Complex Analysis - Analytic Functions

Question

Show that

 

f(z)=[sin(PI/z)]/(z+2)  is nowhere analytic for |z|<a , where a is a some positive number.

 

Can you help me to solve this problem?

Answer 1

Use definition of the analytic function and check if f(z)=[sin(PI/z)]/(z+2)

meets the requirement given in definition. You should be aware that there are several ways to define the function to be analytic, such as expanding by power series, Cauchy-Riemann condition, etc. (in principle all equivalent). You can use that you learned best.

 

I guess by z you mean complex variable. But we can verify analyticityof  real functions as well.