# Integration related to Bessel function

Sir,

While studying on the convolution theorem using Green's function (using weyl identity) and doing some problems related to this, I have faced the following integration and I want to find out the closed form answer: $$I=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\exp{isk\Big[\sqrt{1-p^2-q^2}\Big]}}{\sqrt{1-p^2-q^2}}\exp{-ik(px+qy)}\,dp\,dq$$ Where, $$s \in \mathbb R(+)$$, $$k$$ and $$\beta \in \mathbb R(+)$$ are constants.

After searching some texts related to Weyl identity and Sommerfeld identity, what I did is as follows:

I took $$k_r^2= k^2(p^2+q^2)$$; where $$kp=k_r \cos \phi$$ and $$kq=k_r \sin \phi$$. Therefore, the above integral becomes, $$I=\frac{1}{k^2}\int_0^{\infty}\int_0^{2\pi}\frac{\exp{is\sqrt{k^2-k_r^2}}}{\sqrt{k^2-k_r^2}}\exp{ik_r(x\cos \phi+y\sin \phi)} \times k_r \, dk_r \,d\phi$$ After doing the $$\phi$$ integral first it gives, $$I=\frac{2\pi}{k^2}\int_0^{\infty}\frac{\exp{is\sqrt{k^2-k_r^2}}}{\sqrt{k^2-k_r^2}} J_0(k_r\sqrt{x^2+y^2}) k_r \, dk_r$$ Now two cases will arise here 1) when $$k^2>k_r^2$$ and 2) when $$k_r^2\geq k^2$$ i.e. $$\sqrt{k^2-k_r^2}=i\sqrt{k_r^2-k^2}$$ Accordingly, I can write, $$I=I_1+I_2$$, Where, $$I_1=\lim{a\rightarrow k}\int_0^a\frac{\exp{is\sqrt{k^2-k_r^2}}}{\sqrt{k^2-k_r^2}} J_0(k_rR) k_r \, dk_r$$ where, $$R=\sqrt{x^2+y^2}$$and $$I_2=\int_k^{\infty}\frac{\exp{-s\sqrt{k_r^2-k^2}}}{\sqrt{k_r^2-k^2}} J_0(k_rR) k_r \, dk_r$$ Now, if I consider the further substitution i.e. for $$I_1$$, such that $$k^2-k_r^2=w^2$$, then, it becomes, $$I_1=\int_0^k\exp{i(sw)} J_0(\sqrt{k^2-w^2}R) \, dw$$ and similarly, taking substitution, $$k_r^2-k^2=w^2$$, $$I_2=\int_k^{\infty}\exp{(-sw)} J_0(\sqrt{k^2+w^2}R) \, dw$$ Would you kindly check the procedure I did and then suggest me what to do further or give me any relevant references such that I can get the closed form answer of this integral.

Thanking you..

• In case (1), the limits of the $p$ integral are $-\infty$ to $\infty$, which means $p^2+q^2 < 1$ definitely won't be satisfied over the whole domain of integration. – eyeballfrog Apr 22 '20 at 5:40
• Sure Sir, then, the limit may be broken from $-\infty$ to $a$ and then $a$ to $\infty$, where, $a^2= 1-q^2$ and in the later case $\sqrt{a^2-p^2}=i\sqrt{p^2-a^2}$, we get the case 2. Sir, is my assumption correct? – R. Bhattacharya Apr 22 '20 at 5:56
• ( To eyeballfrog) Sir, the question is edited considering the valuable comment, you posted in this regard. Would you please give any further suggestion. – R. Bhattacharya Apr 22 '20 at 6:45