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Questions tagged [complex-analysis]

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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Finding $\frac{d^{n-1}}{dx^{n-1}}(1+\sqrt x)^{2n}|_{x=1}$

How can we find $$\frac{d^{n-1}}{dx^{n-1}}(1+\sqrt x)^{2n}\Big|_{x=1},$$where $n\in\mathbb{N}$? Attempt of complex integration $$\begin{aligned}&\frac{d^{n-1}}{dx^{n-1}}(1+\sqrt x)^{2n}\Big|_{x=1}...
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48 views

What about the solutions of $z^{1/3} +1 = 0$?

I'm trying to find the zeros of the equation $$z^{1/3} +1 = 0.$$ My professor said that the solutions are the third roots of unity multiplied by $-1$. My problem is that when I calculate the cubic ...
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1answer
20 views

Complex functions and residues

I know that, since the complex function $$f(z)=\frac{1}{z(e^z-1)}$$ have a pole of 2nd order in z=0, I should be able to represent it as: $$f(z)=\frac{g(z)}{(z-0)^2}$$ Being g(z) a function that is ...
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1answer
27 views

Laurent Series in powers of $z$ and $\frac{1}{z}$

I'm working on a few problems from my textbook and have a bit of trouble figuring out a few things. In a particular case, suppose $R$ is a rational function all of whose poles in the plane have ...
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9 views

Complex Analysis on Holomorphic Anti-derivatives

Question: Let $U_{1} \subseteq U_2 \subseteq U_{3} \subseteq ... \subseteq \mathbb{C}$ be connected open sets and let $U = \cup_{i = 1}^{\infty} U_i$. Let $f$ be holomorphic on $U$. Suppose for each $...
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1answer
20 views

Prove the complex series converge absolutely

Prove the complex $$\sum_{k=0}^\infty (k+k^2i)^{-1}$$ series converge absolutely Solution: by triangle inequality, we have $$|k+k^2i|\ge k^2-k \ge {k^2\over 2}$$ if $$k \ge 2 $$ how to understand ...
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25 views

Analytic continuation of the logarithm

This is an example from Serge Lang Complex Analysis book, it says Let us start with the function $log(z)$ defined by the usual power series on the disc $D_0$ which is centered at $1$ and has ...
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1answer
16 views

Line integral depending on a parameter is entire

Suppose you have a continuous function: $$\phi:[0,1]\rightarrow \mathbb{C}$$ define the complex function: $$f(z)=\int_0^1\phi(t)e^{itz}dt$$ prove that it is entire and calculate it's Taylor ...
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11 views

Sequence of entire function that converges uniformly over on sets with empty interior

I have to prove that the sequence of entire functions: $$f_n(z)=\frac 1n \sin(nz)$$ converges uniformly over $\mathbb{R}$ (and this I managed to verify) but doesn't on every set with non-empty ...
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1answer
23 views

Identity theorem for a holomorphic funtion defined near zero

I have to show, whether there is a holomorphic funtion $f$ defined in an open neighborhood of zero, such that: $$ f\left(\frac{1}{n}\right)=(-1)^n \frac{1}{n^3}$$ for all positive integer $n$. My ...
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21 views

Analytic function $f$ either constant or surjective on unit disk.

Suppose $f$ is analytic on unit disk and continuous on it. Assume $|f(z)|=1$ on $\{x;|x|=1\}$. Then $f$ is constant or $f(U)=U$ where $U=\{x;|x|<1\}$. My attempt: By maximum principle, $|f(z)|\...
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1answer
18 views

Entire function invariant by translation is constant [duplicate]

I need to apply Liouville theorem ("entire bounded complex functions are constant") to prove that an entire function satisfying: $$f(z)=f(z+1)=f(z+i)$$ for all complex numbers $z$ is constant. I'm ...
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16 views

complex conjugate line integral

i have $\gamma(t)=e^{it}$, $t\in[0,2\pi]$ and the line integral $\overline{\int_{\gamma} f(z)\>dz}$. It is to show that $\overline{\int_{\gamma} f(z)dz}$=$-\int_{\gamma} \frac{\overline{f(z)}}{z^...
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1answer
37 views

