Thread: How can I solve the integral with singularity at endpoint?

I have a question about a integral that have singularity at the endpoint. The singularity will bring oscillation to the solution. I am wondering if there is a way to approximate this integrand with a recurrence relation so that we can smoothe out the oscillation? Here is the integral.
integral exp(-k*L) /((Ro*k)^2 + Ro^2*(U*k-1)^2-1) *(exp(-Ro*k*z/(1-Ro^2*(U*k-1)^2)^0.5 ) *exp(i(kx-t)) dk from (Ro-1)/Ro/U to (Ro+1)/Ro/U. Here Ro(=0.707), U(=0.625), t(=0.25) are const; we can set x, z as arrays from -10 to 10 and from 0 to 10. As we notice that the singularity exists at endpoint (Ro+1)/Ro/U. My question is if it is possible to modify the integral to a suitable form which can smoothe out the singularity? For example, if we can simplify it to the form as integral exp(t)/t dt from const to infinity, it will be solvable.
Thanks.

I'm not sure if this is something I could help you with or not, but the notation you used isn't very clear to me.

If you want to make an integral sign, you can do it like this:
&#8747 ;
without the space in between the 7 and ;

If you want to make a superscript, as you would if you would raise something to a power or show the upper limit to an integrand, you can type
< sup > 5 < / sup >
without any spaces

If you want to make a subscript, as you would perhaps to show the lower limit of an integrand
< sub > 5 < / sub >
without any spaces.

To me, your "exp" I think could either refer to e or a power so its confusing to me.

Am I right in thinking that this is your integrand?

First, for the sake of clarity, I will temporarily make Ro = r

Your lower limit looks to be
(r-1)/r/u
u(r-1)/r
u - ( 1 / r )

Your upper limit looks to be using the same steps as above:
u + ( 1 / r )

_u-(1/r)∫^u+(1/r) (-k L)e(r k)^-2 + ( r²(u k - 1)² - 1)e(-r k z)( (1 -r²)( (u k - 1)²)^1/2)^-1e( i ( kx - t ) ) dk

If this is the case (as it most likely isn't), this integrand could be greatly simplified.