Consider the function defined by: \begin{equation} G_\mu(z, \psi) = i\int_0^{\infty}\mathrm{d}u\,\mathrm{e}^{-i z \sin( u + \psi) - i \mu u}, \end{equation} where $z$ and $\psi$ are real variables and $\mu$ a non-integer complex constant. The function is obviously periodic in $\psi$ of period $2\pi$, so: \begin{equation} G_\mu(z, \psi) = G_\mu(z, \psi + 2\pi) = i\mathrm{e}^{2\pi \mu i}\int_0^{\infty}\mathrm{d}u\,\mathrm{e}^{-i z \sin( u + \psi + 2\pi) - i \mu (u + 2\pi)} = i\mathrm{e}^{2\pi \mu i}\int_{2\pi}^{\infty}\mathrm{d}\lambda\,\mathrm{e}^{-i z \sin( \lambda + \psi) - i \mu \lambda} = \mathrm{e}^{2\pi \mu i}\left( G_\mu(z, \psi) - i\int_0^{2\pi}\mathrm{d}\lambda\mathrm{e}^{-i z \sin( \lambda + \psi) - i \mu \lambda}\right). \end{equation} Hence $G_\mu(z,\psi)$ can also be represented as: \begin{equation} G_\mu(z, \psi) = \frac{\pi\mathrm{e}^{\pi \mu i}}{\sin\pi \mu}\int_0^{2\pi}\frac{\mathrm{d}\lambda}{2\pi}\,\mathrm{e}^{-i z \sin( \lambda + \psi) - i \mu \lambda}. \end{equation} It is not difficult to show by simple algebraic manipulations that \begin{equation} G_\mu(z, 0) = \frac{\pi\mathrm{e}^{\pi \mu i}}{\sin\pi \mu}\int_0^{2\pi}\frac{\mathrm{d}\lambda}{2\pi}\,\mathrm{e}^{-i z \sin \lambda - i \mu \lambda} = \frac{\pi}{\sin\pi \mu} \mathbf{J}_\mu(z), \end{equation} where \begin{equation} \mathbf{J}_\mu(z) = \int_0^{\pi}\frac{\mathrm{d}\lambda}{\pi}\,\cos(\mu\lambda - z \sin \lambda ) \end{equation} is the Anger function (see DLMF 11.10.1).
On the other hand, starting from the original definition of $G_\mu(z,\psi)$ we have: \begin{equation} G_\mu(z, 0) = i\int_0^{\infty}\mathrm{d}u\,\mathrm{e}^{-i z \sin u - i \mu u} = \int_0^{i\infty}\mathrm{d}t\,\mathrm{e}^{-z \sinh t - \mu t} = \int_0^{\infty}\mathrm{d}t\,\mathrm{e}^{-z \sinh t - \mu t}, \end{equation} where in the last passage I've exploited Cauchy's theorem and the fact that the integrand is an entire function of $t$ for changing the integration path into the positive real line. This should be legitimate as long as $z>0$ and $\mathrm{Re}(\mu t) > 0$, since in that case the integrand calculated on the portion of circle at infinity enclosing the first quadrant of the complex $t$ plane vanishes. Now, using the Schlaefli integral representation of the Bessel function $J_\mu(z)$ (valid for generic $\mu$), namely (see DLMF 10.9.6): \begin{equation}\label{eq:bessel-schlaefli-integral} J_{\mu}(z) = \mathbf{J}_\mu(z) - \frac{\sin\pi\mu}{\pi}\int_{0}^{\infty} \mathrm{d}t\,\mathrm{e}^{-z\sinh t - \mu t}. \end{equation} It apparently comes out that: \begin{equation} G_\mu(z, 0) = \frac{\pi}{\sin\pi \mu}(\mathbf{J}_\mu(z) - J_{\mu}(z)). \end{equation} I'm pretty sure that \begin{equation} G_\mu(z, 0) = \frac{\pi}{\sin\pi \mu} \mathbf{J}_\mu(z) \end{equation} is correct - it can be demonstrated by other means - so there's something wrong with my second argument, but I can't figure out where the error is. Any hint?
By the way, in DLMF 11.10.4 the same integral appearing in my calculations (as well as in the Schlaefli representation) is taken as the definition of the Anger-Weber function $\mathbf{A}_\mu(z)$, namely: \begin{equation} \mathbf{A}_\mu(z) = \int_0^{\infty}\frac{\mathrm{d}t}{\pi}\,\mathrm{e}^{-z \sinh t - \mu t}. \end{equation} The Anger-Weber function is usually defined as (see e.g. "Theory of Incomplete Cylindrical Functions and their Applications", M.M. Agrest and M.S. Maksimov"): \begin{equation} \mathbf{A}_\mu(z) = \int_0^{\pi}\frac{\mathrm{d}\lambda}{\pi}\,\mathrm{e}^{iz \sin\lambda - i\mu \lambda} = \mathbf{J}_\mu(z) - i \mathbf{E}_\mu(z), \end{equation} where $\mathbf{E}_\mu(z)$ is the Weber function: \begin{equation} \mathbf{E}_\mu(z) = \int_0^{\pi}\frac{\mathrm{d}\lambda}{\pi}\,\sin(\mu\lambda - z \sin \lambda ). \end{equation} I can't manage to reconcile the two definitions, but I suspect this is strictly related to my problem. Also in this case, any hint is welcome.