User:Dvaselaar/Lommel-Weber function

The Lommel-Weber function $$\Omega_n(x)$$ is defined by


 * $$ \Omega_n(x) = \frac{1}{\pi} \int_{0}^{\pi} \sin(x \sin\phi - n\phi) d\phi$$.

It can be found in problems such as calculating the current in a thin wire circular loop. Also simply called a Weber function, although this name is conflicting with that sometimes given to the Bessel function of the second kind. Notable is its inclusion in the Bessel function of the second kind in integral form with integer order as



\begin{alignat}{2} Y_n(x) & =  \frac{1}{\pi} \int_{0}^{\pi} \sin(x \sin\phi - n\phi) d\phi - \frac{1}{\pi} \int_{0}^{\infty} \left[ e^{n t} + (-1)^n e^{-n t} \right] e^{-x \sinh t} dt. \\ & = \Omega_n(x) - \frac{1}{\pi} \int_{0}^{\infty} \left[ e^{n t} + (-1)^n e^{-n t} \right] e^{-x \sinh t} dt. \end{alignat} $$

Properties
If $$n$$ is an even integer the Lommel-Weber function can be expressed as


 * $$\Omega_n(x) = \frac{2}{\pi} \int_{0}^{\pi/2} \sin(x \sin\phi)cos(n\phi) d\phi $$      (for even $$n$$).