Complex Analysis : Complex Functions

Study concepts, example questions & explanations for Complex Analysis

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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.

Possible Answers:

Correct answer:

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.

Possible Answers:

Correct answer:

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 

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