Math and stuff

Proposition A function $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^p$ is bilinear if for $x, x_1, x_2 \in \mathbb{R}^n$, $y, y_1, y_2 \in \mathbb{R}^m$, and $a \in \mathbb{R}$ we have \[\begin{align*} f(ax, y) &= af(x, y) = f(x, ay), \\ f(x_1 + x_2, y) &= f(x_1, y) + f(x_2, y), \\...

Proposition Suppose $f$ has an isolated singularity at $z_0$. Show that $f’$ has an isolated singularity at $z_0$. Find $\Res_{z = z_0}(f’)$. Solution 1 Since $f$ has an isolated singularity at $z_0$, there exists a punctured disk with the radius $r$ such that $f$ is holomorhpic in it. Then for...

Proposition Compute $\int_{C[2, 3]}\frac{\cos(z)}{\sin^2(z)} dz$. Solution $\sin(z) = 0$ if and only if $z = k\pi$ for some $k \in \mathbb{Z}$. Thus $0, \pi$ are the only singularities of $f$ inside $C[2, 3]$. As shown before, the Laurent series for $\frac{1}{\sin^2(z)}$ at $z = 0$ does not contain odd terms....

Proposition Define $f: D[0, 1] \rightarrow \mathbb{C}$ through \[\begin{align*} f(z) = \int_{[0, 1]} \frac{dw}{1 - wz}. \end{align*}\] Prove that $f$ is holomorphic in the unit disk $D[0, 1]$. Solution Let $\gamma$ be a closed, piecewise smooth curve in $D[0, 1]$. \[\begin{align*} \int_{\gamma} f(z) &= \int_{\gamma} (\int_{[0, 1]} \frac{dw}{1 - wz})...

Proposition Suppose that $f: (0, 1) \rightarrow \mathbb{R}$ is a non-negative continuous function. Show that $\int_{(0, 1)} f$ exists if and only if $\lim_{\epsilon \rightarrow 0} \int_{\epsilon}^{1-\epsilon} f$ exists. Let $A_n = [1 - 1 / 2^n, 1 - 1 / 2^{n + 1}]$. Suppose that $f: (0, 1) \rightarrow...