# Summation problem

Started by December 27, 2006
```In the course of understanding a cross-correlation problem (this is not
homework!), I need to evaluate:

sum (n = 0 to N) exp (j * n * theta)

or equivalently:

sum (n = 0 to N) cos(n*  theta) + j * sum(n = 0 to N) sin(n *
theta)

where N is 150 or so and theta is something rather small, on the order
of 0.001 * pi.

I'd guess these summations are somewhat common and I could find them in
some listing or other, but my lone math handbook lacks them, and so far
I haven't found any such listing on the web.  Can anyone give me a
pointer to them?  Many thanks.

cheers,
jerry wolf
spaceflight systems corp.

```
```Jerry Wolf wrote:
> In the course of understanding a cross-correlation problem (this is not
> homework!), I need to evaluate:
>
>      sum (n = 0 to N) exp (j * n * theta)
>
> or equivalently:
>
>      sum (n = 0 to N) cos(n*  theta) + j * sum(n = 0 to N) sin(n *
> theta)
>
> where N is 150 or so and theta is something rather small, on the order
> of 0.001 * pi.
>
> I'd guess these summations are somewhat common and I could find them in
> some listing or other, but my lone math handbook lacks them, and so far
> I haven't found any such listing on the web.  Can anyone give me a
> pointer to them?  Many thanks.
>
> cheers,
>   jerry wolf
>   spaceflight systems corp.

Maybe you could approximate the sum with a definite integral and get a
formula that way, since you'd be integrating sines and cosines which
are easy to solve.

John

```
```I appealed for help a little too soon.  Further creative googling and

Weisstein, Eric W. "Exponential Sum Formulas." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/ExponentialSumFormulas.html

and also
http://mathworld.wolfram.com/Cosine.html
http://mathworld.wolfram.com/Sine.html
where the needed formulas were found.

cheers,
jerry

```
```"Jerry Wolf" <jjwolf22@verizon.net> writes:

> In the course of understanding a cross-correlation problem (this is not
> homework!), I need to evaluate:
>
>      sum (n = 0 to N) exp (j * n * theta)
>
> or equivalently:
>
>      sum (n = 0 to N) cos(n*  theta) + j * sum(n = 0 to N) sin(n *
> theta)
>
> where N is 150 or so and theta is something rather small, on the order
> of 0.001 * pi.
>
> I'd guess these summations are somewhat common and I could find them in
> some listing or other, but my lone math handbook lacks them, and so far
> I haven't found any such listing on the web.  Can anyone give me a
> pointer to them?  Many thanks.