Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

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For which alpha does this system of equations spiral towards the orgin?

I am stuck with a math problem and I hope that somebody could help me in the right direction. Question: For which values of $\alpha$ do the trajectories of the solutions of the systems $x^{\prime}=A x$...
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How objectively do we evaluate likelihood of two very rare events?

I've been pondering the concept of objectiveness to evaluate likelihood and predictability of extremely rare events. I'd like to demonstrate my questions over trivial hypothethical examples below. ...
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Proof of Bloch's theorem: continuity of $h(r)$

Bloch's Theorem. Let $f$ be an analytic function on a region containing the closure of the disk $D= \lbrace z ; |z|<1 \rbrace$ and satisfying $f(0)=0$ and $f'(0)=1$, then there is a disk $S\subset ...
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How to show that $||\int_{a}^{b}f(t) dt||_2 \leq \int _{a}^{b}||f(t)||_2 dt $?

Let $f: [a,b] \to \mathbb{R}^2$ be continuous . Why is $||\int_{a}^{b}f(t) dt||_2 \leq \int _{a}^{b}||f(t)||_2 dt $ $\quad$ (the first integral is defined componentwise and $||.||_2$ is the ...
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Prove that there isn't a nonconstant monic polynomial $p(z)$ of degree $n$ so that $|p(z)| < R^n$ on $|z| = R$, where $R > 0$

Prove that there does not exist a nonconstant polynomial $p(z)$ in the complex variable $z$ so that $|p(z)| < R^n$ on $|z| = R$, where $R > 0$ and $p(z)$ is monic and of degree $n$. I know that ...
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Confusion regarding pole of a complex function.

I am a graduate student.I am studying complex analysis.I encountered the following problem in a lecture: Find the residue of $f(z)=\frac{z-\sinh(z)}{z^2\sinh(z)} $ at $z=\pi i$. Now,this problem is ...
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1 answer
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Finding unknown from the given complex integral.

Find real number a such that $\oint_c \frac{dz}{z^2-z+a}=π $ where c is the closed contour |z-i|=1 taken in the counter clockwise direction. This is a question that has been asked in the 2021 NBHM ...
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2 votes
1 answer
57 views

How do I obtain this expression for the Taylor series of$\frac{z^2}{\sin ^2(z)}$?

Reading about Complex Analysis, I came across the following: Consider first the representation $\frac{\pi ^2}{\sin ^2(\pi z)}=\sum_{n\in \mathbb{Z}}\frac{1} {(z+n)^2}$, which applies for all $z\in \...
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What fails in the maximum value principle if the domain is not connected?

We have discussed the maximum value principle: Let $\Omega \subset \Bbb{C}$ be an open connected subset and $f:\Omega \rightarrow \Bbb{C}$ be an analytic function. Assume that there exists $z_0\in \...
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Hilbert Transforms and Analytic signals in signal processing

Hilbert transforms are used in signal processing for creating "analytic" signals (see section 4.4 here https://www.di.ens.fr/~mallat/papiers/WaveletTourChap1-2-3.pdf for example) which are ...
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Squaring complex number equation with absolute values

I don't understand how you go from the first to the second line in this problem : $$|(a-k)+i(7-2a)|=|(a-2)+i(9-2a)|$$ $$(a-k)^2+(7-2a)^2=(a-2)^2+(9-2a)^2.$$ Firstly, squaring i should make it -1 I ...
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2 answers
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Showing the integral $\int_{C_N} \frac{1}{(2z-1)\sin{\pi z}}dz$ converges to zero as $N \to \infty$

I have a question about the 10th question part b of Chapter 11 of the Complex Analysis by Bak & Newman. The question says that Show that $1-1/3+1/5-1/7+...=\pi/4$ by using the integral of $\frac{1}...
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Properties of the critical strip [closed]

I'm searching for the known properties of the critical strip. I have searched a bit on the web but did not find the awnsers i need. Regards, Justin Edit: I’m searching for facts that are known ...
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$z_0+r*e^{2\pi imt}$ meaning of $m$

