I find myself dealing with the following integral $$ \int_{-1} ^1 dt \,e^{it} K_0(b t), $$ and I was thinking if this can be somehow expressed in terms of something nice, like other Bessel functions for instance. $b $ is a constant and $K_0$ is the modified Bessel function of the second kind.
Thanks for your help!
I am not sure (at all) that there is a solution to this problem and I am afraid that only numerical integration could be the only way to go.
What I would do is to generate tables for a series of values of $b$ and use any method of interpolation (independently for the real and imaginary parts of the result); the functions are really smooth.
I give below a few values $$\left( \begin{array}{ccc} b & \text{real part} & \text{imaginary part} \\ 0.25 & +2.98678 & -2.65532 \\ 0.50 & +1.83156 & -2.69092 \\ 0.75 & +1.14941 & -2.75129 \\ 1.00 & +0.64764 & -2.83803 \\ 1.25 & +0.23109 & -2.95345 \\ 1.50 & -0.14555 & -3.10066 \\ 1.75 & -0.50939 & -3.28370 \\ 2.00 & -0.87976 & -3.50766 \\ 2.25 & -1.27265 & -3.77889 \\ 2.50 & -1.70300 & -4.10521 \\ 2.75 & -2.18611 & -4.49626 \\ 3.00 & -2.73868 & -4.96380 \\ 3.25 & -3.37976 & -5.52223 \\ 3.50 & -4.13165 & -6.18909 \\ 3.75 & -5.02096 & -6.98583 \\ 4.00 & -6.07974 & -7.93861 \\ 4.25 & -7.34693 & -9.07937 \\ 4.50 & -8.87001 & -10.4472 \\ 4.75 & -10.7072 & -12.0898 \\ 5.00 & -12.9298 & -14.0658 \end{array} \right)$$