Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Lommel function

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated September 11, 2026
Image credit is listed at the end of this article.

In mathematics, the Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation

z^{2}{\frac {d^{2}y}{dz^{2}}}+z{\frac {dy}{dz}}+(z^{2}-\nu ^{2})y=z^{\mu +1}.

Its solutions are given by the Lommel functions s_{\mu ,\nu }(z) and S_{\mu ,\nu }(z):

s_{\mu ,\nu }(z)={\frac {\pi }{2}}\left(Y_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }J_{\nu }(x)\,dx-J_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }Y_{\nu }(x)\,dx\right),
S_{\mu ,\nu }(z)=s_{\mu ,\nu }(z)+2^{\mu -1}\Gamma \left({\frac {\mu +\nu +1}{2}}\right)\Gamma \left({\frac {\mu -\nu +1}{2}}\right)\left(\sin \left[(\mu -\nu ){\frac {\pi }{2}}\right]J_{\nu }(z)-\cos \left[(\mu -\nu ){\frac {\pi }{2}}\right]Y_{\nu }(z)\right),

where J_{\nu }(z) is a Bessel function of the first kind and Y_{\nu }(z) a Bessel function of the second kind.

The function s_{\mu ,\nu } can also be written as

s_{\mu ,\nu }(z)={\frac {z^{\mu +1}}{(\mu -\nu +1)(\mu +\nu +1)}}{}_{1}F_{2}{\bigg (}1;{\frac {\mu }{2}}-{\frac {\nu }{2}}+{\frac {3}{2}},{\frac {\mu }{2}}+{\frac {\nu }{2}}+{\frac {3}{2}};-{\frac {z^{2}}{4}}{\bigg )},

where {}_{1}F_{2} is a generalized hypergeometric function.

Watch videos about Lommel functionExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Lommel function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

Anger function

In mathematics, the Anger function, introduced by C. T. Anger, is a function defined as J ν ( z ) = 1 π ∫ 0 π cos ⁡ ( ν θ − z sin ⁡ θ ) d θ {\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta } with complex parameter ν {\displaystyle \nu } and complex variable ⁠ z {\displaystyle {\textit {z}}} ⁠. It is closely related to the Bessel functions.

Lommel polynomial

A Lommel polynomial Rm,ν is a polynomial in 1/z giving the recurrence relation J m + ν ( z ) = J ν ( z ) R m , ν ( z ) − J ν − 1 ( z ) R m − 1 , ν + 1 ( z ) {\displaystyle \displaystyle J_{m+\nu }(z)=J_{\nu }(z)R_{m,\nu }(z)-J_{\nu -1}(z)R_{m-1,\nu +1}(z)} where Jν(z) is a Bessel function of the first kind. They are given explicitly by R m , ν ( z ) = ∑ n = 0 [ m / 2 ] ( − 1 ) n ( m − n ) !

Struve function

In mathematics, the Struve functions Hα, are solutions y(x) of the non-homogeneous Bessel's differential equation: x 2 d 2 y d x 2 + x d y d x + ( x 2 − α 2 ) y = 4 ( x 2 ) α + 1 π Γ ( α + 1 2 ) {\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}} introduced by Hermann Struve (1882). The complex number α is the order of the Struve function, and is often an integer.