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==Power series expansion==
The '''Anger function''' has the power series expansion<ref name=DLMF>{{dlmf|id=11.10|title=Anger-Weber Functions|first=R. B. |last=Paris}}</ref>
:<math>\mathbf{J}_\nu(z)=\sum_{k=0}^\infty\frac{\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sum_{k=0}^\infty\frac{\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}</math>
While the '''Weber function''' has the power series expansion<ref name=DLMF/>
:<math>\mathbf{E}_\nu(z)=\sum_{k=0}^\infty\frac{\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}-\sum_{k=0}^\infty\frac{\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}</math>
==Differential equations==
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