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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41 views

Poles of Complex Functions or One-Forms?

The function $f(z) = \frac{1}{\sqrt{z}\sqrt{1-z}}$ with branch cuts chosen so that $f$ is analytic on $\mathbb C-[0,1]$ has a pole at infinity according to this walkthrough of a branch cut contour ...
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2answers
74 views

Compute $\oint \Bigl[ z e^{3/z} + \frac{\cos z}{z^2 (z - \pi )^3} \Bigr] \, dz$ [closed]

Compute $$\oint \left[ z e^{3/z} + \frac{\cos z}{z^2 (z - \pi )^3} \right] \, dz$$ $$|z| = 5$$ My question is how to do residue at $$\oint ze^{3/z} \, dz $$
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1answer
266 views

Abscissa of Convergence for the Laplace Transform of $f(t)=e^t \sin(e^t)$

I am trying to solve the following question: Show that the abscissa of convergence for the function $f(t)=e^t \sin(e^t)$ is zero, i.e the unique number $\sigma$ such that the integral $\int_0^\infty ...
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1answer
66 views

Calculating a contour integral

I want to evaluate the integral $$\int_{\gamma} \sin{(2z)} \ {\rm d}z$$ where $\gamma$ is the line segment joining the point $i+1$ to the point $-i$. Thus $\gamma(t) = -i+t(2i+1)$ for $0\le t\le1$. ...
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1answer
70 views

A condition under which $f:\mathbb{R}^2 \rightarrow \mathbb{R^2}$ is analytic

I have $f:\mathbb{R}^2 \rightarrow \mathbb{R^2}$ which can also be thought of as $f:\mathbb{C} \rightarrow \mathbb{C}$ in the usual way. $f$ is $C^{(2)}$ on open connected domain $\Omega$ and ${Df|_z}^...
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1answer
186 views

Evaluate $\int_c {{{\tan z} \over z}dz}$ using residue theorem

Using residue theorem, evaluate the following; $C:\left| {z - 1} \right| = 2$ $$\int_c {{{\tan z} \over z}dz}$$ I want you guys to check my answer.Is it correct? $$\displaylines{ {\mathop{\rm ...
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1answer
33 views

Roots of a sequence of function in certain disk

I have a sequence of functions defined by $$ g_n(z)=\sum_{k=1}^n \frac{z^{-k}}{k!}$$ Given $r>0$ show that we can find $M(r)$ such that if $n>M(r)$, then all the zeros of $g_n(z)$ lie within $D(...
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1answer
36 views

density on smooth boundary in several variables complex analysis

Assume bounded domain (open and connected) $\Omega\subset \mathbb{C}^n$, and a smooth function $\rho:\mathbb{C}^n\longrightarrow \mathbb{R}$ such that $\rho(x) = 0$ for all $x\in \partial \Omega$, $\...
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2answers
68 views

${{{z^4} + z - 2i} \over {{z^{15}} + i}}$ is the function continuous at every point?

${{{z^4} + z - 2i} \over {{z^{15}} + i}}$ is the function continuous at every point in the complex plane? I tried to do like this but it is right? $$\eqalign{ & {z^{15}} + i = 0 \cr & {...
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3answers
373 views

Identify and sketch the locus ${\mathop{\rm Re}\nolimits} \bigl( {{{z + i} \over {z - i}}} \bigr) = 1$.

Identify and sketch the locus of the point z on a complex plane which satisties the equation ${\mathop{\rm Re}\nolimits} \left( {{{z + i} \over {z - i}}} \right) = 1$. I tried to do it but I stuck. ...
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2answers
43 views

Evaluate $\oint_c {4z - 1}\,dz$ along the circle $|z| = 1$

Evaluate $\displaystyle\oint_c {4z - 1}\,dz$ along the circle $|z| = 1$ from the point $(0,-1)$ to $(1,0)$ My question is how to do a contour integration in the circle? I only know to do it in ...
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1answer
413 views

Does a conformal map take boundaries to boundaries?

I think it is a well-known result that conformal maps between sets in $\mathbb{C}$ take boundaries to boundaries. However, I looked around a little and I had trouble finding this result. Is it true? ...
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124 views

Evaluating an alternating sum using contour integrals

Evaluate: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{3n-1}$$ Using contour integration. Normally I would use $\pi\csc(\pi z)f(z)$ and evaluate the residue multiply by (-1) and divide by $2$ if the ...
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2answers
107 views

Property of complex numbers.

