I need to solve this integral from a physical model
which is singular at the upper limit. The reason for the singularity is that the function reaches a maximum at u=1. The model is completely physically based so the integral must have a solution. I was thinking about using Cauchy's theorem but I don't know how to find a proper contour. Even if I get complex numbers in the solution I think I still can interpret them somehow (or make some simplification like dropping the imaginary part).
What is the physical model and what is the question asked of this model?
Thanks for the answer. I totally agree but let me explain better. My problem indeed is to solve
so it is the function u that has a maximum at u=1. It is part of a model with a negative feedback of u on itself. so what I am trying to find here is the time for the maximum. The straight answer is that it is infinite, but my question is: is there any other possibility, as for example, a branch point in u(t)? I can solve it using a numerical method but I need to do it analytically.
Thanks for any help.
That's not correct.
In fact, u has a horizontal asymptote with equation u = 1 ....... Note that as t --> +oo, u --> 1 and du/dt --> 0.
It's not hard to show that curve u does not cut its horizontal asymptote. Therefore it obviously takes an infinite amount of time for u to equal 1 ......
By the way, I know that u has a horizontal asymptote because the solution to the DE (assuming the boundary condition u(a) = 0, where a < 1) is
and a graph of t versus u clearly has a vertical asymptote at u = 1 ....... Try drawing a graph corresponding to a = 1/2, say and you'll see what I mean. So a graph of u versus t clearly has a horizontal asymptote u = 1 .......