Articles of complex analysis

Upper bound for complex polynomial

I have a polynomial $p$ of degree $n$ satisfying $\lvert p(z) \lvert \leq c\ \ \forall z\in\partial B_1(0)$. (Isn’t this true for any polynomial?) Show $\lvert p(z)\lvert \leq c \lvert z\lvert^n \ \ \forall z\in \mathbb{C}\backslash B_1(0)$. The obvious attempt would be $|p(z)|=|p(\lvert z\lvert\frac{z}{\lvert z\lvert})|=|\sum_{i=0}^n a_i |z|^i (\frac{z}{\lvert z\lvert})^i|\leq |z|^n \sum_{i=0}^n |a_i (\frac{z}{\lvert z\lvert})^i|$ which […]

The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive harmonic function $u(z)$ in $\Omega$ satisfies $u(z_2) \leq M u(z_1)$ for any two points $z_1, z_2 \in E$. This […]

Weierstrass product form

How to show the Weierstrass product form of the entire function $f(z)= \sinh z$ This question seem so interesting. I would like to write my some ideas, but I dont want to direct incorrectly. Please help me to learn correctly and explicitly. I asked to question in fact in order to learn the topic precisely.

Sketch complex curve $z(t) = e^{-1t+it}$, $0 \le t \le b$ for some $b>0$

Sketch complex curve $z(t) = e^{-1t+it}$, $0 \le t \le b$ for some $b>0$ I tried plotting this using mathematica, but I get two curves. Also, how do I find its length, is it just the integral? This equation doesn’t converge right? Edit: I forgot the $t$ in front of the $1$ so it’s not […]

Does convergence of power series on radius of convergence imply absolute convergence?

Let $R$ be radius of convergence of power seires $\displaystyle\sum_{k}a_kz^k$. If the power series converges for all $|z|=R$, can we say that it converges absolutely on the radius of convergence? I tried to show that ${a_k}=\frac{e^{ik^2q}}{k}$ where $q$ is an irrational, serves as a counter example. In this case, $R=1$, and the series diverges absolutely […]

Average value of a complex valued function on a circle.

The following is an exercise from Complex Analysis by Stephen Fisher. Fix a complex number $a$ and a positive real number $R$. Suppose $u$ is a function defined on the circle of radius $R$ centered at $a$. Let $C$ denote this circle. Show that the average value of $u$ on $C$ is given by $\frac{1}{2\pi}\int_{0}^{2\pi} […]

Can we prove the Cauchy-Riemann equations using the matrix form of a complex number?

The definition of being complex-differentiable at $z$ can be stated as the existence of $a\in\mathbb C$ such that: $$f(z+h)-f(z)=ah+r(h)|h|$$ For all $h$, where $r(z)\to0$ as $z\to0$. Thinking of complex numbers as elements of $\mathbb R^2$ in the obvious way, this is asserting the existence of a differential for $f:\mathbb R^2\to\mathbb R^2$ at $z$, where that […]

Show that $\frac{P(z)}{Q(z)} = \sum_{k=1}^{n}\frac{P(\alpha_k)}{Q'(\alpha_k)(z-\alpha_k)}$

Here, $Q$ is a polynomial with distinct roots $\alpha_1, \ldots, \alpha_n$ and $P$ is a polynomial of degree $<n$. Once again, the task is to show $$\frac{P(z)}{Q(z)} = \sum_{k=1}^{n}\frac{P(\alpha_k)}{Q'(\alpha_k)(z-\alpha_k)}$$ For reference, this is page 32 exercise 2 in Complex Analysis by Ahlfors. I’m having a ton of trouble making connections between what is presented in […]

How to evaluate $\int_{-\infty}^{\infty}\frac{\sin^3(x)}{x^3}\mathrm{d}x$ via the complex epsilon method?

Exercise. Evaluate the improper integral $$\int_{-\infty}^{\infty}\frac{\sin^3(x)}{x^3}\mathrm{d}x$$ using the complex epsilon method. First we add the epsilon term. $$\lim\limits_{\varepsilon\to0^+}\int_{-\infty}^{\infty}\frac{\sin^3(x)}{x^3+\varepsilon^3}\mathrm{d}x$$ Using the fact that $$\sin^3(x)=\frac{3\sin x-\sin(3x)}{4},$$ we are left to evaluate $$\lim\limits_{\varepsilon\to0^+}\int_{-\infty}^{\infty}\frac{3\sin x-\sin(3x)}{4(x^3+\varepsilon^3)}\mathrm{d}x=\lim\limits_{\varepsilon\to0^+}\int_{-\infty}^{\infty}\Im\left[\frac{3\operatorname{e}^{\mathrm{i}z}-\operatorname{e}^{3\mathrm{i}z}}{4(z^3+\varepsilon^3)}\right]\mathrm{d}z.$$ There are a couple of singularities, namely $$z_k=(-\varepsilon^3)^{1/3}=(\operatorname{e}^{\mathrm{i}\pi}\varepsilon^3)^{1/3}=\varepsilon\operatorname{exp}\left(\mathrm{i}\frac{\pi+2\pi k}{3}\right),\ \ \ k\in\{0,1,2\}.$$ Considering the singularities on the upper complex plane (including the real number […]

solving for z in $|e^z| = 2$

How would I solve for z in the following case: $|e^z| = 2$, now I know that $|e^z| = e^a$ if we let $z = a+bi$ so then equating moduli we get $a = \ln{2}$ But what about $b$? $2 = 2e^{(0+2\pi n)i}$ so what is the value of b? Can be just be an […]