Let $z \in \mathbb{C}$ such that $Re(z^{n})\geq0, \forall n\in\mathbb{N}$, where $Re(z^{n})$ is the real part of $z^{n}$. Show that $z\in\mathbb{R}^{+}$. If $z=a+bi$, $a,b\in\mathbb{R}$, then for $n=...
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60 views

Countour integral $\int {{{(\overline z )}^2}dz} $

Evaluate $\int {{{(\overline z )}^2}dz} $ along the straight line segment from $z=0$ to $z=2+i$. My attempt to this question is I change z into $x+iy$ and do the integration; $$\int_0^{2 + jy} {(x -...
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1answer
88 views

Cauchy Principal Value Integral calculation

How can i resolve this integral in Cauchy principal value? $$\int_{-\infty}^ \infty \! \frac{x+\sin x}{x(x^2+4j-4)^2} \ \mathrm{d}x $$ Then $$\int_{-\infty}^ \infty \! \frac{1}{(x^2+4j-4)^2} \ \...
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0answers
30 views

Showing that an integral of a curve in $\mathbb{C}$ vanishes when the parameter approaches infinity

I'm trying to solve a problem where you have to use the residue theorem in order to get the value of a certain integral, but I cannot go on from this point: I need to show that $\int_{0}^\pi f(R+it)...
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1answer
138 views

Evaluating$ \int_{-\infty}^{\infty} \frac{x^6}{(4+x^4)^2} dx $using residues

I need help to solve the next improper integral using complex analysis: $$ \int_{-\infty}^{\infty} \frac{x^6}{(4+x^4)^2} dx $$ I have problems when I try to find residues for the function $ f = \...
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1answer
84 views

Use $\sum\limits_{n=-\infty}^{\infty}\frac{1}{n^2+a^2}$ to evaluate $\sum\limits_{n=0}^{\infty}\frac{1}{n^2+a^2}$

I want to evaluate the series $\displaystyle \sum_{n=0}^{\infty}\frac{1}{n^2+a^2}$ with $a>0$. I know that $\displaystyle \sum_{n=-\infty}^{\infty}\frac{1}{n^2+a^2}=\frac{\pi}{a}\coth \pi a$ with ...
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82 views

approximating to delta sequence

I want to prove the following: Let $h_{n}(t)=\sqrt {n}e^{-nt^{2}}$. The area under the graph of $h_{n}(t)$ is $\sqrt {\pi}$ and for any $\epsilon>0$ $h_{n}(t) \rightarrow0$ uniformly outside $]-\...
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69 views

How to integrate this integral by contour integral? [closed]

How to show that \begin{equation*} \int_{-\infty}^{\infty} \frac{x\sin x}{x^2+a^2}dx=\pi e^{-a}\text{ for all }a>0 \end{equation*} by using contour integral?
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1answer
46 views

Solutions to $z^4-4z+\lambda=0$ with complex parameter $\lambda$, $|\lambda|<2$.

For a complex parameter $\lambda$, $|\lambda|<2$, consider the solutions to the equation \begin{equation*} z^4-4z+\lambda=0 \end{equation*} $(1)$ Show that there is exactly one ...
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1answer
70 views

Find the Laurent expansion of $1/(\cos(z)-1)$ in the regions $|z|<2\pi$ and $2\pi<|z| <4\pi$

Find the terms of Laurent expansion of $\frac{1}{\cos(z)-1}$ valid for the regions: a) $|z|<2\pi$, b) $2\pi<|z| <4\pi$. I tried to find the coefficients of Laurent series using the ...
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1answer
42 views

Question about the proof of convergence of power series for complex numbers

I am looking at the proof for convergence of a power series if $|z|< R$, in the proof we defined $l = \limsup_{n\to\infty} (|a_n|)^{1/n}$, the lecturer then states that, $$l = \limsup_{n\to\infty} (...
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60 views

Find an argument for complex number $ \frac{1}{(1+2wi)^2} $

Let $ w $ be a real number. Find an argument for $$ \frac{1}{(1+2wi)^2} $$ Answer should be $ -2\arctan(2w) $. I keep getting $\arctan\left(\frac{-4w}{1+4w^2}\right) $ as an answer. My steps ...
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2answers
137 views