Complex Analysis: evaluating a definite integral using contour integration

When I am evaluating the definite integral $$\int_{0}^{\infty}\frac{\cos x-e^{-x}}{x}dx$$ using contour integration, I find a conclusion $$\int_{0}^{\pi/2}e^{iRe^{i\theta}}d\theta\to0$$ as $R\to\infty$...
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Proving there exists a real functions satisfying two integral equalities

I would like to show there exists a real function $H(x)$ ($C^{\infty}$) increasing with near infinity $H(x)=kx$ ($k$ is real and can be chosen) such that: $$ \int\limits_{\epsilon}^\infty H'(x)^{\...
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55 views

Find the residue of $\frac{\zeta'(1+s)}{\zeta(1+s)}\frac{x^s}{s}$ at $s=0$

Let, $\displaystyle f(s)=\frac{\zeta'(1+s)}{\zeta(1+s)}\frac{x^s}{s}$. Prove that $Res(f,s=0)=A-\log x$ , for some constant $A$. At $s=0$ , $\zeta(s)$ has a pole of order $1$ and $\zeta'(s)$ has a ...
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36 views

Questions about conditionally convergent series and rearrangement of [on hold]

According to Riemann Series Theorem or Riemann Rearrangement Theorem a conditionally convergent series - with a clever rearrangement of terms - can converge to any desired value, or even can be shown ...
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1answer
51 views

Holomorphic function on an annulus

Let $f$ be a holomorphic function on the set $U=\{z \in \mathbb{C}: 1 \leq |z| \leq \pi \}$. Assume that $max_{|z|=1}|f(z)| \leq 1$ and $max_{|z|=\pi}|f(z)| \leq \pi^{\pi}$. How to prove that $...
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acculation point [on hold]

If $(x_n)$ is a complex sequence and we know that $a$ is point of $(x_n)$.By the definition of accumulation point in topological spaces ,for any neighborhood $U_a$ of $a$, $(x_n)\cap( U_a\setminus\{a\}...
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28 views

How do you expand a taylor series about a complex number?

Normally a Taylor series is constructed along real numbers. However, for practical purposes mathematics often heralds that commonly known continuous functions in the real plane are equivalent to their ...
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Prove that there exists a holomorphic function $g$ on $ \Bbb C$ s.t $f=e^g$ [on hold]

The goal of this exercise is to prove that there exists no holomorphic function $f$ on $\Bbb C$ s.t $f(f(z))$ = $e^z$ on $\Bbb C$ We assume that a such function f exists Prove that $f(C)=C^\ast$ ...
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1answer
26 views

Compute the limit for a harmonic function given two known limits

This problem is from a set of exercises that I have. It states: Let $u\in C(\overline{\mathbb{R}_+^2})$ be a bounded harmonic function in the upper half plane $\mathbb{R}_+^2$, satisfying $u(x,0) \...
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1answer
36 views

How do we compute $\sum_{n=1}^{\infty}\frac{1}{K^{Sn}}$, for $K$ an integer and $S$ a complex number?

Consider the series: $$\sum_{n=1}^{\infty}\frac{1}{K^{Sn}}$$ where $K$ is some integer constant and $S$ is a complex number. How we evaluate (compute sum of) this series where S is a complex ...
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65 views

Concerning the ring of continuous functions on $\mathbb{R}$

It is not difficult to check that the set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ is a ring (an $\mathbb{R}$-algebra), and similarly (if I am not wrong), the set of continuous ...
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1answer
19 views

Maximum Modulus Principle and Open Mapping Theorem

The Maximum Modulus Principal states: let $f$ be a function holomorphic on some connected open subset $D$ of the complex plane $\mathbb{C}$ and taking complex values. If $z_{0}$ is a point in $D$ such ...
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31 views

Let $g$ be real valued and continuous on [0,1]. Show that if $f(z) = \int_0^1 g(t) (\cos zt) dt $ is 0 for all complex $z$ then $g$ is identically 0.

