Math and stuff

Proposition Compute $\int_{C[0, 2]} z^{1/2} dz$. Solution $\gamma(t) = 2\exp(it)$ is a parametrization of $C[0, 2]$ where $0 \leq t \leq 2\pi$. \[\begin{align*} \int_{\gamma} z^{1/2} dz &= \int_{0}^{2\pi} (2e^{it})^{1/2} (2ie^{it}) dt \\ &= 2^{3/2}i\int_{0}^{2\pi} e^{3it/2} dt \\ &= 2^{3/2}i\frac{2}{3i}e^{3it/2} \Big\vert^{2\pi}_0 \\ &= \frac{2^{5/2}}{3}(e^{3\pi i} - 1) \\ &= \frac{-2^{7/2}}{3}. \end{align*}\]...

Proposition Show that $\int_{C_{R, n}} d\theta = 2\pi n$, and use Stokes’ theorem to conclude that $C_{R, n} \ne \partial c$ for any 2-chain $c$ in $\mathbb{R}^2 - 0$. Solution \[\begin{align*} \int_{C_{R, n}} d\theta &= \int_{C_{R, n}} \frac{-y}{x^2 + y^2}dx + \frac{x}{x^2 + y^2} dy & \text{(P.93, Spivak)} \\ &=...

Proposition Let $S$ be the set of all singular $n$-cubes, and $\mathbb{Z}$ the integers. An $n$-chain is a function $f: S \rightarrow \mathbb{Z}$ such that $f(c)$ for all but finitely many $c$. Define $f + g$ and $nf$ by $(f + g)(c) = f(c) + g(c)$ and $nf(c) = n...

Proposition Prove that $\int_{\gamma} z\exp(z^2) dz = 0$ for any closed path $\gamma$. Solution Let $f(z) = 1$ and $g(z) = \exp(z^2)/2$. Then $fg’ = z\exp(z^2)$. By integration by parts, $\int_{\gamma} fg’ = f(\gamma(b))g(\gamma(b)) - f(\gamma(a))g(\gamma(a)) - \int_{\gamma}f’g = 0 - \int_{\gamma} 0 = 0$.

Proposition Prove the following integration by parts statement: Let $f$ and $g$ be holomorphic in $G$ and suppose $\gamma \subset G$ is a piecewise smooth path from $\gamma(a)$ to $\gamma(b)$. Then \[\begin{align*} \int_{\gamma} fg' = f(\gamma(b))g(\gamma(b)) - f(\gamma(a))g(\gamma(a)) - \int_{\gamma} f'g. \end{align*}\] Solution \[\begin{align*} \int_{\gamma} (fg)' &= \int_{\gamma} f'g +...