State and prove Green’s Theorem for rectangles. Use this to prove Cauchy’s Theorem for functions that are analytic in a rectangle.
(Variant)
Suppose \(f\in C_\CC^1(\Omega)\) and \(T\subset \Omega\) is a triangle with \(T^\circ \subset \Omega\).
- Apply Green’s theorem to show that \(\int_T f(z) ~dz = 0\).
- Assume that \(f'\) is continuous and prove Goursat’s theorem.
Hint: Green’s theorem states \[ \int_{T} F d x+G d y=\int_{T^\circ}\left(\frac{\partial G}{\partial x}-\frac{\partial F}{\partial y}\right) d x d y .\]
Solution
Green’s theorem: if \(\Omega\) is a domain with positively oriented boundary with \(u, v\) continuously differentiable in \(\bar\Omega\), then \[ \int_{\bd \Omega} u\dx + v\dy = \iint_{\Omega}\qty{v_x - u_y}\dx\dy .\] Now use that if \(f = u+iv\) is analytic in a region, it satisfies Cauchy-Riemann: \[ u_x = v_y \qquad u_y = -v_x .\]
Now integrating \(f\): \[ \oint_{\bd\Omega} f(z) \dz &= \oint_{\bd\Omega} (u+iv)(\dx + i\dy )\\ &= \oint_{\bd\Omega} \qty{u\dx - v\dy} + i\oint_{\bd\Omega} \qty{v\dx + u\dy} \\ &= \iint_\Omega\qty{v_x + u_y}\dx\dy + \iint_\Omega\qty{u_x - v_y}\dx\dy \\ &= \iint_\Omega\qty{v_x -v_x }\dx\dy + \iint_\Omega\qty{u_x - u_x}\dx\dy \\ &= 0 .\]