# Finite product of complex terms

1. Sep 12, 2006

### Gazouille

Can one hint me towards finding a general formula for

$$P_n = \prod_{x=0}^{n} \left( \sqrt{x} + i\right)$$

I need a direction because now i'm stuck with it after having struggled to formulate it.

Either the real or imaginary part would be enough but i guess i won't get one without the other. I'm working out now the term to renorm the result, i'll post it asap, it's nothing complicated.

This current formulation I made up for a geometrical problem i have :).
It is to evaluate exactly a spiral that interpolates a sequence of vertices in orthogonal triangles that you would stitch together (by hypothenuse to the variable length side of the next). I can make a drawing if necessary.

I need to sum up angles for each triangle to get a polar coordinate of the n-th vertex. Hopefully evaluable through a simple continuous function to get the in-betweens aswell.

I started with trying to work in log space but what i naturally get is a sum of arc-cosinuses that i can't find an interpolating function for...

Thanks for any hint :)

Last edited: Sep 12, 2006
2. Sep 12, 2006

### neutrino

I guess you should use letters of the smaller case, i.e. [tex], and not [TEX], for the LATEX to be displayed.

3. Sep 12, 2006

### Gazouille

yes i thought so, i changed it, crossing fingers :)
in case it doesn't like me, it's : P(n) equals product for x=0 to n of (sqrt(x)+i)

4. Nov 11, 2007

### Gazouille

Well, by now I still haven't solved this problem but I found it's called the Spiral of Theodorus and there is no simple closed form solution althought the problem looks so simple :).
In addition, the solutions are converging series that, well... converge very very slowly.
So, I'm gonna use a table of sampled values for my function and that's gonna be all.