Limit with complex numbers

I've never calculated limits with complex numbers before. Why does $$ \lim_{z\rightarrow \infty} \frac{e^{3iz}-3e^{iz}}{z^3} =0?$$ This is contrary to my intuition, since exponentials grow faster ...
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1answer
105 views

Lines of Level Curves of an Analytic Function's Real and Imaninary Parts

I want to solve the following question: Let $g(z)$ be analytic at $z_{0}$ and let $g'(z_{0})=0$ and $g''(z_{0})\neq 0$ so that near $z_{0}$,$g(z)-g(z_{0})=[w(z)^{2}]$ for $w$ analytic, $w'(z_{0})\...
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5answers
88 views

Complex numbers problem: $ |\frac 1z - \frac 14 | = \frac 14 $

How do you go about solving the following equation $ |\frac 1z - \frac 14 | = \frac 14 $ where $ z = a + bi $. A hint is provided, and apprently the equation can be simplified to $ | {z-4\over z} | = ...
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1answer
113 views

Prove that |f|-|g| attains minimum on the boundary

I am trying to prove the following conjecture: Let $f(z)$ and $g(z)$ be holomorphic functions defined on a simplify connected subregion $\Omega$ of the complex plane, where $\forall z\in \Omega$, $f(...
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2answers
60 views

Elliptic function $f(z)=\frac{a_{-2}}{z^2}+a_0+a_1z+a_2z^2+\dots$ must be even

Let $f:\mathbb{C}\longrightarrow \mathbb {C}$ be a nonconstant elliptic function such that $$f(z)=\frac{a_{-2}}{z^2}+a_0+a_1z+a_2z^2+\dots$$ How to prove that $f(z)$ is even. Notice that $f'(z)+...
3
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4answers
125 views

Value of $\lim\limits_{z \to 0}\bigl(\frac{\sin z}{z}\bigr)^{1/z^2}$

Find the value of $$\lim\limits_{z \to 0}\left(\dfrac{\sin z}{z}\right)^{1/z^2}$$ So I took a log: $$\frac{1}{z^2}\log\left(\frac{\sin z}z\right)$$ If I could expand it something like $\log(1+x)$ .. ...
2
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1answer
95 views

Integral $\int_{-\infty}^{+\infty} \operatorname{sech}^2(x) \cos (2x) \, dx=\frac{2\pi}{\operatorname{sinh}(\pi)}$

Show that $$\int_{-\infty}^{+\infty} \operatorname{sech}^2(x) \cos (2x) \, dx=\frac{2\pi}{\operatorname{sinh}(\pi)}$$ I could not start because the integrand does not have any singularities. I know ...
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1answer
64 views

Residue of $\frac{z^3}{(z-1)(z-2)(z-3)}$ at $z=\infty$

Residue of $\dfrac{z^3}{(z-1)(z-2)(z-3)}$ at $z=\infty$. Well, the total sum of residues is zero. But the given answer says 6. We calculate residues with $Res\;f(a)=\dfrac{1}{(n-1)!}\left[\dfrac{d^{...
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0answers
277 views

Enumerating Automorphisms of Upper Half Plane

I'm trying to find all conformal automorphisms of the upper half plane $\{\Im[z] \gt 0\}$, known to be $f(z) = \frac{az + b}{cz + d}$ where $a, b, c, d$ are real and $ad - bc \gt 0$. The main work ...
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25 views

Monotone analytic continuation

Let $f\colon\mathbb{C}\rightarrow\mathbb{C}$ be analytic in some ball $B$ around $a+0i$. Furthermore, suppose $f\left(B\cap\mathbb{R}\right)\subset\mathbb{R}$ and $f$ restricted to $B\cap\mathbb{R}$ ...
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81 views

Proving that a finite number of poles of mermorphic $f$ exists inside a compact $K \subseteq \mathbb{C}$

I've got some small questions regarding the following proof: If $f$ meromorphic on a compact $K\subseteq \mathbb{C}$ then it has finite many poles in $K$. Proof Say $f$ has infinite many poles ...
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1answer
38 views

Is it possible to alter a holomorphic function such that it has a specific pole? (Complex Variables - Flanigan)

