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LommelS1

the Lommel function s

LommelS2

the Lommel function S

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

LommelS1(mu, nu, z)

LommelS2(mu, nu, z)

Parameters

mu

-

algebraic expression

nu

-

algebraic expression

z

-

algebraic expression

Description

• 

The LommelS1(mu, nu, z) function is defined in terms of the hypergeometric function

FunctionAdvisor( definition, LommelS1);

LommelS1a,b,z=za+1hypergeom1,3212b+12a,32+12b+12a,14z2ab+1a+b+1,Anda+b10,a+b+10,3212b+12a::Notnonposint,32+12b+12a::Notnonposint

(1)
  

and LommelS2(mu, nu, z) is defined in terms of LommelS1(mu, nu, z) and Bessel functions.

LommelS2(mu,nu,z) = convert(LommelS2(mu,nu,z), LommelS1);

LommelS2μ,ν,z=LommelS1μ,ν,z+2μ1Γ12μ12ν+12Γ12μ+12ν+12sin12μνπBesselJν,zcos12μνπBesselYν,z

(2)
• 

These functions solve the non-homogeneous linear differential equation of second order.

z^2*diff(f(z),`$`(z,2))+z*diff(f(z),z)+(z^2-nu^2)*f(z) = z^(mu+1);

z2ⅆ2ⅆz2fz+zⅆⅆzfz+ν2+z2fz=zμ+1

(3)
  

The Lommel functions also solve the following third order linear homogeneous differential equation with polynomial coefficients.

FunctionAdvisor( DE, LommelS1(mu,nu,z));

fz=LommelS1μ,ν,z,ⅆ3ⅆz3fz=μ2ⅆ2ⅆz2fzz+ν2z2+μⅆⅆzfzz2+μ1z2ν2μ+1fzz3

(4)

Examples

The AngerJ and WeberE, StruveH and StruveL functions can be viewed as particular cases of LommelS1.

FunctionAdvisorrelate,AngerJ,LommelS1

AngerJa,z=sinπaLommelS10,a,zaLommelS11,a,zπ

(5)

FunctionAdvisorrelate,WeberE,LommelS1

WeberEa,z=a1cosπaLommelS11,a,z+1cosπaLommelS10,a,zπ

(6)

FunctionAdvisorrelate,StruveH,LommelS1

StruveHa,z=2LommelS1a,a,zΓa+12π2a

(7)

FunctionAdvisorrelate,StruveL,LommelS1

StruveLa,z=2ILommelS1a,a,IzzaΓa+12π2Iza

(8)

A MeijerG representation for the Lommel functions.

LommelS1μ,ν,z=convertLommelS1μ,ν,z,MeijerG

LommelS1μ,ν,z=2μ1Γ12μ+12ν+12Γ12μ12ν+12MeijerG12μ+12,,12μ+12,12ν,12ν,14z2

(9)

LommelS2μ,ν,z=convertLommelS2μ,ν,z,MeijerG

LommelS2μ,ν,z=12MeijerG12μ+12,,12μ+12,12ν,12ν,,14z22μΓ12μ+12ν+12Γ12μ12ν+12

(10)

The series expansion of the Lommel functions is not computable using the series command because it would involve factoring out abstract powers, leading to a result of the form z^mu1*series_1 + z^mu2*series_2 + .... This type of extended series expansion, however, can be computed using the Series command of the MathematicalFunctions package.

withMathematicalFunctions,Series

Series

(11)

SeriesLommelS1μ,ν,z,z,4

zμ1μν+1μ+ν+1z1μν+1μν+3μ+ν+1μ+ν+3z3+Oz5,Andμ+ν::NotAndnegint,odd,μν::NotAndnegint,odd

(12)

SeriesLommelS2μ,ν,z,z,4

zν124νΓ12μ12ν+12Γ12μ+12ν+12cscπνcos12μνπ2μνΓ1ν184νΓ12μ12ν+12Γ12μ+12ν+12cscπνcos12μνπ2μνΓν+2z2+Oz4+zν12Γ12μ12ν+12Γ12μ+12ν+12cscπνcos12πμ+ν2μνΓν+1+18Γ12μ12ν+12Γ12μ+12ν+12cscπνcos12πμ+ν2μνΓν+2z2+Oz4+zμ1μν+1μ+ν+1z1μν+1μν+3μ+ν+1μ+ν+3z3+Oz5,Andν::Notinteger,12μ12ν+12::Notnonposint,12μ+12ν+12::Notnonposint,12μ12ν+32::Notnonposint,12μ+12ν+32::Notnonposint

(13)

References

  

Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover publications.

  

Gradshteyn, and Ryzhik. Table of Integrals, Series and Products. 5th ed. Academic Press.

  

Luke, Y. The Special Functions and Their Approximations. Vol. 1 Chap. 6.

See Also

AngerJ

FunctionAdvisor

hypergeom

MathematicalFunctions

MeijerG

Struve Functions

WeberE