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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17 views

Finding an example of an analytic function, 0 on set of points (1/n)

As per the title, I want a function $f(z)$ which is analytic on $\mathbb{C}$ on the set of points $\{1/n\}$ for $n \in \mathbb{Z}^+$ and with $f(z) \neq 0$. What would this look like if the function ...
5
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2answers
63 views

Solve $e^{z-1}=z$ with $|z| \leq 1$

I'm looking for solutions to $$e^{z-1}=z$$ when $z \in \mathbb{C}$ with $|z| \leq 1$. The obvious solution is $z=1$, but I don't know how to show that there aren't any others. This question is ...
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0answers
56 views

Application of Rouché's theorem to $e^{z-1}=z$

I am reviewing my complex analysis and I got stuck with an exercise about Rouché's theorem. It states: for $0 \leq C \leq \frac{1}{e}$, show that $Ce^z=z$ has exactly one root in the closed unit disc. ...
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2answers
19 views

Singularities of $z^{2/3}$

I am trying to figure out whether $\infty$ is an essential singularity of $z^{2/3}$. Letting $z=\frac{1}{w}$, we have $f(1/w)=(1/w)^{2/3}$, which has a singularity at $w=0$. To determine the order, I ...
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1answer
13 views

Searching for a bound on a Mobius functions defined on $B(0,r)$.

Let $\alpha \in \mathbb C$ such that $|\alpha| \in (0,1)$. Prove that if $z\in \mathbb C$ is such that $|z| \le r < 1$ then : $$ \frac{\alpha+|\alpha| z}{\alpha(1-\overline{\alpha} z)}\le ...
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1answer
53 views

Any algorithm or theorem to decide whether two functions are equivalent? [duplicate]

Any algorithm or theorem to decide whether two functions that are polynomials,rationals and analytic over $\mathbb{N}$ or $\mathbb{Q}$ or $\mathbb{R}$ or $\mathbb{C}$ are equivalent ?
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1answer
36 views

Laurent series for $f(z)= \frac{1}{ (z-i)(z+2i)}$

I'm struggling with this question. I tried to break $f(z)$ using partial fractions and modify each equation so it looks like $\dfrac{1}{1-z}$ series but that's where I get stuck. Any help would be ...
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1answer
32 views

Zeros of a finite sum

I want to answer the question that, suppose $$g_{n}(z)=\sum_{k=0}^{n} \frac{1}{k!z^{k}}$$ and let $\epsilon>0$, for large enough $n$, are all the zeros of $g_{n}$ in the disk $D(0;\epsilon)$? I ...
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1answer
25 views

Analyzing Singularities of a Complex Function

I have the function $f(z) = \frac{1}{e^z - 1} - \frac{1}{z}$. I need to determine wither the singularities $z_0$ of the function are removable, a pole, or essential. $f(z) = \frac{1}{e^z - 1} - ...
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1answer
23 views

$ \sum_{m=1}^\infty\sum_{n=1}^{\infty}\frac{1}{|m+in|^p}<\infty $

I need some help with this problem: $$ \text{If} \;p>2 \Longrightarrow \sum_{m=1}^\infty\sum_{n=1}^{\infty}\frac{1}{|m+in|^p}<\infty $$ Note that ...
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33 views

Question about Cauchy-inequality

Let $f:\mathbb{C} \to \mathbb{C}$ a holomorphic function and assume that there exist $M > 0$ and $r>0$ such that $$ |f(z)| \leq M |z|\ln |z| $$ $\forall z \in \mathbb{C}$ with $z \geq r$. I ...
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0answers
35 views

What are the possible expansions for $f(z)$ at $0$ for disks and annuli?

