We want to prove the following formula
For any $ z \in \mathbb{C} $, we have
\begin{equation} \int_{\mathbb{R}} e^{z p} I_{|p|}(x) d p \\ =e^{x \cosh (z)} H(\pi-|\operatorname{Im} z|)+\frac{1}{2 \pi i} \int_{-\infty}^{\infty} e^{-x \cosh (u)}\left(\frac{1}{z+u+i \pi}-\frac{1}{z+u-i \pi}\right) d u \end{equation}
where $ H $ is the Heaviside step function $ H(x) $ equals $1$ for $ x>0 $ and $0$ otherwise, and $ I_{\nu}(x) $ is the modified Bessel function \begin{equation} I_{\nu}(x)=\frac{1}{\pi} \int_{0}^{\pi} e^{x \cos (s)} \cos (\nu s) d s-\frac{\sin (\nu \pi)}{\pi} \int_{0}^{\infty} e^{-x \cosh (s)-\nu s} d s \end{equation}
The following are my calculations：We put $I_v(x))$ into $\int_{\mathbb{R}} e^{z p} I_{|p|}(x) d p,$
\begin{equation} \begin{array}{l} \int_{\mathbb{R}} e^{z p} I_{|p|}(x) d p \\ =\int_{-\infty}^{\infty} e^{z p}\left(\frac{1}{\pi} \int_{0}^{\pi} e^{x \cos (s)} \cos (|p| s) d s-\frac{\sin (|p| \pi)}{\pi} \int_{0}^{\infty} e^{-x \cosh (s)-|p| s} d s\right) d p \end{array} \end{equation} We first compute \begin{align} &\frac{1}{\pi} \int_{-\infty}^{\infty} e^{z p} \int_{0}^{\pi} e^{x \cos (s)} \cos (|p| s) d s d p \\ \nonumber &=\frac{1}{\pi} \int_{-\infty}^{\infty} e^{z p} \int_{0}^{\pi} e^{x \cos (s)} \frac{e^{i p s}+e^{-i p s}}{2} d s d p \\ \nonumber &=\frac{1}{2 \pi} \int_{0}^{\pi} e^{x \cos (s)} \int_{-\infty}^{\infty}\left(e^{(z+i s) p}+e^{(z-i s) p}\right) d p d s \\ \nonumber &=\frac{1}{2 \pi} \int_{-\infty}^{\infty} H(\pi-s) e^{x \cos (s)} \int_{-\infty}^{\infty}\left(e^{-i(-s+i z) p}+e^{-i(s+i z) p}\right) d p d s \\ \nonumber &=\frac{1}{2 \pi} \int_{-\infty}^{\infty} H(\pi-s) e^{x \cos (s)} 2 \pi(\delta(-s+i z)+\delta(s+i z)) d s \\ \nonumber &=\int_{-\infty}^{\infty} H(\pi-s) e^{x \cos (s)}(\delta(-s+i z)+\delta(s+i z)) d s \\ \nonumber &=H(\pi-i z) e^{x \cos (i z)}+H(\pi+i z) e^{x \cos (-i z)} \end{align} Notice that $$ \cos (i z)=\cos (-i z)=\cosh (z) . $$ The third formula from the end seems only hold for pure image $z$, doesn't it?
We secondly compute \begin{align*} &-\int_{-\infty}^{\infty} e^{z p} \frac{\sin (|p| \pi)}{\pi} \int_{0}^{\infty} e^{-x \cosh (s)-|p| s} d s d p \\ &=-\frac{1}{\pi}\left(\int_{0}^{\infty} e^{z p} \sin (p \pi) \int_{0}^{\infty} e^{-x \cosh (s)} e^{-s p} d s d p+\int_{-\infty}^{0} e^{z p} \sin (-p \pi) \int_{0}^{\infty} e^{-x \cosh (s)} e^{s p} d s d p\right) \\ &=\frac{1}{\pi} \int_{0}^{\infty} e^{-x \cosh (s)}\left(-\int_{0}^{\infty} e^{(z-s) p} \sin (p \pi) d p+\int_{-\infty}^{0} e^{(z+s) p} \sin (p \pi) d p\right) d s \\ &=\frac{1}{\pi} \int_{0}^{\infty} e^{-x \cosh (s)}\left(\int_{-\infty}^{0} e^{(z+s)} p \frac{e^{i p \pi}-e^{-i p \pi}}{2 i} dp-\int_{0}^{\infty} e^{(z-s)} p \frac{e^{i p \pi}-e^{-i p \pi}}{2 i} d p\right) d s \\ &=\frac{1}{2 i \pi} \int_{0}^{\infty} e^{-x \cosh (s)}\left(\left.\frac{e^{(z+s+i \pi) p}}{z+s+i \pi}\right|_{p=-\infty} ^{p=0}-\left.\frac{e^{(z+s-i \pi) p}}{z+s-i \pi}\right|_{p=-\infty} ^{p=0}\right)-\left(\left.\frac{e^{(z-s+i \pi) p}}{z-s+i \pi}\right|_{p=0} ^{p=\infty}-\left.\frac{e^{(z-s-i \pi) p}}{z-s-i \pi}\right|_{p=0} ^{p=\infty}\right) ds \\ &=\frac{1}{2 i \pi} \int_{0}^{\infty} e^{-x \cosh (s)}\left(\frac{1}{z+s+i \pi}-\frac{1}{z+s-i \pi}\right) d s-\frac{1}{2 i \pi} \int_{0}^{-\infty} e^{-x \cosh (-s)}\left(\frac{1}{z+s+i \pi}-\frac{1}{z+s-i \pi}\right)ds \\ &=\frac{1}{2 i \pi} \int_{-\infty}^{\infty} e^{-x \cosh (s)}\left(\frac{1}{z+s+i \pi}-\frac{1}{z+s-i \pi}\right) d s \\ \end{align*} but where we need $$ \left.e^{(z+s+i \pi) p}\right|_{p=-\infty}=\left.e^{(z+s-i \pi) p}\right|_{p=-\infty}=\left.e^{(z-s+i \pi) p}\right|_{p=\infty}=\left.e^{(z-s-i \pi) p}\right|_{p=\infty}=0 $$ which is not available for any $ z \in \mathbb{C} .$
I do not know how to solve this.
REFERENCES
[1] A. $Erd\acute{e}lyi$, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Tables of integral transforms. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman.
[2] G. N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.