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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79 views

Evaluating the sum $\sum_{k=0}^\infty \frac{1}{(3k+1)^2}$

I am looking for a closed form to: $\displaystyle \sum_{k=0}^\infty \frac{1}{(3k+1)^2} $
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2answers
1k views

Calculating residue of pole of order $2$

Is there a good way to compute the residue of $f(z)=\dfrac{1+z}{1-\sin z}$ at $z=\pi/2$, which is a pole of order $2$? Using the residue calculation formula yields ...
3
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2answers
300 views

Calculating residue in exponential fraction

I want to calculate the residue of $$\dfrac{e^{it}}{e^t+e^{-t}}$$ at $t=\pi i/2$. To calculate the residue at $\pi i/2$, I write ...
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1answer
212 views

When can Jordan's lemma be applied to contours less than a complete semicircle?

This paper on the evaluation of the Fresnel integral $\int_0^\infty \cos (x^2)\, dx$ claims that the Jordan lemma (see the bottom of page 3) shows that if $C_R$ is the eighth-circle contour $\{R e^{i ...
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0answers
48 views

Taylor series convergence

$$f(z)=\int^z_0 \frac{\zeta-\sin(\zeta)}{\zeta^2+4} \, d\zeta$$ I am supposed to find the convergence radius of its Taylor series at point $a=2$. I can find the radius in simple cases by finding ...
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2answers
79 views

If $f = u + iv$ is a complex function, is $|f| = (u^2 + v^2)^{1/2}$?

If $f = u + iv$ is a complex function, is $|f| = (u^2 + v^2)^{1/2}$? Where, $|f|$ is the modulus or absolute of $f$. I thought this should be correct because for any complex number $z = a + ib$, $|z| ...
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46 views

Wiman-Valiron theory

In Wiman-Valiron theory we have the comparison method which compares an etire function with a fully indexed power series to get estmates for the terms of entire function. In the case of functions that ...
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1answer
64 views

solving the equation ($\sin(z)/z$)

I want to find all set of $\omega \in \mathbb{C}$ such that the equation $\omega=\frac{\sin(z)}{z}$ has solutions $z$ arbitrary close to $0$. I started with Taylor series for $\sin(z)$ but as it's ...
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0answers
31 views

Find Limit of Complex integration

How should we prove that $\int_{\mathbb{C}}\frac{|G(w)|}{|z-w|}dA(w)\longrightarrow0$ as $|z|\longrightarrow\infty$where $G(z)\in L^{1}(\mathbb{C})$ and $|G(z)|\leq\frac{C_{0}}{|z|^{1+\varepsilon}}$ ...
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1answer
51 views

Is holomorphic functions on (0, 1) (vanishing at endpoint) dense in $C_0((0, 1))$?

Here is my argument, please let me know if it works or not. By Stone-Weierstrass Theorem (Complex Version), functions in $C_0((0, 1))$ can be uniformly approximated by polynomials in z and $\bar{z}$ ...
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123 views

Improper integral $\int_0^{\infty}\frac{x^{k-1}}{1+x^n}dx$

What is the convergence condition of the following integral $$\int_0^{\infty}\frac{x^{k-1}}{1+x^n}dx$$ and how prove that if integral is convergent then ...
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1answer
42 views

what are the value of these complex integrals

(i) I think it will $|z|={\pi\over 2}$, so the value will be $0$ as on that specified domain $\tan z$ is analytic. (ii) I have no idea!Could any one help me? (iii) at $z=2\pi i$ the integrand has ...
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0answers
25 views

$f,g$ are entire function satisfies $|f|\le |g|$ what is the relation between them? [duplicate]

$f,g$ are entire function satisfies $|f|\le |g|$ what is the relation between them? I did $h(z)={f\over g}$, Then $|h|\le 1$ so $h$ is bounded, entire(?) and so constant? $h$ could possibly have ...
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795 views

Convergence of complex sequences

Do the following sequences converge? 1) $(-1)^n\frac{n}{n+i}$ 2) $\frac{n^2+in}{n^2+i}$ I don't really understand how to decide whether a complex sequence converges, and I don't have much ...
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1answer
42 views

Homotopy cauchy

-If $D$ is convex (part of $\mathbb{C}$) and if we have two paths $\gamma_1$ and $\gamma_2$ in $D$ with $\gamma_1(a)=\gamma_2(a)$, and $\gamma_1(b)=\gamma_2(b)$. Proof that $\gamma_1$ and $\gamma_2$ ...
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3answers
171 views

top journals in analysis [closed]

as an undergraduate I find analysis as my favorite.I want to read journals regarding that. give me top 5 journals in analysis(real,complex)? top 5 journals in differential geometry? and generally some ...
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246 views

Improper Integral: $\int_{-\infty}^{\infty}\frac{\log(1+x^4)}{x^4}dx$

How can I prove this? $$\int_{-\infty}^{\infty}\frac{\log(1+x^4)}{x^4}dx=\frac{2\sqrt{2}}{3}\pi$$
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1answer
172 views

Harmonic conjugates and holomorphic functions

I've been researching complex analysis recently and come across a couple of questions that I'm a little confused about. This is one of them. Let $u:U\to \mathbb R$ be a harmonic function on an open ...
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1answer
144 views

Is image of boundary a boundary of image?

