Articles of complex analysis

special values of zeta function and L-functions

I was reading in some lectures notes about the Riemann zeta-function which takes on special values: $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$ In fact, we can compute even values of the zeta function $\zeta(2k) \in \pi^{2k}\mathbb{Q}$. We can twist with a quadratic character and get more special values that way: $$ L(\chi_4, 2k+1) = \sum_{n […]

Maximum value of the modulus of a holomorphic function

I’m looking for the maximum value of the modulus of a holomorphic function, and I am getting a bit stuck. The function is $$(z-1)\left(z+\frac{1}{2}\right)$$ with domain $\,|z| \leq 1\,$ Now, I know by the maximum modulus principle the max value will occur on the boundary. So by multiplying the two expressions I get: $$\left|z^2 – […]

Evaluate $\int_0^{2\pi} \frac{d\theta}{(1-a\cos(\theta)+a^2)}$

Evaluate $\displaystyle \int_0^{2\pi} \frac{d\theta}{(1-a\cos(\theta)+a^2)}$ Super general. I get to a step: $\displaystyle \frac{2}{i}$ multiplied by Path integral $\displaystyle \frac{z}{[(2-a)z^2 + 2(a^2 z) + a]}.$ No idea if I’m on the right track. Maybe distribute the $i$? Wondering if I can get some help.

Conformal mapping between regions symmetric across the real line

In Conway’s Functions of One Complex Variable, the section on the Riemann Mapping Theorem has the following exercise: Let $G$ be a simply connected region which is not the whole plane, and suppose that $\bar{z}\in G$ whenever $z\in G$. Let $a\in G\cap\mathbb{R}$ and suppose that $f:G\rightarrow D=\{z:|z|<1\}$ is a bijective analytic function with $f(a)=0,\ f'(a)>0$. […]

Bounds of $|f(z)|$ for specific z via Schwarz Lemma and the such

I’m trying to brush up on some complex analysis for a qualifying exam. I’m quickly realizing that I have never done many applications of Schwarz. In particular I don’t have much experience with those types of questions that ask to find a bound for $|f(z_0)|$ for some specified $z_0$. So my questions are (1) Can […]

Find all entire functions such that $|f(z)|\leq |z^2-1|$ for all $z\in\mathbb C$.

Find all entire functions such that $|f(z)|\leq |z^2-1|$ for all $z\in\mathbb C$. For large $z$ we have $$|f(z)|\leq 2|z|^2$$ so $f$ is a polynomial of degree $\leq 2$. But how to continue? Could someone give me a hint?

Constructing the Riemann Sphere

Problem: In the construction of the Riemann sphere, we begin with the sphere $\mathbb{S}^2$ with two charts: the stereographic projection $\sigma_N : \mathbb{S}^2 \setminus \{N\} \to \mathbb{R}^2 \cong \mathbb{C}$ from the North pole, $N$, given by $$ \sigma_N (x_1, x_2, x_3) := \frac{(x_1, x_2)}{1-x_3}, $$ the stereographic projection $\sigma_S : \mathbb{S}^2 \setminus \{S\} \to \mathbb{R}^2 […]

Verifying $|F(r)| \geq \frac{1}{1-r}\log(\frac{1}{1-r}) $ and $|F(re^{i \theta})| \geq c_{q/r}\frac{1}{1-r}\log({\log(\frac{1}{1-r})})$

I’m attempting to take a Tauberian route in verifying the proposition in $(1)$ below, which is from Complex Analysis, by Elias M Stein and Rami M. Shakarchi. Let $F(z)$ be the following series: $$F(z) = \sum_{n=1}^{\infty}d(n)z^{n} \, \, \text{for} \, |z| < 1$$ $$\text{Remark}$$ One can also observe the following relationship: $$\sum_{n=1}^{\infty}d(n)z^{n} = \sum_{n=1}^{\infty} \frac{z^{n}}{1-z^{n}}$$ […]

Evaluate the definite integral $ \int_{-\infty}^{\infty} \frac{\cos(x)}{x^4 +1} \ \ dx $

I am having more trouble with this problem then I feel like I should be. I set $ \ \cos(x) = e^{ix} \ $ and $ \ x^4 +1 = e^{\pi i /4} \ $ or $ \sqrt{i} \ $. I think I am suppose to do a residue to evaluate it but I’m not […]

Norm of a functional on square integrable harmonic functions

Let H be the Hilbert space of square integrable (real) harmonic functions on the unit disk of the complex plane. I want to find the norm of the linear functional $$h\mapsto h_x(0)$$ Here is my proof that this functional is bounded. The partial derivative $\frac{\partial h}{\partial x}(z)$ is also harmonic, therefore by the mean value […]