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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2
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0answers
17 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 ...
2
votes
1answer
16 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
votes
2answers
37 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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votes
3answers
44 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. ...
2
votes
1answer
64 views

Prove that $f$ has a simple pole at $z=0$

Let, $f:\{z\in \mathbb C:0<|z|<1\}\to \mathbb C$ be analytic such that $n\le |f(1/n)|\le n^{3/2}$ for $n=2,3,...$. Assume that $z^2f(z)$ is bounded in $|z|<1$. Show that $f$ has a pole of ...
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1answer
55 views

Prove the complex function is entire

The function $$ f(z) = e^{x^2 - y^2} (\cos 2xy + i \sin 2xy )=e^{z^2} $$ Then, how to prove it's analytic everywhere of complex plane of the exp function...?
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votes
0answers
12 views

Fuchs type equation

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 ...
1
vote
0answers
25 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
27 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 ...
0
votes
1answer
68 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
votes
2answers
28 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
votes
0answers
8 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 ...
1
vote
1answer
25 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 ...
1
vote
1answer
61 views

Complex hypersurface globally defined

Let $A$ be a pure one-codimensional analytic subset of a domain $D \subset \mathbb{C}^n$. Is it true that $A$ is defined by one single holomorphic equation $f(z)=0$ if $D$ is bounded and pseudo-convex ...
0
votes
0answers
33 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
vote
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. ...
0
votes
1answer
23 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 ...
2
votes
1answer
34 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?
5
votes
4answers
90 views

Inequality $|e^z -1| \le 2 |z|$ for complex $z$ with $|z|\le1$

I am trying to prove that for $z \in \mathbb C, |z|\le 1$: $$|e^z -1| \le 2 |z|$$ But I'm stuck and I need help. I showed that for all $z$: $|e^z -1| \le |z|e^{|z|}$ but it does not seem useful. ...
1
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2answers
39 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 ...
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) + ...
2
votes
1answer
32 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.
3
votes
0answers
29 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
vote
1answer
44 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 ...
1
vote
1answer
23 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
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 ...
1
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1answer
27 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 ...
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 ...
3
votes
4answers
41 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 ...
0
votes
1answer
25 views

Residues, singularities

For $t\in\mathbb R$ and $n=1,2,3,\dots$ let $$f_n(z)=\frac{z^n}{1-2z\cos t +z^2}$$ Find the singularities of $f_n$ inside $B_2=\{z\in\mathbb C:|z|<2\}$, determine their types, and compute ...
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 ...
0
votes
0answers
25 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 ...
1
vote
1answer
23 views

What is the condition for Morera's theorem to be true?

The answer could be chosen from a) simply connected domain b)connected domain c)no conditions(true for any complex domain) I chose c because the theorem(in our textbook, at least) does not imply ...
1
vote
1answer
24 views

Schwarz Lemma, an onto map with $f'(0)>0$ is the identity

Let $f$ be $1-1$ holomorphic on unit disk onto itself. It satisfies (a) $f(0)=0$, (b) $f'(0)>0$. We need to prove that $f(z)$ is equal to $z$. I am stuck here, because I can prove using Shcwarz ...
1
vote
1answer
27 views

Modular forms: What is $\mathbb{H} / SL_2(\mathbb{Z})$?

I am beginning to understand the very basics of modular forms, in that I understand the concept of a weakly modular function, I have seen the examples of $G_k(z)$ and $E_k(z)$ as weakly modular ...
0
votes
1answer
24 views

Cauchy formula in Polydisks

I don't understand a remark after the proof. Here's the theorem: The proof is done by induction on $n$; starting from $n=1$ on the unitary disk in $\Bbb C$, which is the well known Cauchy formula. ...
0
votes
0answers
28 views

Prove locally uniformly convergence of a sequence

Let for $n = 0,1,2,...$ , $f_n : [0,1] \rightarrow \mathbb{R}$ defined by $f_n (x) = x^n$. 1) Is the convergence of {$f_n$}$_{n=0} ^\infty$ to $f$ locally uniformly on the interval $[0.1]$? 2) And ...
2
votes
1answer
23 views

Radius of convergence of a complex function with a Taylor Series expansion

The function $$f(z)=\frac{1}{1+i-\sqrt{2}z}$$ has a Taylor series expansion around $z_0=0$. What is its radius of convergence? So far I have computed the singularity point to be $z = ...
2
votes
3answers
236 views

Geometry with complex numbers.

Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle, such that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$. Find a simple expression for $d$ in ...
0
votes
1answer
15 views

Two holomorphic functions which have a simple roots at the origin

I am trying to solve the following question: Let $f$ and $g$ be functions holomorphic on the closed unit disk. Assume that f and g have simple zeros at the origin and that g has no any other root in ...
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vote
1answer
143 views

Problem about normal family

Let $D\subset \mathbb{C}^n$ be a region and $a\in D$. If $F$ is the set of all holomorphisms $f$ on $D$ such that $\mathrm{Re}(f)>0,f(a)=1$, Prove that $F$ is a normal family on $D$. ...
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votes
0answers
16 views

contour integral

I am looking for help in find the contour integrals of I want to know what is a good theorem to use in this integral I do not know who to deal with power 1/3 and 2/3 when I need to find the ...
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0answers
15 views

Proof of Contour Integrals and Limits

I've been thinking about the following proof and I'm simply not sure where to start, so any help is appreciated. Thanks in advance. Proof: Let $E$ be a domain in $\mathbb{C}$ and let $f$ be an ...
0
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2answers
16 views

Find the image of the set $\{z \in \mathbb C | -3\lt Re(z) \lt 5, -1 \lt Im(z) \lt 6 \}$, under the function $e^z$

Find the image of the set $\{z \in \mathbb C | -3\lt Re(z) \lt 5, -1 \lt Im(z) \lt 6 \}$, under the function $e^z$ So I know that I should like it as $e^z=e^{x+iy}=e^x(\cos(y)+i \sin(y))$. And I ...
0
votes
1answer
30 views

Complex number, logarithm

Find i)log(e) ii)log(i) I do not know if these issues are of simple fact, that there is something behind. I did i)Since $log$ and $e$ are inverse functions so$$log(e)=log(e^1)=1$$ Knowing that ...
1
vote
1answer
18 views

A little guidance on finding the limit

How do I find the limit of $f(z) = \frac{x^2y}{x^3+y^3} + ixy$ as $z \to0$ ? What I think is if $z\to0$, that implies $x ,y\to0$. But since the $f(z)$ has both variables $x$ and $y$ mixed together, ...
0
votes
0answers
17 views

holomorphic function writen as a serie

To each complexe number $\alpha\in \mathbb{C}$ we associate a function defined by: $$ f_\alpha(z)=1+\sum_{n\geq 1}\frac{\alpha(\alpha-1)...(\alpha-n+1)}{n!}z^n $$ I want to show that this function is ...
1
vote
5answers
339 views

Finding the n-th root of a complex number

I am trying to solve $z^6 = 1$ where $z\in\mathbb{C}$. So What I have so far is : $$z^6 = 1 \rightarrow r^6\operatorname{cis}(6\theta) = 1\operatorname{cis}(0 + \pi k)$$ $$r = 1,\ \theta = \frac{\pi ...
3
votes
2answers
35 views

Finding roots of $z^3 = 8$

I am having trouble finding the cubed roots of $8$ as a complex number. $$z^3 = 8+0i$$ $$z^3 = r^3 e^{3\theta i}=8e^{2i\pi k},\quad k\in \Bbb Z$$ $$\implies r=2,3\theta = 2\pi k\implies \theta = ...
0
votes
2answers
23 views

All solutions of $z^2 + (i-2)z + 3-i =0$

I want to find all solutions of $$z^2 + (i-2)z + 3-i =0$$ Now this is what I do: $$x^2 - y^2 +2xyi +(i-2)(x+iy)+3-i =0$$ $$x^2-y^2 +2xyi + xi-y-2x-2iy+3-i=0$$ $$x^2-y^2-y-2x+3+i(2xy+x-2y-1)=0$$ Now ...