# Analytical form of the answer of a definite integral

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.

• Is that really y = in your difinition of f? Also MMa knows that I is Sqrt[-1], so you don't need to define i if you replace it by I in your code. – Bill Watts Aug 25 '18 at 7:10

Looks like your code works once you clean up the syntax some.

Clear["Global*"]

y=d (N+r^2)^(1/2)
z=Q+2*r*Sin[ϕ]+r^2

f[r_,ϕ_]=(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&&L>0,Integrate[f[r,ϕ],{r,0,L},{ϕ,0,2π}]]//Simplify

-((2*Pi*s*E^(I*k*Sqrt[q])*(E^(I*d*k*Sqrt[N]) - E^(I*d*k*Sqrt[L^2 + N])))/(d^2*q))
`

I have moved definitions for y and z outside your definition for f. Normally you would use Block or Module for multiline definitions. I still have doubts as to whether I changed your code to what you mean, so you should check it.

• Sir, Thanks a lot.If I have any queries further, I shall ask then for further suggestion.... – R. Bhattacharya Aug 25 '18 at 8:29
• Sir, there is a mistake, I overlooked.I am very sorry for that.In the above case, the expression defined as $z$ will be replaced by $q$. While putting this change in the said syntax, no output is come out. Would you kindle check the case once again. I have edited the question. thanks ... – R. Bhattacharya Aug 25 '18 at 15:07