I would like to find the that maximizes this discrete function
where n is 0,1,2,....
Is that even possible?
Hint: Try treating n as a continuous variable and using calculus to get the maximum points and look at answers that are integer values around those points.
With regards to the constraints on R and theta, consider using multi-variable optimization with those constraints.
If you get a small number of turning points, then it will be easy to find the values of N that maximizes the function.
I will write instead of , just to make clear that is continuous.
Now setting equal to zero gives
Is it possible to isolate t? It seems hard but my PC says it is possible, it also gives a result but I haven't found out yet how to do it manually. Any hints?
I think I've got it now. I worked backwards and figured out the trick. I found the trig identity to use and the underlying idea. The idea is to take the trig terms with argument and factor them out in terms of . Then the terms involving can be boiled down to a tangent. The remaining terms does not depend on t. Very nice .
Actually, this max value I have been working on finding an expression for is a special case of a more general problem which I am now more prepared to tackle (a max overshoot value in some digital filters).
Thanks for your feedback.
Although not relevant anymore, I though I just mention that the expression for the derivative I posted is not correct. It has some sign issues. I thought I just mention it. Anyway thanks.