# Question:Need help with a function of complex variables

## Question:Need help with a function of complex variables

Maple

Hi everybody,

Here's what I have:

```After many calculations, I arrive to this expression
> S := Sum(exp(I*Pi*(2*p+1)*(2*m+1)/(2*n)), p = 0 .. n-1);

n - 1
-----    /1                         \
\       |- I Pi (2 p + 1) (2 m + 1)|
)      |2                         |
/    exp|--------------------------|
-----    \            n             /
p = 0
with {n,m}::posint and n > m.  Then I rewrite it this way
> exp(I*Pi*beta)*(Sum(exp((I*Pi*2)*p*beta), p = 0 .. n-1));

/n - 1                   \
|-----                   |
| \                      |
|  )                     |
exp(I Pi beta) | /    exp(2 I Pi p beta)|
|-----                   |
\p = 0                   /
> beta = (2*m+1)/(2*n);

2 m + 1
beta = -------
2 n
if I am not wrong, this sum give
> St := exp(I*Pi*beta)*(1-exp((2*I)*Pi*n*beta))/(1-exp((2*I)*Pi*beta));

exp(I Pi beta) (1 - exp(2 I Pi n beta))
---------------------------------------
1 - exp(2 I Pi beta)
I have try a couples of hours to simplify it so I can get,
using at the end the Euler's identity, the expression
> St1 := -I/sin(Pi*beta);

I
- ------------
sin(Pi beta)
If you could show me the way, I have exhausted all I could think of
so far.  In paper or with Maple would be nice.

```

Thanks

Mario Lemelin

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