For the expression $f(z)$ what are its all possible expansions (I am considering disks and annuli) around the origin and where do they converge? $$ f(z) = z + 2z^2 + 3z^3 + \ldots + nz^n + \ldots = ...
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1answer
18 views

A question on analytic functions

Suppose $f(z)$ is analytic in $\overline{D}_1(0)$. Let $z_n$=$\frac{1}{n}$, and the $f(z_n)=\frac{z_n}{z_n+1}$ for every positive integer $n$. Clearly 0 is the accumulation point of $z_n$ . From this ...
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1answer
17 views

Conformally mapping the unit disk to the upper-half plane

Within $\mathbb{C} \cup \{ \infty \}$, consider the unit-disk $\mathbb{D} = \{ z : |z|\leq 1 \}$ with three points labelled as $a$, $b$, $c$ on its boundary. I want to map $\mathbb{D}$ conformally ...
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26 views

Asymptotic behaviour of Fourier transform: $|F[f]|=|\lambda^{-k}F[f^{(k)}]|$ for absolutely continuous $f$

I read in Kolmogorov-Fomin's (p. 429 here) that if function $f:\mathbb{R}\to\mathbb{C}$ is such that $f^{(k-1)}$ [the $(k-1)$-th order derivative] on any finite interval and if $f,...,f^{(k)}\in ...
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1answer
30 views

Is there any holomorphic function in a unit ball

Is there any holomorphic function in a unit ball such that $f(1/n)=n^{-5/2}$ for $n=2,3,\dots$ Natural candidate is $f(z)=z^{5/2}$ But it isn't holomorphic obviously inside that ball. Can you tell me ...
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2answers
68 views

Why is the derivative of a polar function $dy/dx$ and not $dr/d\theta$?

I don't understand. If $r = 2\cos(\theta)$ then why is the derivative: $dy/dx$? I have a "hypothesis," By the polar equation are you really describing a curve in the cartesian plane? So is that ...
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2answers
20 views

Finding class of analytic function which are constant on cicrcles

Suppose $f(z)=u(x,y)+iv(x,y)$ $u^2+v^2=c$ could anyone tell me how to solve this?
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14 views

$z_0 = 0$ a branch point for $f(z) =(z + i)^{\frac{1}{2}}$?

I seem to have a mental block regarding branch points...I thought that the singularities of a function determined its branch points but then I read that they are irrelevant when deciding if a point is ...
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1answer
20 views

What is the image of this mobius transformation

Consider the standard mapping $w=\frac{1}{z}$. What is the image of the "half" plane above the line whose imaginary part is $c$, for the three cases of $c\gt 0 , c=0 , c\lt 0$? For $c=0$ obviously ...
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82 views

The integral $\int_{|z|=2}\log\frac{z+1}{z-1}dz$

Let $\log$ be the branch of the logarithm that extends the usual real logarithm, and consider on $D=\Bbb C\smallsetminus [-1,1]$ the function $$f(z)=\log\frac{z+1}{z-1}$$ I have to find the integral ...
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43 views

Find the linear fractional transformation that maps the circles |z-1/4| = 1/4 and |z|=1 onto two concentric circles centered at w=0?

I am very close to the solution I think. Since the circles cross the real axis, I want to find mappings from $z \to w$ such that $1/2 \to m$ and $0 \to -m$ and $-1 \to n$ and $1 \to -n$. Using $$ w = ...
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4answers
84 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log ...
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1answer
32 views

Integral from $e^{-itx}$ over $\mathbb R$

As a part of my task considering characteristic functions I have to compute $\int_{\mathbb R} e^{-itx}dx$ The result i get is $\frac{1}{-it}e^{-itx}|^{\infty}_{-\infty}$, but I don't really know ...
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13 views

Proof that the argument is continuous on a cut plane

I am trying to understand the following proof of continuity of the principal argument function $$\text{Arg}:\mathbb{C}\setminus\{x\leq 0 : x \in \mathbb{R}\}\to \mathbb{R}$$ which takes $z \in ...
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39 views

Undergraduate Complex Analysis: Use of Rouche's Theorem

We are asked to prove $ f = z^{3}e^{1-z} = 1 $ has exactly 2 roots inside $|z| = 1$ We've tried creating functions $p$ and $q$ where $p + q = f$, $p$ with 2 roots inside our boundary, and using ...
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39 views

