Anger function

Let $\nu \in \mathbb{C}$. The Anger function $\mathbf{J}_{\nu}$ is defined by $$\mathbf{J}_{\nu}(z) = \dfrac{1}{\pi} \displaystyle\int_0^{\pi} \cos(\nu \theta - z \sin(\theta)) \mathrm{d}\theta.$$

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  Graph of $\mathbf{J}_0$.

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  Graph of $\mathbf{J}_{\frac{1}{2}}$.

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  Graph of $\mathbf{J}_{\frac{3}{2}}$.

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  Graph of $\mathbf{J}_2$.

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  Domain coloring of $\mathbf{J}_0$.

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  Domain coloring of $\mathbf{J}_{\frac{1}{2}}$.

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  Domain coloring of $\mathbf{J}_1$.

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  Domain coloring of $\mathbf{J}_{\frac{3}{2}}$.

Properties

Value of Anger at 0
Anger recurrence relation
Anger derivative recurrence
Relationship between Anger function and Bessel J sub nu
Relationship between Weber function and Anger function
Relationship between Anger function and Weber function

See Also

Bessel J
Weber function

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 12.3.1