Let $g$ be real valued and continuous on [0,1]. Show that if the complex valued function $$f(z) := \int_0^1 g(t) (\cos zt) dt $$ is 0 for all $z \in \mathbb{C}$, then $g$ is identically 0. I managed ...
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1answer
36 views

Let $f(x)$ and $g(x)$ are two complex polynomials such that $f^{-1}(c_{i})=g^{-1}(c_{i})$

Let $f(x)$ and $g(x)$ are two complex polynomials such that $f^{-1}(c_{i})=g^{-1}(c_{i})$ for two distinct complex numbers $c_{i}$, $i=1,2$. Then can we say $f=g$? Here nothing is given about the ...
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1answer
26 views

Integral of the reciprocal of a complex polynomial [duplicate]

For a polynomial P(z) of degree $n\geq2$, show that there exists some $R_{1}>0$ such that for $R>R_{1}$ it holds that: $$\int_{C_{R}}\frac{1}{P(z)}dz=0$$ where $C_{R}$ is a circle of radius R ...
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29 views

What can we say about $f$?

Let $f$ be an entire function. Consider $A= \left \{z \in \Bbb C : f^{(n)} (z) = 0\ \text {for some positive integer}\ n \right \}.$ Then $1.$ if $A=\Bbb C$ then $f$ is a polynomial. ...
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Show that for a function $f$ bounded by $M$ on a disk $D_{r}$ show that $|f^{n}(z)|\leq n!M/\delta^{n}$ for $D_{r-\delta}$

Suppose that a function $f$ is analytic in the open disk $$D_{r}=\{z\in \mathbb{C}:|z|<r\}$$ where $r>0$, and there is a number $M\in\mathbb{R} $ such that $|f(z)| \leq M$ for all $z \...
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33 views

Why is the derevative of meromorphic function is meromorphic?

I know that if $f$ is meromorphic then $\exists A\subset \Omega$ s.t $f$ is holomorphic on $\Omega \setminus A$ and $A$ is discrete, and $A$ are the poles of $f$. I want to show that $f'$ is ...
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22 views

How to take an absolute value or modulus of z?

let's assume the : $$z=\frac{e^{(-jc)}}{(a+jb)}$$ I would like to take the absolute value of z. I started with multiplication z with $\frac{(a-jb)}{(a-jb)}$ and got: $$abs\frac{e^{(-jc)}}{(a+jb)}=\...
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2answers
61 views

Complex Analysis: Prove that an entire function with $\lim_{z\to\infty}f(z)=\infty$ is surjective.

If $f$ is an entire function with the property that $|f(z)|\to\infty$ as $|z|\to\infty$, verify that $f(\mathbb{C})=\mathbb{C}$. This is a problem from my textbook. And I guess the Rouché Theorem ...
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23 views

Contour Integral with shifted pole [on hold]

I am trying to calculate the following contour integral. $$\sum_{s=-\infty}^{\infty}\int_{0}^{\infty} \frac{Exp(i2\pi s x)}{x-a+i\gamma}\;dx$$.
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1answer
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Questions about Cauchy's thm. on complex integration

I have the following integral $$I = \int_{\gamma_{1}} \frac{e^{z^2}}{(z-1)^2}dz,$$ where $$\gamma_{1} : [0,2Pi] \rightarrow \mathbb{C}, \\ \quad t \quad \mapsto \quad 2e^{i t} .$$ I want to show that ...
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1answer
27 views

What type of singularity does $\frac{e^{iaz}-e^{-z}}{z}$ have at $z=0$?

What type of singularity does $\frac{e^{iaz}-e^{-z}}{z}$ have at $z=0$? It is not a pole since the numerator is zero when $z=0$. The derivatives of this function always have a $z$ at the denominator, ...
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How to show that $|\int_0^\infty\frac{e^{is\cos \theta}-e^{-s}}{s}ds| \le 2log \frac{c}{\cos \theta}$, where $c>1$ and $\theta \not=\frac{\pi}{2}$?

