# 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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### Examples in physics or engineering in which choice of branches of complex root or log make differences

Are there any nontrivial examples in physics or engineering in which a choice of the branch of complex root or log matters to obtain the "correct" solution of the problem? Of course, I'll not choose a ...
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### $\int_{\gamma(.,s)}f(z)dz$ is independet from s (homotopy lemma)

Let $U\subset\mathbb{C}$ open, $f:U\to\mathbb{C}$ continuous and complex differentiable. Let $\gamma:[a,b]\times [c,d]\to U$ be in $C^2([a,b]\times [c,d])$. And assume that one of the following ...
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### How is orthogonality between a line and a circle most simply defined?

I was asked to show that if I have a generalized circle C, and two points not in C, then every circle going through these two points is orthogonal to C iff these two points are symmetric. Since I ...
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### Proving this complex function is constant

Suppose that $f$ is an entire function such that $g(z)=f(1/z)$, $z\in C\setminus \{0\}$ has a removable singularity at 0. Show that $f$ is constant. I think this has to be proved by Liouville's ...
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### Q: How to evaluate the contour integral $\int_{z_1}^{z_2}\bar z^ndz$

I'm trying to evaluate $\int_{z_1}^{z_2}\bar z^ndz$ along the straight line $[z_1,z_2]$, where $n \in \mathbb{N}\cup \{0\}$ (I found this problem here). As has been suggested before there, I've tried ...
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### How to find a family of analytic functions with a modulus of boundary value = 1, and no zeros inside the unit disk?

a) Find all $f(z)$ satisfying i) is analytic in $|z|\le 1$ ii) has no zeros in $|z|<1$ iii) $|f(e^{i\theta})|=1$ for $0<\theta<2\pi$ b) What are the possible $f(z)$ if (ii) is replaced by ...
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### Given that $f$ is entire, $f(0) = 1$ and $f\left(\frac{1}{n}\right) = 1 +\frac{i}{n} - \frac{1}{n^2}$, find $f(1 - 2i)$

Let $f$ be an entire function such that $f(0)=1$ and $f\left(\frac{1}{n}\right)=1+\frac{i}{n}-\frac{1}{n^3}$ for each positive integer $n$. Find $f(1-2i)$. Ideas: If ...
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### Finding a harmonic function on half-disk that is equal to $1$ on the semi-circle and $0$ on the diameter

I first showed that the mapping $$z + \frac{1}{z}$$ sends the upper semi-disk, $\{|z|<1, \operatorname{Im} z >0\}$, along with the real line from $-1$ to $1$, to the whole of the real line in ...
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### Given a sequence $\{x_k\}$ in $\mathbb{R}^n$. Let ||x|| denote the Euclidean norm of $x∈\mathbb{R}^n$.

Given a sequence $\{x_k\}$ from k=1 to infinity in $\mathbb{R}^n$. Let ||x|| denote the Euclidean norm of $x∈\mathbb{R}^n$. Recall: (a) $\{x_k\}$ from k=1 to infinity is bounded if there exists ...
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### If $\frac{(x+1+i)^n -(x+1-i)^n}{2i}=k$ and $\cot(Q)=x+1$ prove $k=\csc^n(Q)\cdot\sin(nQ)$

If $$\frac{(x+1+i)^n -(x+1-i)^n}{2i}=k$$ and $$\cot(Q)=x+1$$ prove that $$k=\csc^n(Q)\cdot\sin(nQ)$$ replacing value of $\cot{Q}$ $$\frac{(\cot(Q)+i)^n -(\cot(Q)-i)^n}{2i}$$ and taking out ...
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### Define two sets in $x\in \mathbb{R}^2$ by A:= {(x,y) | y ≥ e^x} and B:= {(x,y) | y=0) [closed]

Define two sets in $x\in \mathbb{R}^2$ by $\mathbb{A}$:= {(x,y) | y ≥ e^x} and $\mathbb{B}$:= {(x,y) | y=0). Prove that $\mathbb{A}$ + $\mathbb{B}$ = {(x,y) | y>0}. Here, $\mathbb{A}$ + ...
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### Showing that $\overline{\int_{\gamma}f(z)dz}=\int_{\overline{\gamma}}\overline{f(\overline{z})}dz$

I am trying to show that Let $\gamma$ be a piecewise-$C^1$ curve, and let $\gamma$ be its image under the mapping $z\mapsto \overline{z}$ (symmetry in the real axis). Then if $f(z)$ is continuous on ...
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### Composition of lower semicontinuous function and continuous function is lower semicontinuous

stuck on this question, I went through my notes and couldn't find any proof to this, it's only shown as a statement. Help? Assume that $f:\mathbb{R}^n\to\mathbb{R}$ is lower semicontinuous at $g(a)$ ...
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### Complex entire function on $D=\{z:Im(z)=0\}\cup\{z:Im(z)=a\}$ for some $a>0$.

This question is a duplicate of this. I really couldn't understand the answer for $(B)$ given in the question. Can anybody explain it to me? It's important and I am not so smart in complex analysis. ...
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### Is this a continuity / connectedness argument or is it an orientation-preservation argument?

Take, for example, the simple linear fractional transformation that sends the upper half plane to the unit disk, and the real line to the unit circle. We know the fact that the UHP maps to the inside ...
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### Asymptotic expansion of Legendre polynomial

I am trying to determine the asymptotic expansion of the Legendre function as described in a Bender and Orszag text, but have been unable to - some solutions are online, but they approach it in ...
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### Is it correct to say that conformal mappings (not just the class of linear transformations) preserve orientation?

