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Let $n$ be an integer and consider the Bessel function of order $n$

$J_n(z)=\frac{1}{2\pi i} \int_{|u|=1} e^{\frac{z}{2}(u-\frac{1}{u})}\frac{du}{u^{n+1}}$

This satisfies the linear differential equation

$\frac{d^2y}{dz}+\frac{1}{z}\frac{dy}{dz}+(1-\frac{n^2}{z^2})y=0$.

Now, there is a second fundamental solution to this equation (Bessel function of the second kind). My question is: does it also has an integral representation of the form

$\int_C e^{\frac{z}{2}(u-\frac{1}{u})}\frac{du}{u^{n+1}}$

for a suitable chosen (non-closed) contour $C$ in the complex plane?

I'm a bit lost with the literature; I found many integral representation but none with exactly the same integrand as in the definition of $J_n(z)$.

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Watson's Treatise has the contour integral formulas you are after in §6.21, pp. 178–179:

Schläfli formulas Schläfli contours

(This expresses the "Hankel" linear combinations $H_n^{(1,2)}=J_n\pm iY_n$ of Bessel functions of the 1st and 2nd kind, hence also indirectly $J_n$ and $Y_n=\frac1{2i}(H_n^{(1)}-H_n^{(2)})$.)

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