Let $p(z)=z^n+a_1 z^{n-1}+…+a_n$ and suppose that $|p(z)|<1$ for $|z|=1$. Show that $p$ has a zero inside $|z|=1.$

Let $p(z)=z^n+a_1 z^{n-1}+...+a_n$ and suppose that $|p(z)|<1$ for $|z|=1$. Show that $p$ has a zero inside $|z|=1.$ Comments: I'm trying to use the Rouché's Theorem. I tried to write the way: ...
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1answer
34 views

Let $\lambda>1$ and show the equation $\lambda - z -e^{-z} = 0$ has exacly one solution in the half plane $\{z:Re(z)>0\}$ [duplicate]

Let $\lambda>1$ and show the equation $\lambda - z -e^{-z} = 0$ has exacly one solution in the half plane $\{z:Re(z)>0\}$. Show that this solution must be real. What happens to the solution as ...
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17 views

Find power series expansion and radius of convergence

I have to find power series of $\frac{1}{(z-1)(z-2)}$ centered at $3+i$ and give its radius of convergence. I just simply transformed it using partial fractions and geometric series expansion and ...
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1answer
25 views

Find all points where function is holomorphic

I have to find all points where function $$\frac{|z^2|\bar{z}}{e^{\bar{z}}}$$ is holomorphic. First thought was to check C-R equations, but it turned out to be a bad idea. It would take ages. There ...
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1answer
84 views

How useful is the Weierstrass representation of minimal surfaces?

Weierstrass representation of minimal surfaces says that if I have a holomorphic function $f: U \rightarrow \mathbb{C}$ and a meromorphic function $g: U \rightarrow \mathbb{C}$ such that $f g^2$ is ...
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1answer
27 views

Sobolev space $W^{1,2}((0,1))$ and boundary ODE - how does integration by parts goes?

As a part of a question about $W^{1,2}((0,1))$, I want to get a boundary ODE on $g$ and don't quite know how to integrate (?) in order to get the equation. let $g\in C^2[0,1]$ be our variable, $f\in ...
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1answer
71 views

Evaluating $\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$ using complex analysis

how do I compute $$\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$$ with complex analysis? I feel like im calculating the residue wrong and I cant get to the answer correctly. I tried to branch cut ...
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0answers
16 views

Poles of analytic functions are isolated

Can the set of poles of an analytic function $f:G\rightarrow \mathbb{C}$ contain a limit point? I know that the answer is no for open $G$, but after thinking more I have become paranoid about ...
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1answer
33 views

If $|\alpha|\leq 1$ and $|\beta|\leq 1$, prove that $|\alpha+\beta|\leq |1+\overline{\alpha}\beta|$

Note $\alpha$ and $\beta$ are complex numbers and $\overline{\alpha}$ is the conjugate of $\alpha$. I've tried using variations of the triangle inequality and I couldn't find anything to work.
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1answer
73 views

How prove exists $z_{0}$ such $|z_{0}|=1$,and $|f(z_{0})|\ge\frac{1}{2^{n-1}}\prod_{j=1}^{n}(1+|a_{j}|)$

let $a_{1},a_{2},\cdots,a_{n}\neq 0$ be given complex numbers and $$f(z)=\prod_{j=1}^{n}(z-a_{j})$$ I need to show that there exists a complex number $z_{0}$ such that $|z_{0}|=1$ and ...
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25 views

circular contour integral with complex numbers [closed]

Let gamma(w,R) denote the circular contour t maps to w + Re^it where 0 < t < 2Pi. Evaluate the integral of 1/1+z^2 when gamma is gamma(i; 1)
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24 views

Solution of gaussian integral with hyperbolic cotangent

I was wondering if the integral $$I=\int_{-\infty}^{\infty}d\omega \omega e^{-(\omega/a)^2}\coth(\frac{b\omega}{2})\cos(\omega c)$$ where $a,b,c>0$ can be solved using complex countour ...
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1answer
45 views

When is a path integrals just “ordinary integration”?