We know that conformal mappings preserve angles and orientation of any two intersecting curves in the z-plane. Is this fact alone enough to conclude that the region (domain?) to the right of some ...
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### Value of $\gamma(0)$ if $\gamma(t)=e^{2\pi it\sin(1/t)}$

Let's take a look at problem 18 on chapter 6 in PMA Rudin. Let $\gamma(t)$ be curve in the complex plne, defined $[0,2\pi]$ by $$\gamma(t)=e^{2\pi it\sin \frac{1}{t}}.$$ It's defined on ...
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### Is this the right way to find the image of the interior of a half-disk, under the mapping z+1/z

I am trying to find the image of the interior of the half disk {|z|<1, Im z>0} under the mapping $$z + \frac{1}{z}$$ and the problem statement also asks to find the images of specific points A, ...
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### (Non)density of the set of rational functions in the set of meromorphic ones

Let R (resp. M) be the set of rational (resp. meromorphic) functions from $\mathbb{C}$ to $\mathbb{C}$. What can we say about the relation among R and M? E.g. is R nowhere dense in M (w.r.t. ...
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### Can I think of the conformal mapping w = (z+1/z) as a linear fractional transformation?

The mapping $$w = z + \frac{1}{z}$$ looks linear in $z$. However, it would not be in the form $$\frac{Az+B}{Cz+D}$$ since putting the two terms together gives $$\frac{z^2+1}{z}$$ So my ...
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### Determining a line integral of $\frac{p'}{p}$ with $p$ being a polynomial

Let $p: \mathbb{C} \to \mathbb{C}$ be a polynomial, and $R > 0$ large enough, so that all roots of $p$ lie within $B_R(0)$ (the open ball with radius $R$ around $0$). I now want to determine: ...
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### Find image of $D=\{z: |z|<1\}$ under $f(z)={z\over z-1}$.

Find image of $D=\{z: |z|<1\}$ under $f(z)={z\over z-1}$. I know it is supposed to be $\Re z <{1\over 2}$, but unfortunately I've been trying to show this for too long and my sanity along with ...
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### Prove that for a generalized circle $C$ and $z,w\notin C$, $z=w^*$ iff every cline $C_1$, $z,w\in C_1$, $C\bot C_1$.

Prove that for a cline (a generalized circle) $C$ and $z,w\notin C$, $z=w^*$ iff every cline $C_1$, $z,w\in C_1$, $C\bot C_1$. Hint: show that it is enough to show this for the real line. I tried ...
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### If a locally convergent power series factorises in $\Bbb{C}[[x,y]]$, then each factor is locally convergent

I've just started reading through Miranda's "Algebraic Curves and Riemann Surfaces", and there's one small bit I can't seem to work out: it seems like it should be easy, and indeed the author says ...
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### Tradeoffs Between Mapping Vs. Domain Coloring for Visualizing Functions of Complex Variables

There appears to be a variety of methods for visualizing functions of complex variables. I'm considering the function: $f(z) = z^2+1, z \in \mathbb{C}$ As I understand it, one way to think about ...
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### The maximal domain of the function $f(z)=z^3$ [closed]

How to approach to quastion like: What is the maximal domain of the function (for example) $f(z)=z^3$ What are the steps? I know that $z^3=x^3-3y^2x+(3yx^2-y^3)i$
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### A problem with Fourier Transform about the growth condition with a constraint.

A problem in Stein & Shakarchi Complex Analysis Suppose $\hat f(\xi)=O(e^{-a|\xi|^p})$ as $|\xi|\rightarrow\infty$, for some $p\gt 1$. Then $f$ is holomorphic for all $z$ and satisfies the ...
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### Problem on entire function which is reduced to constant

Suppose $f$ is an entire function satisfying any one of the following conditions for all $z\in \mathbb C$ (1) im$f(z)$ has no zeros (2)$|f(z)|\geq 1$. Then f is constant. My thought: For(2) ...
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### (Anti-) Holomorphic significance?

What are holomorphic and anti-holomorphic components? Why don't we call them complex components and their conjugates? What is holomorphic coordinate transformation?
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### If $f(z)$ analytic in the domain $D$ and $f'(z)=0$ so $f(z)=$constant?

I've been thinking how to prove that an analytic function $f$ in the domain $D$ is a constant if $f'(z)=0$ in every point in $D$, but I haven't figured it out yet. What I was thinking is to use ...
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### Write a formula for $z^*$ with respect to line $l=\{ax+by=c\}$.

Write a formula for $z^*$ with respect to line $l=\{ax+by=c\}$. ($z^*$ is the symmetric point of $z$ with respect to $l$) At first, I thought of solving it like I used to in High-School, but then ...
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### Why does the principle value of $z^{\frac{1}{2}} := \sqrt{r}e^{i\frac{\theta}{2}}$ need to be restricted to one rotation in order to be continuous?

I understand that the function $f(z) = z^{\frac{1}{2}}$ is multi-valued, since it has exactly two solutions for every $z$. However if I take the principle value square root function, expressed as ...
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### Existence of zeroes of an analytic function in closed unit disk

Let f be a non constant analytic function in the closed disk $D=\{z:|z|\leq1\}$. Suppose that $|f(z)|=K$ (K is constant) for all $z$ on the circle $|z|=1.$ Show that f has no zero in $D$ I think if ...
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### Show that $\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right)$

I'm trying to show that $$\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right)$$ using Jordan's lemma and contour integration. MY ATTEMPT: The function in ...
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### Limiting behaviour of a real Möbius sequence

Consider the fractional linear transformation of the real variable $t$, transformation $$f(t)=\frac{at+b}{ct+d}$$ where $a,b,c,d\in\mathbb{R}$. Define $t_{n+1}=f(t_n)$ where $t_0\in\mathbb{R}$. There ...
In the book Complex Variables and Applications by Churchill page 32: An open set S is connected if each pair of points $z_1$ and $z_2$ in it can be joined by a polygonal line, consisting of a ...