All Complex Analysis Resources
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.
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....
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.
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