Cauchy's Theorem: Analytic Functions and Integrals

[Daniiel]

Hey,

This is just a small question about Cauchys theorem.

If there is a function f(z) such that int f(z)dz = 0 can you conclude f is analytic in and on the region of integration?

What I mean is can you work the theorem in reverse?

For example if the above is true over a region C which is a simple closed curve, is f(z) analytic both inside and on C?

 

[mathwonk]

check our morera's theorem.

 

[HallsofIvy]

That depends on exactly what you mean by "int f(z) dz". If you mean simply that $\oint f(z)dz= 0$ for some closed path, no. For example, f(z)= 1 for Im(z)> 0, f(z)= -1 for Im(z)< 0, the integral around any circle centered on the origin is 0 but that function is not analytic. Morera's theorem, that mathwonk suggests, says that if the integral around every closed path in a region is 0, then the function is analytic in that region.