Two questions that I need help with


1) State a condition on a closed path gamma that will guarantee that the contour integral (over gamma) of (1/z)dz equals zero

if gamma is a closed path contained in the complex plane, and does not cross contain any negative real numbers, then ln(z) is defined everywhere on gamma, and so, the integral is zero. But is there another condition, like a condition that would give an if and only if statement. (not sure if that condition I mentioned is one) (what if gamma contains negative real numbers, then could the integral still turn out to be zero?



2) Calculate the contour integral (over gamma) of (z^i) dz, where gamma(t) = (t^3 + 1 ) + i sin(t) , t is in [0,pi]


I could try using the definition of the contour integral, but it seems it would become pretty messy, usually there are some kinds of tricks, or theorems that can be used... can someone help me out?

Not going too far into details, see for example :

http://en.wikipedia.org/wiki/Cauchy's_integral_theorem

1) If Gamma is included into Omega, a simply connected open subset of the complex plane without zero, then f(z) = 1/z is holomorphic on Omega, and you can use Cauchy's theorem .
The idea is that, following the path of gamma, you don't make a whole turn around zero.

2) Since gamma doesn't contain any negative real number, you can consider the principal definition of the complex logarithm (where Log (1) = 0) and write z^i = exp ( i. Log(z) ).
This shows that f(z) = z^i is holomorphic in a simply connected open subset of C.
For example, you can use the open disk Omega with center z2 = pi^3 +1 and radius R = pi^3 + 1/2

Thus, z^i have primitives functions on Omega.

Precisely, F(z) = (i/(i+1)).exp ( (i+1). Log(z) ) is a primitive of f.
In a more simple form : F(z) = z^(i+1) / (i+1)

Thus, your integral over gamma is F(z2) - F(z1), with z2 = (pi^3 + 1) + i sin(pi) = pi^3 + 1 and z1 = (0^3 + 1) + i sin(0) = 1

Integral = [ (pi^3 +1) ^ (i+1) - 1 ] / (i+1) = [ (pi^3 +1).Exp(i.ln(pi^3+1)) - 1 ] / (i+1)

I hope it will help.