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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1answer
14 views

Transforming a function to use method of residues

Suppose I have an integral $$I=\iiint_{\mathbb{R}^3}\dfrac{d^3\textbf{k}}{(k^2+\gamma)^2}$$ where $\gamma$ is independent of k. $d^3\textbf{k}$ is given as the 3 components of a vector. I am asked ...
0
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2answers
18 views

Stone Weierstrass and Runge

Suppose $E\subset\{z:|z|=1\}$ and let $f(z)$ be a continuous function on the set $E$. I want to show that $f(z)$ can be approximated by polynomials on $E$. I am not exactly sure how to solve this ...
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1answer
31 views

$f$ is differentiable on $U\setminus\{p_1,\dots,p_r\}\implies$ $f$ is holomorphic on $U$

Let $U\subset\mathbb C$ be open and $p_1,\dots,p_r$ be finite number of points in $U$. If $f:U\to\mathbb C$ is continuous function that is complex-differentiable in any point of ...
1
vote
1answer
37 views

Basic complex integration question

If I have an integral: $$\int_{0}^{2\pi} \frac{1}{3+2cos(t)^2} = \frac{a\pi}{b}$$ How can I find a and b? What formula do I use?
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0answers
18 views

Confusion with $\int_{C}\frac{ze^z}{z^6 - 1}$

I must solve $\int_{C}\frac{ze^z}{z^6 -1}dz$ where $c:= \{ z \; : \; |z-a|=a\}$ and $a > 1$. I wish to apply the Cauchy Integral Formula (or generalized). The only singularities inside the ...
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2answers
30 views

$\int_{\gamma}\left(\frac{1}{z}-\frac{1}{z-1}\right)dz=0$ on $\mathbb C\setminus[0,1]$

$\int_{\gamma}\left(\frac{1}{z}-\frac{1}{z-1}\right)dz=0$ on $U:=\mathbb C\setminus[0,1]$ for a closed path with image in $U$ For any analytic function $f$ and a closed path $\gamma$: ...
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0answers
20 views

Why can we write $\displaystyle u(x+h_1,y+h_2)-u(x,y)=\frac{\partial{u}}{\partial{x}}h_1+\frac{\partial{u}}{\partial{y}}h_2+|h|\psi_1(h)$?

In this image, why can we write $\displaystyle u(x+h_1,y+h_2)-u(x,y)=\frac{\partial{u}}{\partial{x}}h_1+\frac{\partial{u}}{\partial{y}}h_2+|h|\psi_1(h)$ ? [I borrowed link to the image uploaded by ...
2
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1answer
27 views

Show that $z^n+nz-1$ has $n$ zeros in $D(0,R)$

Let $n\geq 3$. Show that the polynomial $z^n+nz-1$ has $n$ zeros in $D(0,R)$, where $$R=1+\left(\frac{2}{n-1}\right)^{1/2}.$$ I was hoping to use Induction and Rouche's Theorem. For the base case ...
0
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0answers
15 views

construct a function which is differentiable at infinitely many points in the complex plane but is nowhere analytic

I tryed many times. but It is very hard to me to find such a function. I need some help or hint. It is not the infinitely differentiable but the infinetly many points and nowhere analytic in the ...
1
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1answer
25 views

Poles of power series

This may be a trivial question, but I haven't been able to find an answer. Given a power series about $x_0$ $F(x)=\sum_{n=0}^\infty a_n (x-x_0)^n$, how do we find its (complex) poles? What about the ...
1
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2answers
33 views

Fourier transform of $e^{-4\pi ^2 x^2}$

How do you prove $$\int_{-\infty}^{\infty}e^{-(2\pi x + i\xi/2)^2}dx=\int_{-\infty}^{\infty}e^{-(2\pi x)^2}dx$$ for $\xi \in \mathbb{R}$. The Question arises from calculating the Fourier Transform ...
0
votes
1answer
48 views