I have a question which appers in problems concerning Möbius transformation, for example Let $A=\{ z\in \mathbb{C} : \|z\| <1, \Re(z)>0\}$ and $f(z)=\frac{z+i}{z-1}$ Determine $f(A)$. Often ...
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2answers
572 views

complex conjugates of holomorphic functions

I came across this question whilst doing some research into complex analysis, and I just can't see what to do! Let $f(z)$ be a holomorphic function on $\mathbb{C}$. Show that ...
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1answer
72 views

Complex integral square

Let $\alpha$ be the closed curve along the square with vertices at $1, i, -1, -i$. Give an explicit parametrization for $\alpha$ and calculate $$\frac{1}{2\pi i}\int_\alpha\frac{dz}{z}$$ I ...
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63 views

Is $\sum_{n=3}^\infty \dfrac{z^n}{n \ln n}$ uniform coverge on $\lvert z\rvert \ <1$??

I tried to solve this problem using Cauchy's convergence criterion. Maybe that problem's answer is not uniform converge. But I didn't solve it. Please reply why this power series is not uniform ...
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1answer
59 views

What does it mean for a function $u : D \to \mathbb{C}$ to be harmonic, $D \subset \mathbb{R^2}$?

On page 167 of David Ullrich's "Complex Made Simple", he defines $u : D \to \mathbb{C}$ to be harmonic, $D \subset \mathbb{R^2}$, to be harmonic in $D$ if it is twice continuously real differentiable ...
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1answer
106 views

An application of Runge's Theorem to approximate analytic functions by polynomials

Apply the following form of Runge's Theorem: if $X\subset \mathbb{C}$ is an open subset,and if $\mathbb{C}\setminus X$ is connected,then $\mathbb{C}[z]$ is dense in $\mathcal{O}(X)$ in the topology of ...
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1answer
406 views

Extended Proof of the Theorem that a bounded analytic function is constant.

I am having trouble feeling convinced by my proof and more importantly - feeling confident in my working out. The question reads (a) Let $f$ be an entire function such that there exist real ...
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0answers
274 views

A conformal mapping onto a region bounded by convex contours (Ahlfors)

I want to solve the following exercise (from Ahlfors' text, page 261) *3. Using Ex. 2, show that $p + q$ maps $\Omega$ in a one-to-one manner onto a region bounded by convex contours. Comments: ...
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2answers
53 views

Over Cosin Squared Integral

Show (using complex analysis) that $$\int_{0}^\pi \frac{d\theta}{(a+\cos\theta)^2} = \frac{\pi.a}{(a^2-1)^{3/2}}$$ I choose a unite semi-circular path C. So I did: $$z = e^{i.\theta}$$ $$dz = ...
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1answer
47 views

Question on Rudin's Proof of the Residue Theorem

The Theorem in question is Theorem 10.42.: If $f$ is meromorphic in $U$, $A$ is the set of poles of $f$ and $\Gamma$ is a cycle in $U-A$ so that $Ind _{\Gamma}=0$ in $U^c$ then \begin{equation}\frac ...
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75 views

Constructing an L2 function from an entire function bounded on R

I have an entire function $f(z)$ of exponential type $\tau\geq0$ that is bounded on $\mathbb{R}$ and zero at every member of the complex sequence $\{\lambda_n\}$. What I want is an entire function of ...
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3answers
334 views

Where does the function $f(z)=z\bar z+z/\bar z$ satisfy the Cauchy-Riemann equations?

Where does the function $f:\mathbb{C}\setminus\{0\}\to\mathbb{C}, f(z)=z\bar z+z/\bar z$ (where $\bar z$ is the complex conjugate of $z$) satisfy the Cauchy-Riemann differential equations? I tried ...
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1answer
61 views

Continuity and complex derivative

Investigate the continuity and complex differentiability of the following functions $f$. Find the derivatives at points where they exist. $f(z)=z \ \text{Re}(z)$. To show that this ...
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1answer
74 views

Partial derivatives in $\mathbb{C}^n$

I'm trying to figure out an equality from a proof by Griffiths and Harris to the holomorphic inverse function theorem (in Principles of Algebraic Geometry). They state: $$\frac{\partial}{\partial ...
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1answer
156 views

Proving a sufficient condition for complex differentiability

I'm trying to show that given $f=u+iv:\mathbb{C}\to\mathbb{C}$ and $z_{0}\in\mathbb{C}$ if $u,v$ are differentiable (as functions $\mathbb{R}^{2}\to\mathbb{R})$ at ...
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2answers
62 views

Locus in complex z-plane given eqn

I have the question: $ \text{Find the locus in the complex }z\text{-plane that satisfies the equation: } z-c=\rho\dfrac{1+it}{1-it}, \text{where }c\text{ is complex, }\rho\text{ is real, and }t\text{ ...
3
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1answer
81 views