I was wondering about this: If you have the function $f(x)=x$ in the complex plane, then the path integral $\int_{\gamma}x dx $ for any path $\gamma$ connecting 0 and $z$ by a straight-line can be ...
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1answer
18 views

Understanding connectedness argument in proof of Analytic Fredholm Theorem

Let $X$ be a complex Banach space, and let $D \subset \mathbb{C}$ be a domain. Let $\mathcal{L}(X)$ denote the Banach space of bounded linear transformations $X \to X$. The Analytic Fredholm Theorem ...
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1answer
69 views

How prove this $|z|>1$ with $1+z+\frac{z^2}{2!}+\cdots+\frac{z^n}{n!}=0$

For give the postive integer $n$,and $z\in C$ such this $$1+z+\dfrac{z^2}{2!}+\cdots+\dfrac{z^n}{n!}=0$$ show that $$|z|> 1$$ maybe we Assmue that exst $z$ such $$|z|\le 1$$ then we ...
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1answer
10 views

Locally vs uniformly bounded in Montel's theorem

Wikipedia gives Montel's theorem saying essentially $\mathcal F$ is a normal family in $\mathbb H(G)$ iff $\mathcal F$ is uniformly bounded. My book say the same thing but iff $\mathcal F$ is locally ...
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1answer
21 views

Analytic map from punctured disk to punctured plane

If $\mathbb{D}$ is the open unit disk, we know from Liouville's theorem that there cannot exist a 1-1 analytic map from $\mathbb{D}\to\mathbb{C}$? But could there exist one from ...
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6 views

Entire function and biorthogonal set.

Let us suppose that $\{e^{i\alpha_n x}\}$ has a biorthogonal set and let us consider $$H(w)=\int_{-\pi}^{\pi}h_0(x) e^{iw x} dx, \ \ \ \ w\in\mathbb C$$ such that $H(\alpha_n)=0$, $n\neq 0$ where ...
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1answer
16 views

Entire function $\phi(z)$

Let us assume $\phi(z)$ entire function (see enter link description here). Is $$\frac{\phi(z)}{\prod_{n=1}^{+\infty} \left(1-\frac{z^2}{\alpha_n^2}\right)}$$ still a entire function? Where ...
5
votes
2answers
60 views

Prove that $\zeta(4)=\pi^4/90$

I am asked to "use the calculus of residues" to prove that $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{1}{n^4}=\frac{\pi^4}{90}$$ I think I can do this given the Laurent series for $\cot z$ ...
3
votes
1answer
108 views

conformal mapping, regions of the complex plane marked +/-, find the function f,

The picture shows what the function f: $\mathbb{C}\to\mathbb{C}\cup\infty$ does to the plane. The values 0 at 0, 1 at $\pm$1, and $\infty$ at $\pm i$ are specified. To elaborate on the picture: ...
3
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0answers
80 views

Approximation of holomorphic functions and topological properties

So, in the last couple of lectures of my complex analysis class we've proved some approximation theorems of holomorphic functions. Eventually, we showed the following propositions: Theorem 1. Let ...
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3answers
27 views

Questions about poles

Find order of pole for $$f(z)=\frac{1}{e^{z}-1}$$ at $z=0$. Now I turned the function into this: $$\frac{1}{\sum_{1}^{\infty}x^k/k!}$$ I think the pole has order $1$ but $\lim(z(f(z)))$ seems to be ...
0
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0answers
21 views

removable singularity and injective function

Let $U \subset \mathbb{C} $ a conected open subset, $ a \in U $ and $ f:U- \{a\} \to \mathbb{C}$ a holomorphic function such that $ V=f (U-\{a\}) $ is a open bounded subset. (A) Show that $ f $ has a ...