Prove: $\int_a^b e^{z_0t}dt=\frac{1}{z_0}e^{z_0t}|_a^b$

From a complex variables online course, and I need to prove that $$\int_a^b e^{z_0 t}dt=\frac{1}{z_0} e^{z_0 t}|_a^b$$ For every $0\neq z_0\in \mathbb{C}$ and for every $a,b\in\mathbb{R}$. Do I need ...
-1
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0answers
33 views
0
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0answers
15 views

Contour integration along a contour containing two branch points

I need to compute following contour integrations: $$f(u)=\oint_\alpha dz \sqrt{z^3+z+u} \qquad ; \qquad g(u)=\oint_\beta dz \sqrt{z^3+z+u}$$ In which $\alpha$ and $\beta$ are two contours in ...
1
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0answers
23 views

Finite groups of Mobius Transformations

Let $M_2(\mathbb{C})$ be the group of all Mobius transformations $z\mapsto \frac{az+b}{cz+d}$ from $\mathbb{C}\cup\{ \infty\}$ to itself. Let $PSU(2,\mathbb{C})$ be the group of all Mobius ...
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0answers
20 views

Derive chain rule for complex functions, from the chain rule for real functions

I'm trying to obtain the chain rule for complex (not necessarily holomorphic) functions $\mathbb{C} \to \mathbb{C}$, using the known chain rule for functions $\mathbb{R}^2 \to \mathbb{R}^2$. The ...
3
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1answer
33 views

$\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$ - different answers depending on value of $t$?

After taking an inverse Laplace transform I have the following - $$y = \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$$ In my notes it says if $t ...
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0answers
8 views

Composition of a holomorphic function with a normal family of holomorphic functions.

Let $\Omega_1$ and $\Omega_2$ (open) $\subset \mathbb{C}$ and $\mathcal{F}$ a normal family of holomorphic functions in $\Omega_1$ such that $f(\Omega_1) \subset \Omega_2$ $\forall f \in \mathcal{F}$. ...
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votes
2answers
50 views

Showing $f$ is a constant function

If $f$ is an entire function across the complex plane, how can I show that $Im(f(z)) \gt Re(f(z))^2 - 2$ is constant?
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0answers
12 views

Find domain of $f$ where $f(z)=\ln(z^2-5z+6)$

$\ln$ is the principal branch of the complex natural logarithm. I think I've solved it, but I don't know if I covered everything. Here's what I did: $Im(z^2-5z+6)>-\pi$ and $Im(z^2-5z+6)\leq\pi$ ...
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1answer
31 views

Develop the Taylor series of $\ln(z^2-5z+6)$ in $z=0$

Also, determine the radius of convergence. $\ln$ is the principal branch of the complex logarithm. What I've tried is splitting the function into $\ln(z-3)+\ln(z-2)$ and then finding the formula for ...
11
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2answers
71 views

If $f(a+re^{it})\in \Bbb{R}$ for all $t\in \Bbb{R}$ then $f$ is constant.

I would like to prove that if $f(a+re^{it})\in \Bbb{R}$ for all $t\in \Bbb{R}$ then $f$ is constant. Of course $f$ is holomorphic on a domain $U$ and $r>0$ such that $\overline{D(a,r)}$ is ...
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1answer
24 views

which of the following is/are true for the entire function $f$?

Let , $f$ be an entire function. Let, $g(z)=\overline{f(\bar z)}$. Let, $D=\{z:Im(z)=0\}\cup\{z:Im(z)=a\}$ for some $a>0$. Then which are correct ? (A) If $f(z)\in \mathbb R$ for all $z\in \mathbb ...
2
votes
1answer
25 views

If $\lim_{|z|\to \infty}\frac{f(z)}{g(z)}$ exists then either $f\equiv0$ on $\Bbb C$ or $f(z)\not =0$ for all $z\in \mathbb C$.

Let , $f,g:\mathbb C\to \mathbb C$ be analytic such that $g(z)\not =0,\forall z\in \mathbb C$. If $\lim_{|z|\to \infty}\frac{f(z)}{g(z)}$ exists then prove that either $f\equiv0$ on $\Bbb C$ or ...
4
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2answers
44 views

Proving that a doubly-periodic entire function $f$ is constant.

Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: ...
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0answers
13 views

Fuchs type equation [on hold]

How to show for any second order equation $u''+p(z)u'+q(z)=0$, with finitely many singularities at $z_0,\ldots,z_n,\infty$ of Fuchs type is of the form $$p(z)=\sum_{j=0}^n\frac{p_j}{z-z_j}, \quad ...
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votes
3answers
53 views

Evaluate $\int_0^{2\pi} e^{ e^{i\pi} } d\theta $ by rewriting this as an integral about a suitable contour. [on hold]

Evaluate $\int_0^{2\pi} e^{ e^{i\pi} } d\theta$ by rewriting this as an integral about a suitable contour. The integral became intense. I would write my work here, but I am not the best at LaTeX. ...
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0answers
29 views

a function defined as an integral can be continued analytically

I am trying to solve the following question: Verify that the integral $\int_{0}^{\infty} \, \frac{t^{z}}{e^{\,t\,}+1}dt$ represents an analytic function in the half plane $Re(z)>-1$. Show also ...
2
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0answers
31 views

integral of harmonic function

I'm having trouble with this one: Let $u$ be a real-valued harmonic function on $D(0,1)$, and let $\gamma$ be a closed curve in that disk. Then $\int_\gamma u=0.$ I'm supposed to prove or disprove ...
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2answers
30 views

Evaluating a complex integral (Hints please)

I am supposed to be able to show that, given $f(z)=\frac{1}{\pi}\int_0^1r\int_{-\pi}^\pi\frac{d\theta}{re^{i\theta}+z}dr$ then $f(z)=\overline{z}$ for $|z|<1$ and $f(z)=1/z$ if $|z|\geq1$. (This ...
1
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1answer
26 views

Use Cauchy's Integral Formula to evaluate the following integrals.

Use Cauchy's Integral Formula to evaluate the following integral: $$\int\limits_\Gamma \frac{1}{{(z-1)^3}{(z-2)^2}}dz$$ where $$\Gamma$$is a circumference of radius $4$ centered at $-2+i$ and ...
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0answers
13 views

Covariance and cross spectrum

A bivariate process $(x_t, y_t)$ is called stationary if each component is a univariate stationary process and $cov (x_s , y_{s+j}) =cov (x_t , y_{t+j}), \forall s,t,j$. The autocovariance function ...
2
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0answers
23 views

Finding the Laurent series given the poles and residues

I am working on the following problem, suppose that $f$ has a simple pole at $-1$ with $Res(f,-1) = 1$. A double pole at $2$ with $Res(f, 2) = 2$. Also $f(0) = 7/4$ and $f(1) = 5/2$. I am supposed ...
0
votes
1answer
80 views

The importance of being real

Let $\Sigma$ be a collection of holomorphic, one-to-one function from some simply connected region $\Omega$, which map $\Omega$ into the open unit disc $U$. Fix $z_0 \in \Omega$ and put $$\eta = ...
0
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0answers
34 views

Solve the complex euqtions

I have a question from complex calculus. How to solve this two equations: a) $$ sin(z)=2015 $$ I know that sin(z) equals to $$ \frac{e^{iz}-e^{-iz}}{2i} $$ And i don't know whats next. b) $$ ...
1
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1answer
24 views

Calculate complex integral $\int_\Gamma\frac{\ln(z+5)}{z^3+iz^2+6z}$

$\Gamma$ is a circle of radius 2 around the point $1+i$. I've parametrized the circle as $\gamma(t)=2e^{it}+1+i$ substituting $z$ in te integral for that expression gets really ugly really quickly. ...
2
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1answer
44 views

Is there any condition while applying law of exponents?

${[(-3)^2]}^\frac{1}{2}$ = ${(-3)^2}^\frac{1}{2}$ = $-3^1$ = $-3$ But counted other way it is $9^\frac{1}{2} = \surd{9} = 3$ where I went wrong?
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1answer
25 views

Corollary of Riemann Mapping Theorem

I was trying to prove the uniqueness of the map in the Riemann mapping Theorem. I'm not sure if the proof I wrote is right. Let $\Omega \subset \mathbb{C}$ be a simply connected open subset such that ...
3
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0answers
35 views

Is there a spherical coordinates system for vectors of complex numbers?

