# Tagged Questions

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### Prove: $|f(z)| ≤ \frac{2A|z|}{1−|z|}$.

Let $f$ be a holomorphic function on the unit disc $\{z : |z| < 1\}$ satisfying $f(0) = 0$ and $Ref(z) ≤ A$ for some positive number $A > 0.$ Prove: $|f(z)| ≤ \frac{2A|z|}{1−|z|}$. Not sure how ...
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### Weird conformal map problem

Construct a conformal map from the region $\omega$ = open disk of radius 1 centered at 0 minus the closed disk of radius 0.5 centered at 0.5 to $\mathbb{D}$ = disk radius 1 centered at 0. I really ...
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### Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$.

Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$. $(a)$ Show that $f(V) \subset V.$ $(b)$ Let $f_n$ be ...
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### Separating a Complex Valued Function

Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $f(z)=u(x,y) + iv(x,y)$? Thank You, C.A
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### holomorphic function with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
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### To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
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### holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
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### How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$?

I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of ...
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### How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
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### If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
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### Neighbourhood of a disc

I'm a bit confused on how to write down precisely a neighborhood on an example. My question is the following: Suppose I have a disc $\Omega=\lbrace x\in\mathbb{C}, |x-1|<2.5\rbrace$ and its ...
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### How to show that if möbius transformation has an inverse, then it is injective?

Let $f(z)$ be möbius transformation. How to show that if möbius transformation has an inverse, then it is injective? I mean why don't you use this definition to show injectivity of möbius ...
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### Asymptotic expansion of $z^{-x}$

Consider the function $z\mapsto z^{-x}$ for $x>1$ (real) and $z$ in the cut complex plane $\mathbb C\backslash\{z\leq 0, \text{ real}\}$. Does this function have an asymptotic expansion of the form ...
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### If $f$ is holomorphic, then there is a holomorphic function $h$ such that $e^{h(z)}=f(z)$

Let $f:G\to\mathbb{C}$ denote a holomorphic function over a star-shaped domain $G$ and $f\ne 0$ on $G$. I want to show that it holds $\frac{f'}{f}$ is holomorphic There is a holomorphic function ...
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### $\lim\limits_{R \rightarrow \infty}{|\cos z|}$ in complex numbers

Let $z=Re^{i \alpha}$ $\alpha$ is given number and $\alpha \in (0, \frac{\pi}{2})$ Find (if exist) $\lim\limits_{R \rightarrow \infty}{|\cos z|}$ What happen if function $f(z)=|\cos z|$ when ...
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### Given a family $\mathcal{F}$ of complex differentiable functions, prove that the derivative at a point is bounded.

We were asked to prove the following claim as part of an alternative proof of the Riemann mapping theorem. Let $G$ be a (proper) simply connected domain, and $z_0\in G$. Consider the family ...
### Laurent series of $\frac{e^{iz}}{z^2+p^2}$, $p>0$.
I need help finding the main part of the laurent series of $f(z)=\frac{e^{iz}}{z^2+p^2}$ in $ip,-ip$ since these are the two poles of $f$. Due to the orders of the poles are 1 I just have to find ...