I was told to post this kind of question on the homework help section by one of the mentors,even though I'm not sure it is appropiate.
Anyway,I'm doing complex integration now,so I need to get some important concepts cleared.
I'll post my doubts in points...
1.firstly,in complex integration,only curves with nonzero and continuous derivatives are considered....but usually,for integration,all we need,is to take a continous curve,with a derivative defined at each point,not necessarily a non-zero derivative.
2. Whenever I read about complex integration,it's always line integrals....isn't the riemann sum concept of integration applicable to complex functions,where we're simply calculating the area under a curve?
3.Why is simple connectedness a necessary condition for complex integration?
(my book says it's necessary everywhere,without stating any specific reason.)
4.In complex integration,does integration refer to 'integration over a certain curve',or over a certain 'domain area'?
5. It is found that the complex integrals between two fixed points taken over different paths are not always equal...this is the same in real functions isn't it? In that respect,we could perhaps introduce the concept of 'conservative fields as we do in vector calculus' to complex integrals,....but we don't..why? I f we did,we could find out those complex functions that are path independant by using curl=0.