I know that $f:[0,1] \to \mathbb{C}, f(t)=z_0+r*e^{2\pi imt}, r>0, m\in\mathbb{Z},z_0\in\mathbb{C}$ parameterizes the circle around $z_0$ with radius $r$. But how does $m$ affect the ...
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Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$

Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$ using residues. So I have a theory how to calculate $PV \int_{-\infty}^{\infty} f(x)e^{iax}dx$ a>0, but I don’t know how to ...
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Where does $\;f(z)=\displaystyle \frac{1}{z} +e^{2i \gamma}z\;$ maps the unit disk $|z|<1$ in $w$- plane? given that $0 \leq \gamma \leq 2\pi$

Where does mapping $\;f(z)=\displaystyle \frac{1}{z} +e^{2i \gamma}z\;$ maps the unit disk $|z|<1$ in $w$- plane? given that $0 \leq \gamma < 2\pi$ Solution i tried - Given mapping is $$\;f(z)=\...
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Mapping the common part of the disk $|z|<1$ and $|z-1|<1$ on the inside of the unit circle

Question: Map the common part of the disk $|z|<1$ and $|z-1|<1$ on the inside of the unit circle. This is question 1 on page 96 of Ahlfors book. This question is also asked (and answered) here:...
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1 answer
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Unsure how to resolve two contradicting identities in finding the residue of a complex function

In my complex analysis class I was solving the integral: $$I=\int_{-\infty}^\infty\frac{\sin(2x)}{x^2+x+1}dx$$ using contour integration. I initially tried the integral over the semicircle of radius $...
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Find Taylor series for $\cosh z \cos z$

Find Taylor series for $\cosh z \cos z$. $\cos z = \cosh iz$ and $\cosh z \cosh iz = \dfrac{1}{2}(\cosh (i+1)z + \cosh (i-1)z)$ and finally $$\cosh z \cos z = \dfrac{1}{2}\sum_{n=0}^\infty {\dfrac{(i+...
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2 answers
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When will the quadratic equation $z^2+z_1z+z_0=0$ have a root lie on the unit circle

Very similar to this question, but what if the coefficients are complex? Is there a necessary and sufficient condition to guarentee that there is at least a root on unit circle for $z^2+z_1z+z_0=0$ ...
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$f:D(0,1)\to \mathbb{C}$ , $e^{f(z)}$ is constant $\Rightarrow f$ is constant

Let $f:D(0,1)\to \mathbb{C}$ continuous and $e^{f(z)}$ is constant $\forall z\in D(0,1)$ show $f$ is constant. $\cdot $ $e^f=c=e^{x_0+iy_0}$ and $c\neq 0$ $\Rightarrow f-(x_0+iy_0)=k(z)2\pi i$, ...
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how to find the table integral that contains factorial of n?

I read in this page that" There are lots of definite integrals that depend on a parameter n∈N and whose result contains factorials of n " I am really struggle with this integral \begin{align*...
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An asymptotic in the hint of Stein's Complex Analysis Chapter 6 Exercise 12(a).

The entire statement of Stein's Complex Analysis Chapter 6 Exercise 12(a) is Show that $1/\lvert\Gamma(s)\rvert$ is not $O(e^{c\lvert s\rvert})$ for any $c>0$. And the hint of this problem is If $...
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1 answer
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Complex polynomial that sends real line to real line, positive imaginary part to positive imaginary part, negative imaginary part to...

Let p be a polynomial that sends points on the real line to points on the real line, sends points with positive imaginary part to points with positive imaginary part, and sends points with negative ...
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Set $T^{\mathbb N}x$ dense in $\mathbb S^1$ (Poincaré recurrence theorem)

Let $Ω =\mathbb S^1$ be the unit circle in $\mathbb R^2 = \mathbb C$, and let $T : Ω → Ω$ be multiplication by $e^{i\alpha}$. For $α \notin π\mathbb Q$ and every $x ∈ Ω$, is the set $T^{\mathbb N}x$ ...
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2 answers
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How do we conclude that $\int_{-\infty}^\infty x/(1+x^2)dx$ is not convergent?

