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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+1hypergeom1,32+b2+a2,32b2+a2,z24ab+1a+b+1,a+b10a+b+1032b2+a2::¬?32+b2+a2::¬?

(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+2μ1Γμ2ν2+12Γμ2+ν2+12sinμνπ2BesselJν,zcosμνπ2BesselYν,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,zaLommelS1−1,a,zπ

(5)

FunctionAdvisorrelate,WeberE,LommelS1

WeberEa,z=a1cosπaLommelS1−1,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=2μ1Γμ2+ν2+12Γμ2ν2+12MeijerGμ2+12,,μ2+12,ν2,ν2,z24

(9)

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

LommelS2μ,ν,z=MeijerGμ2+12,,μ2+12,ν2,ν2,,z242μ2Γμ2+ν2+12Γμ2ν2+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,μ+ν::¬?oddμν::¬?odd

(12)

SeriesLommelS2μ,ν,z,z,4

zν4νcscπνcosμνπ2Γμ2ν2+12Γμ2+ν2+122μν2Γν+1184νcscπνcosμνπ2Γμ2ν2+12Γμ2+ν2+122μνΓν+2z2+Oz4+zνcscπνcosπμ+ν2Γμ2ν2+12Γμ2+ν2+122μν2Γν+1+18cscπνcosπμ+ν2Γμ2ν2+12Γμ2+ν2+122μνΓν+2z2+Oz4+zμ1μν+1μ+ν+1z1μν+1μν+3μ+ν+1μ+ν+3z3+Oz5,ν::¬?μ2ν2+12::¬?μ2+ν2+12::¬?μ2ν2+32::¬?μ2+ν2+32::¬?

(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