Just a quick True-False question from Complex Variables - Francis J. Flanigan Could someone confirm my reasoning? True or false? If $f(z)$ holomorphic at the origin, then there exists an ...
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1answer
59 views

find an asymptotic expansion by using the Watson's theorem

I want to apply the Watson's theorem to find an asymptotic expansion for the function $$f(z)=\int_{- \infty}^{\infty} e^{-z \frac{y^{2}}{2}} \sin(y^{2})dy$$ (Assume $z \rightarrow \infty, z>0$). ...
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1answer
62 views

Sum of Residues of $\psi^2(-z)$

Compute the Sum of residues of $f(z) = \psi^2(-z)$, where $\psi(z)$ is the digamma function. There are singularities for $z= 1, 2, 3, \ldots$, i.e. for all natural numbers. But how do I compute the ...
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1answer
45 views

Prove that Cauchy transform diverges at 1

I found this problem on Lang Complex Analysis chapter VIII. Le consider the function $g$ defined on the unit circle by: $g(e^{ix})= \begin{cases} -\frac x{\pi\log(4/\pi)} & -\pi\leq x\leq 0\\ \...
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2answers
110 views

Calculating Harmonic Sums with residues.

Evaluate: $$\sum_{n=1}^{\infty} \frac{H_n}{(n+1)^2}$$ A user stated: "most of the time sum up the residues of $(\gamma+\psi(z))^2\cdot r(z)$. To determine the residues, just expand the digamma ...
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0answers
99 views

Evaluating a sum $-\zeta'(2)$

Is it possible to obtain any closed-form expression for the infinite sum $$\sum_{n=1}^{\infty}\frac{\log(n)}{n^{2}}$$ by Residue calculus? My thought was to try to integrate $$f(z) =\frac{\pi\log(z)}{...
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2answers
133 views

Branch cuts of $\frac{1}{\sqrt{z^2 + m^2}}$

I am reading up about Quantum Field theory and the integral of the following function pops up: $$\frac{1}{\sqrt{z^2 + m^2}}$$The details are actually explained in this question. The book says that ...
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1answer
294 views

Branch cut of $\sqrt{z}$ along the *positive* real axis

Consider the function $\sqrt{z}$, with $z\in\mathbb{C}$. Writing $z = re^{i\theta}$, the imaginary part of $\sqrt{z}$ can be expressed as: $$Im({\sqrt{z}) = r^{\frac{1}{2}}\sin({\frac{\theta}{2} + k\...
3
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2answers
90 views

Iteration of analytic function

Suppose $f$ is analytic on the unit disc $D$ with $f(0)=0$ and $f(D)\subset D$. Define $f_n=f\circ f\circ\dots\circ f$. If $f$ is not a rotation, can we say $f_n\to 0$ uniformly on compact subsets of ...
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0answers
73 views

if $w=e^{\pi z}$ find all values of $|z|=r$ for which $w=i$

Question(s) below: $(1)$. if $w=e^{\pi z}$ find all values of $|z|=r$ for which $w=i$ $(2)$. Without finding the equation of the curve obtained from transforming $|z|=r=constant$, what are ...
2
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1answer
74 views

Uniform convergence in the Laurent coefficients proof, why?

Using the Laurent-expansion of $f(z)$ around $0$ $$f(z) = \sum_{n=0}^{+\infty} a_n z^n + \sum_{n=1}^{+\infty} b_n z^{-n} \tag{1}$$ Theorem The coefficients of the Laurent series are given by ...
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2answers
154 views

Why does $\Gamma(1-z)\Gamma(z) = \frac{\pi}{\sin \pi z}$ imply $\Gamma(z) \not = 0$

I'm reading on the extension of $\Gamma$ to the complex plane and there is written: Corollary $$\Gamma(z) \not = 0 \qquad \forall z \in \mathbb{C}\setminus\{0,-1,-2, \dots\}$$ Proof ...
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1answer
99 views

Integrate using residue theorem

This was a question on my complex analysis take home final. Since the semester is over and grades have been posted I believe I can post it now. Let $a > 0$ and $b > 0$. Verify that $$\int_{-\...
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1answer
96 views

solve integral with residue theorem [duplicate]

I want to show that for positive $a$ $$\int_{-\infty}^{\infty}{\frac{\cos(x)}{x^2+a^2}} dx = \frac{\pi e^{-a}}{a}$$ I'm not even sure how to define a smart contour… I guess it can't be a half circle,...