Suppose I have a scalar field $f(\vec{x})$, where $\vec{x}\in\mathbb{R}_3$, and I wish to average $f$ over a sphere $|\vec{x}|=R$: $\displaystyle\langle f\rangle_{R} = \frac{\int_{S} f(\vec{x})\, ...
1
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3answers
22 views

Some complex logarithms: please could somebody check my work?

I am doing some exercises from my book, this one asks me to find suitable $z \in \mathbb C$. Please could someone check my work? 1) $z$ such that $e^{z}=-2$: This means that $-2 = iArg(z) + ...
1
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1answer
53 views

Show that f and e^f can not have a common pole

Let $f$ be holomorpic on a punctured neighborhood of $z_o$. Show that $f$ and $e^f$ can not have a common pole. My attempt at solution is WLOG let $z_o =0$ be a pole of $f$. Then the Laurent series ...
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2answers
40 views

How do I find $\frac{d}{dz}\left(\frac{2z-i}{z+2i}\right)\text{?}$

How do I find: $$\frac{d}{dz}\left(\frac{2z-i}{z+2i}\right)\text{?}\quad\quad z\in\Bbb C$$ Do I turn it into an $x+iy$ form and use the Cauchy-Riemann equations? I couldn't get it into such a ...
0
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0answers
14 views

Where $|f| <\infty$ a.e. condition is used in Vitali Convergence Theorem

Vitali convergence theorem_Wiki Here above is a Wiki article about Vitali convergence theorem, which is referred to Rudin, Real and Complex Analysis. And I'm wondering where the fourth condition is ...
2
votes
1answer
33 views

Divergent succession, but with convergent sum average.

An example of a sequence $a_n$ such that: $$a_n\rightarrow\pm\infty$$ but $$b_n=\frac{\sum_{k=1}^{n}a_k}{n}$$ converge.
1
vote
1answer
28 views

Showing $f(z)=x^2+iy^3$ is not analytic anywhere

I want to show that the following function is not analytic anywhere. $$f(z)=x^2+iy^3$$ Now I don't really understand the Cauchy-Riemann equations, but it seems we take: $$u(x,y)=x^2,v(x,y)=y^3$$ as ...
3
votes
4answers
42 views

Finding $\lim \limits_{z\to i} \frac{1}{(z-i)^2}$ rigorously

I want to find the limit of the following: $$\lim \limits_{z\to i} \frac{1}{(z-i)^2}$$ And to me, I can see that the denominator is clearly $0$, and since we are in the extended complex plane, I feel ...
2
votes
5answers
51 views

Solving $\cos z = i$ for $z$

Solve $\cos z = i$ for $z$. What I have tried: $$\cos z = i$$ $$\frac{e^{-zi}+e^{zi}}{2}=i$$ $$e^{-zi}+e^{zi}=2i=2e^{\frac\pi 2 + 2\pi k},\quad k\in \Bbb Z$$ I would take logs, but then I would ...
1
vote
1answer
28 views

Where does the imaginary unit dissapear in the Fourier transform of $f(t)= \exp(iat)$?

So I make the Fourier transform of$ f(t)= e^{iat} $on $[- \pi, \pi]$ for some real $a$ and i get: $$a_n=\frac{2a \sin(a \pi)(-1)^n}{\pi(a^2-n^2)}$$ $$b_n=\frac{2i(n\sin(a \pi) (-1)^n)}{\pi(a^2 - ...
0
votes
0answers
26 views

Compute radius of convergence and the first three coefficients of a function

Let $\displaystyle f(z) = \frac{z+1}{(2z+1)(1+ \sin z)}$, with serie expansion $\sum_{n=0} ^\infty a_n z^n$ around zero. Now I want to compute the radius of convergence and the first three ...
0
votes
0answers
18 views

Harmonic Function Cauchy implication

Let $b$ be harmonic real valued on unit disk. Then I wish to prove that $\int_\alpha b =0$. I know that there exists $f$ holomorphic such that $\Re(f)=b$, and I know from Cauchys result that ...