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AngerJ

The Anger function

WeberE

The Weber function

 Calling Sequence AngerJ(v, x) WeberE(v, x)

Parameters

 v - algebraic expression (the order or index) x - algebraic expression (the argument)

Description

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

${x}^{2}y\text{'}\text{'}+xy\text{'}+\left({x}^{2}-{v}^{2}\right)y=\frac{\left(x-v\right)\mathrm{sin}\left(v\mathrm{\pi }\right)}{\mathrm{\pi }}$

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

${x}^{2}y\text{'}\text{'}+xy\text{'}+\left({x}^{2}-{v}^{2}\right)y=\frac{\left(v-x\right)\mathrm{cos}\left(v\mathrm{\pi }\right)-\left(v+x\right)}{\mathrm{\pi }}$

Examples

 > $\mathrm{AngerJ}\left(0,0\right)$
 ${1}$ (1)
 > $\mathrm{evalf}\left(\mathrm{AngerJ}\left(4,3\right)\right)$
 ${0.1320341839}$ (2)
 > $\mathrm{WeberE}\left(1.5-I,2.6+3I\right)$
 ${4.135197692}{-}{11.77785498}{}{I}$ (3)
 > $\mathrm{series}\left(\mathrm{AngerJ}\left(\frac{1}{3},x\right),x,4\right)$
 $\frac{{3}{}\sqrt{{3}}}{{2}{}{\mathrm{\pi }}}{+}\frac{{9}}{{16}}{}\frac{\sqrt{{3}}}{{\mathrm{\pi }}}{}{x}{-}\frac{{27}}{{70}}{}\frac{\sqrt{{3}}}{{\mathrm{\pi }}}{}{{x}}^{{2}}{-}\frac{{81}}{{1280}}{}\frac{\sqrt{{3}}}{{\mathrm{\pi }}}{}{{x}}^{{3}}{+}{O}{}\left({{x}}^{{4}}\right)$ (4)
 > $\mathrm{series}\left(\mathrm{WeberE}\left(\frac{1}{2},x\right),x,6\right)$
 $\frac{{2}}{{\mathrm{\pi }}}{+}\frac{{4}}{{3}}{}\frac{{1}}{{\mathrm{\pi }}}{}{x}{-}\frac{{8}}{{15}}{}\frac{{1}}{{\mathrm{\pi }}}{}{{x}}^{{2}}{-}\frac{{16}}{{105}}{}\frac{{1}}{{\mathrm{\pi }}}{}{{x}}^{{3}}{+}\frac{{32}}{{945}}{}\frac{{1}}{{\mathrm{\pi }}}{}{{x}}^{{4}}{+}\frac{{64}}{{10395}}{}\frac{{1}}{{\mathrm{\pi }}}{}{{x}}^{{5}}{+}{O}{}\left({{x}}^{{6}}\right)$ (5)
 > $\mathrm{diff}\left(\mathrm{AngerJ}\left(v,x\right),x\right)$
 ${-}{\mathrm{AngerJ}}{}\left({v}{+}{1}{,}{x}\right){+}\frac{{v}{}{\mathrm{AngerJ}}{}\left({v}{,}{x}\right)}{{x}}{-}\frac{{\mathrm{sin}}{}\left({v}{}{\mathrm{\pi }}\right)}{{\mathrm{\pi }}{}{x}}$ (6)
 > $\mathrm{diff}\left(\mathrm{WeberE}\left(v,x\right),x\right)$
 ${-}{\mathrm{WeberE}}{}\left({v}{+}{1}{,}{x}\right){+}\frac{{v}{}{\mathrm{WeberE}}{}\left({v}{,}{x}\right)}{{x}}{-}\frac{{1}{-}{\mathrm{cos}}{}\left({v}{}{\mathrm{\pi }}\right)}{{\mathrm{\pi }}{}{x}}$ (7)

References

 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.