# Tagged Questions

The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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### Contour integral independant of parametrisation

I have a question about the definition of contour integrals in $\mathbb{C}$. The same question could be applied to line integrals in $\mathbb{R}^n$ though. $\Gamma \subseteq \mathbb{C}$ is called a ...
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### Square root of several variables analytic function

The question is as follow: Let $H\subset \mathbb{C}^n$ be an simply-connected region. If $f$ is a nowhere vanishing analytic function on $H$, with $f(z)>0$ for all $z\in H\cap\mathbb{R}^n$, ...
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### The identity $\sqrt[n]{z}\sqrt[n]{w} = \sqrt[n]{zw}$ for complex numbers

In the general case, when $z$ and $w$ are two complex numbers, we have that $(1) \sqrt[n]{z}\sqrt[n]{w} \neq \sqrt[n]{zw}$ For example, $\sqrt{-1}\sqrt{-1} \neq \sqrt{-1.-1} = 1$. However, there ...
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### Mean Value Property to show that entire function is a constant

Let $f(z)$ be an entire function so that, $$\int \frac{|f(z)|}{1 + |z|^3} dA(z) < \infty$$ where the integral is taken over the entire complex plane. Show that $f$ is a constant. I believe ...
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### Holomorphic functions and complex conjugation

Suppose I have given two holomorphic functions $g,f:\mathbb{C}\backslash(-\infty,1]\rightarrow \mathbb{C}$ and I know that $\overline{ g(z)}=f(z)$ for all $\vert 2-z \vert <1.$ I am wondering if ...
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### inequality by taking reciprocal or other way to check if pole lies inside unit circle

If $a^2$ <1 is given in the problem then how do we prove that the pole z=1/a lies outside the unit circle ?
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### Finding a branch of the complex logarithmic function $\log(1-z).$

I have a question that asks me to find the holomorphic branch $L(1 − z)$ of $\log(1 − z)$ valid in the cut-plane $z \in \mathbb{C}\setminus [1, ∞)$ and such that $L(1) = 0.$ We have defined the ...
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### Maximum and minimum modulus principle

Let $U\subset \mathbb C$ be a bounded domain and $f:\overline{U}\to\mathbb C$ continuous and holomorphic $U$. Show that $|f(z)|\leq\max\{|f(w)|:w\in\partial U\}$ for all $z\in U$. Show that ...
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### Show $\frac{\pi^2}{8}=\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}$ [closed]

Knowing $$\frac{\pi^2}{sin^2(\pi z)}=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n)^2}$$ how can I prove $$\frac{\pi^2}{8}=\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}\qquad?$$
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### How to simplify the following expression involving Jacobian elliptic functions?

I would like to show that a certain elliptic function $F(x)$ (that is periodic, say with some period $h$) has exactly two zeroes in $[0,h)$. Let us recall some notation. Given a parameter $m \in [0,1]$...
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### Function satisfying inequality has no root

Let $f$ be an entire function such that, for all $z \in \mathbb{C}$ with $|z| > 1$, $$|f'(z)| < \frac{|f(z)|}{|z|^2} < 1$$ Show that there is no $a \in \mathbb{C}$ such that $f(a) = 0$. ...
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### Relationship between the residues $Res(g\circ\varphi,z_0)$ and $Res(g,w_0)$

Let $\varphi:U\rightarrow\mathbb{C}$ be holomorphic with $\varphi'(z_0)\neq 0$ for some $z_0\in U$. Let $g$ be another function having a pole of order $1$ in $w_0=\varphi(z_0)$. What is the ...
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### Polynomial has not roots on disc D($0$,$1$) [duplicate]

Let p($z$)=$a_n$$z^n+..+a_0 with 0< a_n \le a_{n-1} \le...\le$$a_0$ .Show that p($z$) has not roots on D={ ℂ $\exists$ $z$ : $|z|<1$ }. thanks
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### Why is a absolutely and uniformly convergent series in the complex plane holomorphic?

Suppose I have some series $f(z) = \sum_{k = 0}^\infty a_n(z)$, with $a_i$ holomorphic on $\mathbf{H}$, that is absolutely convergent for all $z \in \mathbf{H}$ and uniformly convergent on compact ...
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### Katok book Exercise 2.8.3 [closed]

Prove that for any holomorphic function w wich is not a polynomial there exist a number $\lambda = exp2\pi i \alpha$, where $\alpha$ is irrational, such that the linearized equation (2.8.3) does not ...
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### Discontinuous complex integral

I would like to compute the integral $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+\infty}\frac{x^{s}}{s(s-1)}ds$$ for $x \geq 1$ and $c > 0$. I know that it should be $\sim x$ and that if $s-1$ ...
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### Series expansion of $1/(1+z^2)$ about the point I
I am trying to find a series representation for the complex function: $1/(1+z^2)$. The text I am reading gives: $1/(1+z^2) = 1/((z+I)(z-I)) = -I/(2(z-I)) +1/4 - I(z-I)/8 - (z-I)^2/16 + ...$ I do not ...
### General method: show subset of $\mathbb{C}$ is connected
Consider the two sets $$A = \{z \in \mathbb{C} : |z^2 - 3| < 1\}, ~~~~ B = \{z \in \mathbb{C} : |z^2 - 1| < 3 \}$$ $B$ is connected, while $A$ is not. However, I have no idea how to prove this....