Hankel H (1) in terms of csc and Bessel J

Theorem

The following formula holds: $$H_{\nu}^{(1)}(z)=i \csc(\nu \pi) \left[ e^{-\nu \pi i} J_{\nu}(z)-J_{-\nu}(z) \right],$$ where $H_{\nu}^{(1)}$ denotes the Hankel function of the first kind, $\csc$ denotes the cosecant, $e^{-\nu \pi i}$ denotes the exponential, $\pi$ denotes pi, $i$ denotes the imaginary number, and $J_{\nu}$ denotes the Bessel function of the first kind.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 9.1.3