Sommerfeld integral

An integral representation of the cylinder functions by a contour integral: The Hankel functions of the first kind are given by

$$ H _ \nu ^ {(1)} ( z) = \frac{1} \pi \int\limits _ {C _ {1} } e ^ {i z \cos t } e ^ {i \nu ( t - \pi / 2) } dt , $$

where $ C _ {1} $ is a curve from $ - \eta + i \infty $ to $ \eta - i \infty $, $ 0 \leq \eta \leq \pi $; the Hankel functions of the second kind are given by

$$ H _ \nu ^ {(2)} ( z ) = \frac{1} \pi \int\limits _ {C _ {2} } e ^ {i z \cos t } e ^ {i \nu ( t - \pi /2 ) } dt , $$

where $ C _ {2} $ is a curve from $ \eta - i \infty $ to $ 2 \pi - \eta + i \infty $, $ 0 \leq \eta \leq \pi $; the Bessel functions of the first kind are given by

$$ J _ \nu ( z ) = \frac{1}{2 \pi } \int\limits _ {C _ {3} } e ^ {i z \cos t } e ^ {i \nu ( t - \pi / 2 ) } dt , $$

where $ C _ {3} $ is a curve from $ - \eta + i \infty $ to $ 2 \pi - \eta + i \infty $, $ 0 \leq \eta \leq \pi $. The representation is valid in the domain $ - \eta < \mathop{\rm arg} z < \pi - \eta $, and is named after A. Sommerfeld [1].

References

[1]	A. Sommerfeld, "Mathematische Theorie der Diffraction" Math. Ann. , 47 (1896) pp. 317–374 Zbl 27.0706.03
[2]	E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)
[3]	G.N. Watson, "A treatise on the theory of Bessel functions" , 1–2 , Cambridge Univ. Press (1952)

Comments

The Hankel functions are also called Bessel functions of the first kind.