# Complex Analysis : Complex Functions

## Example Questions

### Example Question #1 : Complex Functions

What does the sum below equal?

Another way of asking this question is what is the sum of the  roots of unity.

Explanation:

As messy as it looks, this is just a geometric series.

we will use the partial sum formula for the geometric series.

the red part is the only part that matters....the  cancel out leaving....

and...

thus we have...

which gives the answer of zero.

### Example Question #2 : Complex Functions

Consider the function

Find an expression for  (hint: use the definition of derivatve) and where it exists in the complex plane.

Explanation:

Applying the definition of derivative, we have that

If the limits exists, it can be found by letting  approach  in any manner.

In particular, if we it approach through the points , we have that

A similar approach with  implies that

Since limits are unique, these two approaches imply that

, which implies  and  cannot exist when