Bourget function

The function $J_{n,k}(z)$ which may be defined as a generalization of the integral representation of the Bessel functions

$$J_{n,k}(z)=\frac{1}{2\pi i}\int t^{-n-1}\left(t+\frac1t\right)^k\exp\left\lbrace\frac12z\left(t-\frac1t\right)\right\rbrace dt,$$

where $n$ is an integer and $k$ is a positive integer. The integration contour makes one counter-clockwise turn around the coordinate origin. In other words,

$$J_{n,k}(z)=\frac1\pi\int\limits_0^\pi(2\cos\theta)^k\cos(n\theta-z\sin\theta)d\theta,$$

$J_{n,0}(z)\equiv J_n(z)$ is a cylinder function of the first kind. So named after J. Bourget [1], who studied the function with a view to various applications in astronomy.

References

[1]	J. Bourget, "Mémoire sur les nombres de Cauchy et leur application à divers problèmes de mécanique céleste" J. Math. Pures Appl. (2) , 6 (1861) pp. 32–54
[2]	G.N. Watson, "A treatise on the theory of Bessel functions" , 1 , Cambridge Univ. Press (1952) pp. Chapt. 10