Lommel function - Alchetron, The Free Social Encyclopedia

The Lommel differential equation is an inhomogeneous form of the Bessel differential equation:

z 2 d 2 y d z 2 + z d y d z + ( z 2 − ν 2 ) y = z μ + 1 .

Two solutions are given by the Lommel functions s_μ,ν(z) and S_μ,ν(z), introduced by Eugen von Lommel (1880),

s μ , ν ( z ) = π 2 [ Y ν ( z ) ∫ 0 z x μ J ν ( x ) d x − J ν ( z ) ∫ 0 z x μ Y ν ( x ) d x ] S μ , ν ( z ) = s μ , ν ( z ) + 2 μ − 1 Γ ( μ + ν + 1 2 ) Γ ( μ − ν + 1 2 ) ( sin ⁡ [ ( μ − ν ) π 2 ] J ν ( z ) − cos ⁡ [ ( μ − ν ) π 2 ] Y ν ( z ) )

where J_ν(z) is a Bessel function of the first kind, and Y_ν(z) a Bessel function of the second kind.

References

Lommel function Wikipedia

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