In computing the integral $\int_{-\infty}^\infty \frac{x}{1+x^2}dx$ I get the following answers: $0$ due to the symmetry of the integrand. $\left.\ln(\sqrt{1+x^2})\right|_{-\infty}^\infty$ from brute ...
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Is $g(z)=1/f(z)$ a meromorphic function? [closed]

If $f(z)$ is a meromorphic function on $\mathbb{D}$,let $g(z)=\frac{1}{f(z)}$, How can we prove that $g(z)$ is a meromorphic function too?
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Different sequance problem

Let $x \in \Bbb{R}^+$, $a_1=x, a_2=x^{x}$ and $a_n=x^{a_{n-1}} \ \forall n\geq 3 $ $$S=\{x \in \Bbb{R}^+: a_n \rightarrow a \in \Bbb{R} \}$$ find the $sup(S)$ I can show $ 0<x<1 \vee 1<x<\...
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3 votes
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Description of complete analytic function for $\sqrt{4z-\sqrt[3]{z}}$.

The problem is to describe all branches and all the curves of analytic continuation of $\sqrt{4z-\sqrt[3]{z}}$. I started with representing function in a way $\sqrt{4w^3-w}\circ\sqrt[3]{z}$ So, there ...
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Derivation of Area theorem (conformal mapping).

While solving the Area theorem , i'm facing trouble in understanding the equation in these two black boxes, i know how they write $\displaystyle A=\frac{1}{2}\int_{c} R^2 d\phi$, but how they got its ...
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1 vote
1 answer
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Limits of abstract smooth surfaces

For $r >0$, let $K_r \subseteq \mathbb{C}$ be the closed subset, $K_r = \mathbb{C} \setminus D(0,r)$. Define $S_r$ to be the quotient of $K_r$ under the identification: $$ z \sim -z, \hspace{1cm} ...
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Evaluating a complex integral of two variables

In https://mathoverflow.net/questions/423124/expectation-of-complex-random-variable?noredirect=1#comment1087394_423124, I got a clue that the following integral could be computed with a suitable ...
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Sets known with Hausdorff distance

Let A and B be two arbitrary compact sets. Then what property should hold by A and B such that the distance D(A, B) = D(B, A) = h(A, B), where D is the set distance and h is the Hausdorff distance. ...
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1 vote
0 answers
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When do real, analytic, monotonic functions on an interval extend to univalent functions on an open region of the complex plane?

Suppose I have a real, analytic function $f(x)$ which is monotonic on some connected, open subset of the real line $W \subseteq \mathbb{R}$ and such that $f'(x)>0$ on $W$. Let me naturally include ...
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Alternate solution for https://math.stackexchange.com/q/458071/691870

Let $f$ be an entire function with $|f(z)|\le 100\log|z|,\forall |z|\ge 2,f(i)=2i, \text{ Then} f(1)=?$ I am trying to solve this question using Cauchy's Inequality . Now since given that function is ...
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5 votes
1 answer
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Elementary proof that $\text{Re}\big(\frac{z^{n+1} - n z - z + n}{(z-1)^2}\big) \ge \frac{n}2$?

I would like an elementary proof that the real part of $$ f(z) = \frac{z^{n+1} - n z - z + n}{(z-1)^2} $$ is greater than or equal to $n/2$ for any $z \in \mathbb C$, $|z| \le 1$, where $n$ is a ...
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1 vote
1 answer
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Classifying singularities of an entire function at infinity

In preparing for an examination, I have run across the following problem that has me stumped: Let $f$ be an entire function, where $f(0)=\alpha, \alpha\in\mathbb{R}$, and for all $z\in\mathbb{C}\...
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1 answer
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Show the identity

Let $f:\mathbb{R^2}\rightarrow\mathbb{C}$ be a differentiable function. The function $F:\mathbb{C}\rightarrow\mathbb{C}$, $F(x+iy):=f(x,y)$ is given and the "partial derivative" operator is ...
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0 answers
37 views

