QUESTIONS - COMPLEX ANALYSIS

QUESTIONS - ANSWERS - COMPLEX ANALYIS
Question:
Please help~~
The function
f(z) = 1 / z^3
satisfies
integral f(z) dz =
for every closed curve C in the
c
complex plane that does not pass through the point 0. However, this function is not analytic at the point 0. Why does Morera's Theorem not apply?
Answer:
There are 2 stipulations to apply Morera's theorem.
1.f(z) should be continuous in a simply connected domain D.
2.Along any closed path in D integral f(z) should be zero.
Then the function will be anaytic in D.
In this case the second condition is satisfied while first condition is not satisfied as obviously the function is discontinuous at z=0 in D.
Hence,Morera's theorem is not applicable here.