Computing $\int_{0}^\infty\frac{t^a}{1+t^2}dt$ for $-1<a<0$

I am integrating the following integral $$\int_{0}^\infty\frac{t^a}{1+t^2}dt$$ for $-1<a<0$. by computing residues inside some contour. But I'm not sure what contour to use here, since ...
5
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1answer
102 views

Continuity of Green's function

Suppose $\Omega \subset \mathbb C$ is a region (open and connected set) and let $$g(z,z_0)=G(z,z_0)-\log|z-z_0| $$ be its Green's function with pole at $z_0 \in \Omega$. Here $G(z,z_0)$ is the ...
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2answers
333 views

Find the Laurent series for $f(z) = (z^2 - 4)/(z-1)^2 $ for $z=1$

What I understand is that we have to expand $f(z$) in the positive and negative powers of $(z-1)$. Hence I tried factorizing the numerator $(z^2-4)=(z+2)(z-2)$ , which can then be written in terms of ...
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1answer
84 views

special parameter integral

Does anyone know a proof for the following formula ? $$\int_{0}^{\infty} \frac {1}{x^y+1} dx=\frac{\frac{\pi}{y}}{\sin(\frac{\pi}{y})}$$ for $y>1$? If $y$ is an even positive integer than the ...
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1answer
96 views

Two complex integrals of the function $1/z$

Let $a,b \in \mathbb{R}^{*}$ and $\alpha, \beta : [0,1]\rightarrow\mathbb{C}$ be defined by: $$\alpha(t):=a\cos(2\pi t)+ia\sin(2\pi t)$$ $$\beta(t):=a\cos(2\pi t)+ib\sin(2\pi t)$$ Show that ...
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1answer
66 views

How to bound away integral over complex rectangle?

I am integrating the following integral $$\int_{-\infty}^\infty\frac{\cos t}{e^t+e^{-t}}dt$$ by computing residues inside the rectangle with vertices $-R,R,-R+\pi i,R+\pi i$. On the left and ...
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1answer
188 views

Evaluating $\int_0^\infty \frac{\log t}{1+t^2}\,\mathrm dt$ using residues

I want to integrate $$\int_0^\infty \dfrac{\log t}{1+t^2}\,\mathrm dt$$ using the residue theorem. The poles are at $i,-i$. If the integral were from $-\infty$ to $\infty$, I would consider ...
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2answers
149 views

Complex integral with $1-\sin z$ as denominator

I want to evaluate $$\int_{|z|=8}\dfrac{1+z}{1-\sin z}dz$$ using the residue theorem. But I'm not sure what the residues are. In $|z|=8$, for $z$ real, we have $\sin z=1$ for $z=-3\pi/2,\pi/2,5\pi/2$. ...
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1answer
143 views

Proof of $\Gamma(z) e^{i \pi z/2} = \int_0^\infty t^{z-1} e^{it}\, dt$

I am trying to prove the identity $$\Gamma(z) e^{i \pi z/2} = \int_0^\infty t^{z-1} e^{it}\, dt$$ for $0 < \Re(z) < 1$, starting from the integral definition of the gamma function $$\Gamma(z) = ...
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1answer
436 views

Evaluating a Real Improper Integral by Residues

I am having trouble evaluating this improper integral due to its integrand and the singularities that are present. The question reads as Show that ...
3
votes
1answer
56 views

Using residue to compute real fractional integral

Compute the integral $$\int_{-\infty}^\infty \dfrac{t-1}{t^5-1}dt$$ The hint is to use residues. I tried taking a look at the residue theorem, but I don't know which curve in the complex plane I ...
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1answer
61 views

Zeros of polynomials are continuous

For two sets $A,B$, let $d(A,B)=\sup_{x\in A}\inf_{y\in B}|x-y|+\sup_{y\in B}\inf_{x\in A}|x-y|$. Let $p(z)=a_nz^n+\ldots+a_0$, and let $\epsilon>0$. Show that there exists $\delta>0$ such ...
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1answer
42 views

Why do these properties determine this map up to an additive constant?

In Ahlfors' text, he considers a region $\Omega$ bounded by analytic contours $C_1, \dots, C_n$. On page 259 he states: We have thus established the existence of a function $p(z)$ which is ...
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2answers
167 views

Prove that $f$ is constant if $f$ is real when $|z|=1$ [duplicate]

Let $f$ be a holomorphic function in $\mathbb{C}$. Prove that if $f$ is real when $|z|=1$, then $f$ must be a constant function. I honestly do not know how to do this problem, consider using ...
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1answer
65 views

Show that $f=g$, if $f(z)=g(z)$ for $z\in dA$ with $A$ bounded region

Let $A$ be a bounded region, $f$, $g$ continuous functions of $\bar{A}$ in the complex. Suppose that these functions are holomorphic in the region and agree on the border. Prove you are the same. I ...
5
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2answers
364 views

Why does this integral vanish? $\int_C \frac{e^{az}}{1+e^z}dz$

I'm looking for an argument that would prove that the integral $$I=\int_C \frac{e^{az}}{1+e^z}dz$$ vanishes for $R \to \infty$, where $C$ is the horizontal line segment from $(1+i)R$ to $(-1+i)R$, ...