The Anger function AngerJ(v,x) solves the inhomogeneous Bessel equation

The Weber function WeberE(v,x) solves the inhomogeneous Bessel equation

$\mathrm{AngerJ}\left(0\,0\right)$

$1$

$\mathrm{evalf}\left(\mathrm{AngerJ}\left(4\,3\right)\right)$

$0.1320341839$

$\mathrm{WeberE}\left(1.5-I\,2.6\+3I\right)$

$4.135197692-11.77785498\mathrm{I}$

$\mathrm{series}\left(\mathrm{AngerJ}\left(\frac{1}{3}\,x\right)\,x\,4\right)$

$\frac{3\sqrt{3}}{2\mathrm{\pi }}+\frac{9}{16}\frac{\sqrt{3}}{\mathrm{\pi }}x-\frac{27}{70}\frac{\sqrt{3}}{\mathrm{\pi }}{x}^2-\frac{81}{1280}\frac{\sqrt{3}}{\mathrm{\pi }}{x}^3+O\left(x^4\right)$

$\mathrm{series}\left(\mathrm{WeberE}\left(\frac{1}{2}\,x\right)\,x\,6\right)$

$\frac{2}{\mathrm{\pi }}+\frac{4}{3}\frac{1}{\mathrm{\pi }}x-\frac{8}{15}\frac{1}{\mathrm{\pi }}{x}^2-\frac{16}{105}\frac{1}{\mathrm{\pi }}{x}^3+\frac{32}{945}\frac{1}{\mathrm{\pi }}{x}^4+\frac{64}{10395}\frac{1}{\mathrm{\pi }}{x}^5+O\left(x^6\right)$

$\frac{\partial }{\partial x}\mathrm{AngerJ}\left(v\,x\right)$

$-\mathrm{AngerJ}\left(v+1\,x\right)+\frac{v\mathrm{AngerJ}\left(v\,x\right)}{x}-\frac{\sin \left(v\mathrm{\pi }\right)}{\mathrm{\pi }x}$

$\frac{\partial }{\partial x}\mathrm{WeberE}\left(v\,x\right)$

$-\mathrm{WeberE}\left(v+1\,x\right)+\frac{v\mathrm{WeberE}\left(v\,x\right)}{x}-\frac{1-\cos \left(v\mathrm{\pi }\right)}{\mathrm{\pi }x}$

Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover, 1972.

Watson, G.N. A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1966.