I would like to evaluate a complex-valued integral of the form
$$ I_e = \int_0^1 x e^{iax} J_0(b \sqrt{1-x^2}) dx $$
where $a$ and $b$ are positive real numbers and $J_0(z)$ is the Bessel function of the first kind. I am particularly interested in the special case of $a=(c+1)b$ for $c \ll 1$.
The task boils down to evaluating two real-valued integrals
$$ I_s = \int_0^1 x \sin(ax) J_0(b \sqrt{1-x^2}) dx $$ $$ I_c = \int_0^1 x \cos(ax) J_0(b \sqrt{1-x^2}) dx $$
The integral with the sine has a simple form given by Gradshteyn and Ryzhik (6.738.1) which, after simplification, becomes
$$ I_s =\sqrt{\frac{a^2}{a^2 + b^2}}j_1(\sqrt{a^2 + b^2}) $$
where $j_1(z)$ is the spherical Bessel function of the first kind.
I am not exactly sure how this expression was derived. Perhaps it holds a clue. I tried substituting the integral form of the Bessel function and integrating analytically but did not get very far.
By symmetry, I naively expected the integral involving the cosine to be proportional to the spherical Bessel function of the second kind $y_1(z)$ (and thus, the complex-valued integral to be proportional to the spherical Hankel function of the second kind), but that does not appear to be the case.
$$ I_c \approx -\sqrt{\frac{a^2}{a^2 + b^2}}y_1(\sqrt{a^2 + b^2}) $$
The agreement for the special case of $a=b$ is not too terrible, but something is going wrong for small values of $b$. If I take the difference between the true value of the integral and my guess, it is proportional to another Bessel function, but the arguments are crazy and it does not make any sense.
Now, if I assume that it is proportional to the spherical Bessel function of the first kind instead, and compute the ratio between the true value of the integral and my guess, I get the classic graph of the cotangent.
For $b \gg 1$, it appears
$$ I_c \approx \cot(\sqrt{a^2 + b^2} + \phi)) \sqrt{\frac{a^2}{a^2 + b^2}}j_1(\sqrt{a^2 + b^2} + \phi) $$
In practice, this is hardly useful, since the period of the cotangent would have to correspond to zeros of the spherical Bessel function, and these are hard to compute. Additionally, $j_1(z)$ and the actual $I_c$ appear to be slightly off-phase.
Of course, ideally, I would like to solve the problem analytically, but I am not sure how (using a series expansion, perhaps?). I would appreciate any tips or guidance.
Thank you!
Edit: by transforming using $x=\sin{\theta}$, there is an almost perfect match in Gradshteyn and Ryzhik (6.688.2), except that the leading term in my case is $\sin{2\theta}$ rather than $\sin{\theta}$. Additionally, it gives a result in terms of the spherical Bessel function of the first kind rather than the second kind, so it does not explain my empirical approximation.
$$ I_c = \int_0^{\pi/2} \frac{1}{2} \sin{2\theta} \cos(a \cos{\theta}) J_0(b \sin{\theta}) d\theta $$
Using the representation given above, we can expand $\cos(a \cos{\theta})$ in a series, which does give an analytic solution, but it converges rather poorly and does not appear to contain any spherical Bessel terms (just the regular Bessel functions). I am yet to find an expansion in terms of spherical Bessel functions.
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