Proving the Value of a Complex Integral Involving Cosecant and the Unit Circle

[benjamin198]

I need help to solve this problem from Complex variables, Arthur A. Hauser, Ch. 5. pag. 122. Problem 5.42
show that ∫ csc(z)dz/z = 0
where C is the unit circle around the origin.


Solve it without using The Cauchy Integral Formula...

 

[tiny-tim]

hi benjamin198!

you need to prove ∫₀^2π csc(e^iθ) dθ = 0

perhaps there's something symmetric, or anti-symmetric, about the integrand?