How to show that $|\int_0^\infty \frac{e^{is \cos \theta}-e^{-s}}{s}ds| \le 2log \frac{c}{\cos \theta}$, where $c>1$ and $\theta \not = \frac{\pi}{2}$? Why is there a downvote? $\Sigma$ is the ...
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37 views

the index of a closed curve is continuous

Suppose $\gamma:[a,b]\rightarrow\mathbb{C}$ is a closed curve in the complex plane.We know that if $f:\widehat{\gamma}\rightarrow\mathbb{C} $ continuous such that $f(\gamma(t))\not=0 $ $\forall t\in[a,...
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73 views

Is this true? $Re\int{f(z)dz}=\int{Re(f(z))dz}$ [on hold]

I have to say if this is true or not and why. Let f a complex function, then $$Re\int_{\gamma}{f(z)dz}=\int_{\gamma}{Re(f(z))dz}$$
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1answer
13 views

Is there a biholomorphic map between a simply connected domain and non simply connected domain?

Is there a biholomorphic map between a simply connected domain and non simply connected domain? I am not sure how to approach this question. This is not a homework question but one I simply came ...
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9 views

Question about Bidirectional Associative Memory(BAM) testing.

I have designed a BAM for the recognition of the 4x5 binary images. (total of three images) I was testing it, first i gave it inputs ( Xi, where i=1,2,3 etc.) and it gave correct results (meaning it ...
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2answers
42 views

Question about the point at infinity

I have started complex analysis and I am stuck on one definition 'extended complex plane'.Book is saying 'To visualize point at infinity,think of complex plane passing through the equator of a unit ...
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2answers
41 views

A line integral equals zero implies a real integral also is zero

I'm asked to check that the following line integral is zero: $$\int_{C(0,r)} \frac {\log(1+z)}z dz=0$$ (where $C(0,r)$ is the circle of radius $r$ centered at $0$) and then to conclude that for ...
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let $f(x+iy)=u(x,y)+iv(x,y)$ what is $f(x-iy)$ equals to? [on hold]

this is a little question that stuck me to do my homeworks. the question Thanks alot for the helpers.
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46 views

An equation involving Non-Trivial Zeros of the Riemann Zeta function

$\rho$ is a Non-Trivial Zero of the Riemann Zeta function if and only if $$\displaystyle\int_1^{+\infty} \lfloor x\rfloor x^{-2-\rho} dx =\int_1^{+\infty} \lfloor x\rfloor \{ x \}x^{-2-\rho} dx $$ ...
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2answers
55 views

Roots of a polynomial with holomorphic coefficients

Let $f_1,f_2,\dots,f_n : \mathbb{D} \to \mathbb{C}$ be holomorphic functions and consider the polynomial $$ w^n + f_1(z)w^{n-1} + \dots + f_n(z). $$ Suppose, I happen to know that For each $z$, ...
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0answers
26 views

How to calculate contour integration of multivalued functions?

I have a question about contour integral of the multivalude function. The question is from paper arXiv:1211.6767 (page 6-7). I want to calculate the Fourier transformation of a muti-valued function $...
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1answer
34 views

Proof that a finite series expansion of $f(X)$ at $\alpha$ exists iff $Q(X)$ is a power of $(X-\alpha)$, in $f(X)=\frac{P(X)}{Q(X)}$

I'm working through Gouvea's P-adic numbers book, and early on they give the problem Write $f(X)=\frac{P(X)}{Q(X)}$ in lowest terms, so that $P(X)$ and $Q(X)$ have no common zeros. Show that the ...
2
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2answers
47 views

Confusion about radius of convergence of a power series

I'm a bit confused about the following statement from a script. We look at the power series with coefficient $ a_k = (1+1/k)^{k^2}$ and want to compute the radius of convergence $R$ of the series. The ...
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0answers
31 views

If $u$ and $v$ are complex-valued and satisfy the Cauchy-Riemann equations, does it follow that $u$ and $v$ are $C^{\infty}$-smooth?

I am trying to follow an example in my literature and I am pretty lost. It says that if $u$ and $v$ are complex-valued functions that are $C^1$-smooth in some open set, and $$\frac{\partial u}{\...