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 […]

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

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 $f(a)=0,\ f'(a)>0$. […]

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 $z\in\mathbb C$. 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?

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 […]

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: $$\sum_{n=1}^{\infty}d(n)z^{n} = \sum_{n=1}^{\infty} \frac{z^{n}}{1-z^{n}}$$ […]

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 […]

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 […]

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