LommelS1[2, 6, z]]

Neat Examples  (1)

Riemann surface of LommelS1:

Wolfram Language code: With[{reIm = Im, μ = 1 / 3, ν = 1 / 2, ε = 1*^-12}, ParametricPlot3D[...]]
Wolfram Research (2025), LommelS1, Wolfram Language function, https://reference.wolfram.com/language/ref/LommelS1.html.

Text

Wolfram Research (2025), LommelS1, Wolfram Language function, https://reference.wolfram.com/language/ref/LommelS1.html.

CMS

Wolfram Language. 2025. "LommelS1." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LommelS1.html.

APA

Wolfram Language. (2025). LommelS1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LommelS1.html

BibTeX

@misc{reference.wolfram_2026_lommels1, author="Wolfram Research", title="{LommelS1}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/LommelS1.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_lommels1, organization={Wolfram Research}, title={LommelS1}, year={2025}, url={https://reference.wolfram.com/language/ref/LommelS1.html}, note=[Accessed: 13-June-2026]}