I am trying to find out the analytical form of the answer of the following integration,
$$I=\int_0^l r \, dr \int_0^{2\pi} \, d\phi (iks)[ \frac{\exp [ikx_0(N+r^2)^{1/2}]}{[x_0(N+r^2)^{1/2}]}\frac{\exp[ikx_0(Q+2r\sin\phi+r^2)^{1/2}]}{(Q+2r\sin\phi+r^2)}]$$ Where, $x_0,N,Q,s $ are constants. $s$ and $x_0$ have same dimension of length; i.e. their ratio will be a real positive number only as per the physical condition of the problem. $N,Q$ are also dimensionless real positive constants.$k$ is the wave vector, i.e. $k=\frac{2\pi}{\lambda}$. So, this is also a real positive constant.I tried to write the program in the following format,
Clear[f];
i = (-1)^(1/2);
f[r_, \[Phi]_] := (
y =d (N + r^2)^(1/2);
z=Q+2*r*Sin [\[Phi]]+r^2; (i*k*s)*Exp[i*k*y] /(y)* (Exp[i*k*q^(1/2)]/q)*r
);
Assuming[k>0 && Q>0 && d>0 && N>0 && s>0,Integrate[f[r,\[Phi]],{r,0, \
L},{\[Phi],0,2\[Pi]}]]//Simplify
Where, $x_0$ is replaced by $d$. $L$ is the upper limit of the integration of $r$. It is a number say, o.5 Would you kindly suggest me how can I get the answer in analytical form of the above integral and also can improve the skill of programming in mathematica for this type of cases? Thanking you.....
**UPDATE:**The expression denoted by $z$ in the above syntax,will be replaced by $q$. it was typed by mistake.$q$ is involved in the function.THE CORRECT EXPRESSION WILL BE
q=Q+2*r*Sin [\[Phi]]+r^2
While applying the same syntax, answered by Sir Bill Watts, for this new case, nothing comes out.Would you kindly suggest me the correct procedure to solve this.