Müntz-Szasz theorem: assumption measure is concentrated in (0,1]

I'm studying the proof of the Müntz-Szasz theorem of Rudin's book.. They define the function $$f(z)= \int_{I} t^z d\mu(t)$$ and we may assume that the measure is concentrated in (0,1]. But why is this ...
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1 vote
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Formula connecting Fourier transforms of function and its derivative

Suppose $f: \mathbb{R} \to \mathbb{R}$ is continuous, differentiable and suppose $f$ and $f'$ are both summable. Then $$\mathcal{F}(f')(x) = -ix \mathcal{F}(f)(x)$$ (Here $\mathcal{F}$ represents ...
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1 vote
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Contour Integral with square root

I'm a master degree theoretical physics student and while working on my thesis I've encountered the following guy: $$\int_0^{\infty}dx\frac{e^{-ax^2+ibx}}{\sqrt{x}}$$ with $a,b>0$. I wanted to ask ...
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1 vote
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Factorization of modular forms using zeros and poles

In complex analysis, it is possible to decompose any given entire function into a product of linear factors using the zeros, using the Weierstrass factorization theorem. For example, $\sin(z)$ is ...
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1 vote
1 answer
103 views

show $\int_\mathbb{R} \frac{1}{\pi(1+x^2)}dx=1$

I am trying to show that density function of standard Cauchy distribution is well defined: that $$\int_\mathbb{R} \frac{1}{\pi(1+x^2)}dx=1.$$ I tried the following calculation: $$\int_{-N}^N \frac{1}{\...
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3 votes
0 answers
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Contour integration of $\int_{0}^{+\infty}\frac{x^2\cos x}{\cosh x} \,{\rm d}x$

Using contour integration, find $$\int_{0}^{+\infty}\frac{x^2\cos x}{\cosh x} \,{\rm d}x$$ How to calculate it? I never worked with integrals of this type.
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1 vote
1 answer
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Prove that even doubly periodic function satisfies a differential equation

Let $f(z)$ be analytic in $\mathbb{C}\setminus\{m+ni:m,n\in\mathbb{Z}\}$. Assume that $f(z)=f(-z)$ , $f(z)=f(z+m+ni)$ and $f$ has a pole of order $2$ at $0$. Prove that there exist numbers $a_0,a_1,...
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5 votes
3 answers
264 views

Prove that $\sum_{n=0}^\infty \frac{1}{(2n+1)^2}=\frac{\pi^2}{8}$

I am asked to prove that $$\sum_{n=0}^\infty \frac{1}{(2n+1)^2}=\frac{\pi^2}{8}.$$ However, I am asked to prove it using the fact that $$\frac{\pi}{2}\tan\left(\frac{\pi}{2}z\right)=\sum_{m \text{ odd}...
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  • 1,345
0 votes
1 answer
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Convergence of series over a complex lattice

I am currently working on my bachelor thesis that is on the J-Invariant. I am working with Apostol, Modular Functions and Dirichlet Series in Number Theory, and I have encountered a lemma I would like ...
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Some questions on index and single-valued property of Riemann-Hilbert Problems

Let $D \subseteq \mathbb{C} $ be bounded and simply connected domain, $\Gamma := \partial D \in C^2 $, $ g \in C^{0,\alpha}(\Gamma) $ be a complex-valued and nowhere vanishing function defined on the ...
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Evaluating Conditions for Normal Family

I am having troubles dealing with following problems: Suppose $\mathcal{F}$ is a collection of all holomorphic functions defined on a region $\Omega$ with values on the right half plane. If there ...
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Evaluate $\int_0^{\infty}\frac{\log( x)}{x^2+a^2} \,dx$ using contour integration; Re a > 0

Evaluate $\int_0^{\infty}\frac{\log( x)}{x^2+a^2} \,dx$ using contour integration; $Re (a) > 0$ I found two questions where a > 0 but in my case I have the following condition: Re a > 0 (It ...
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