I am a graduate student of mathematics.I am stuck with the proof of Cauchy's theorem for a rectangle.It does not seem intuitive to me although the statement is intuitive:
Let $f$ be analytic in a domain containing rectangle $C$ and its interior then $\int_C f(z)dz=0$.
I am looking for a proof of this theorem that is intuitively obvious.Can someone suggest any book for that.I have already referred to complex analysis by Ponnusamy-Silverman but that did not help me much.
Best Answer
This book gives a few different proofs: https://mtaylor.web.unc.edu/notes/complex-analysis-course/
If you know Stoke's theorem (or just Green's theorem in this case), then you can note that we have the complex valued one form $f\,dz = f(dx + i\,dy) = f\,dx + if\,dy$. Thus $d(f dz) = d(f\,dx + if\,dy) = (if_x - f_y)\,dx \wedge dy$. The Cauchy-Riemann equations say that $f$ is holomorphic if and only if $if_x = f_y$. Thus $f$ is holomorphic if and only if $d(f\,dz) = 0$.