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Finite product of complex terms

  1. Sep 12, 2006 #1
    Can one hint me towards finding a general formula for

    [tex]
    P_n = \prod_{x=0}^{n} \left( \sqrt{x} + i\right)
    [/tex]

    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. jcsd
  3. Sep 12, 2006 #2
    I guess you should use letters of the smaller case, i.e. [tex], and not [TEX], for the LATEX to be displayed.
     
  4. Sep 12, 2006 #3
    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)
     
  5. Nov 11, 2007 #4
    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.
     
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