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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83 views

Prove if $f$ is entire and $|f(z)| \leq |z|^{1/2}$ for all $z$, then $f(z) = 0$ for all $z$.

Prove if $f$ is entire and $|f(z)| \leq |z|^{1/2}$ for all $z$ in the complex plane, then $f(z) = 0$. This was given on a homework assignment, but I have no idea how to do it! A walkthrough would be ...
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1answer
1k views

Sketching variations of $Arg(z)$

I'm asked to describe and sketch the following sets in $\mathbb{C}$: A) $\{ z \in \mathbb{C}: |Arg(z+1)| < \frac{\pi}{4}$ B)$\{ z \in \mathbb{C}: |z| \le arg \ z \ \ \ \ and\ \ 0 \le arg \ z \le ...
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1answer
86 views

integration of $\int \frac{1}{x+ i\:y}\mathrm{d} x$

I can't seem to find where this result comes from $$ \int \frac{1}{x+ i\:y}\mathrm{d} x = \frac{\ln(x^2 +y^2)}{2} - i \: \arctan \left( \frac{x}{y} \right) $$ by my calculation the result should be ...
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1answer
70 views

factorization of an expression involving gamma function

Does the equation $\Gamma(x+1/2)\Gamma(x-1/2)=\Gamma(x+iy)\Gamma(x-iy)$, where $\Gamma(z)$ is the Gamma function and $i=\sqrt{-1}$, have any solution assuming $x,y$ are both real and $x>1/2$? This ...
5
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1answer
97 views

sequence of complex polynomials $p_n$ s.t. $p_n(0) = 1$ for every $n \in \mathbb{N}$ and $p_n(z) \to 0$ for each $\mathbb{C}-\{0\}$?

Is there a sequence of complex polynomials $p_n$ s.t. $p_n(0) = 1$ for every $n \in \mathbb{N}$ and $p_n(z) \to 0$ for each $z \in \mathbb{C} \setminus \{0\}$? Any help with this would be great!
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140 views

Calculating a root of a complex number with euler formula

Let $z= 1+i$. The polar form of $z^{1/5}$ can be easily calculated: $$z = 2^{1 \over 10}\left\{\cos({\pi/4 + 2\pi k \over 5}) + i\cdot \sin({\pi/4 + 2\pi k \over 5})\right\}$$ the "principal root" ...
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190 views

How find the maximum value of $|bc|$

Question: Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$ It is said this is answer is ...
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1k views

Can we prove that all equations can be solved via complex numbers?

$x^2+1=0$ cannot be solved via real numbers. Because of this, we extend the real numbers to complex numbers.We can solve $x^2+1=0$ and $x^2+x+1=0$ equations after we define complex numbers. I ...
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1answer
40 views

Find a certain analytic function

Let $D_1$ and $D_2$ be two disjoint disks.Find an analytic function $f$ defined on the upper half plane such that $f$ takes every value in $D_1$ exactly once and every value of $D_2$ exactly twice. I ...
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0answers
54 views

Extensions of the Hermite Bielher and Hermite-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...
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1answer
65 views

$(\mathbb{C}\setminus \mathbb{R}) \cup (-1,1)$ simply connected?

How does one show simply connectedness? It is painfully obvious to me geometrically, but this isn't a proof.
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1answer
67 views

Basic question about complex function representation u(x,y)+iv(x,y)

I suppose it natural to write $f(x,y) = u(x,y) + iv(x,y)$ but it make me wonder are we losing anything in that process? When we talk about things component-wise, is that limiting anything? Can we ...
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1answer
151 views

Constant function with maximum modulus [duplicate]

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
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189 views

Compactly supported Dolbeault Cohomology: is this True?

nLab states that for $D$ the unit disk in $\mathbb C$, the cohomology of the complex $$ (\Omega_c^{1,\ast}(D),\overline{\partial})$$ is the continuous dual of the space of holomorphic functions ...
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3answers
248 views

Integrating over Branch Cuts

I'm having problems following the solution for b). The main problem is finding the interval which you integrate over, which for some reason in this case is $(-i,i)$. To be frank I don't really get the ...
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2answers
114 views

Simplify result of $\int_0^{\infty} \frac{1}{1+x^n}dx$

It is quite easy to show that (by using residue theorem) $$\int_0^{\infty} \frac{1}{1+x^n}dx = \frac{2 \pi i^{1+2/n}}{n(e^{2 \pi i / n} - 1)} $$ for $$n \ge 2$$ Is there any possibility to simplify ...
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1answer
38 views

Pole and residue of $f(z) = \frac{1}{1+z^n}$

Let $f(z) = \frac{1}{1+z^n}$ for $n \ge 3$. How can we compute poles and residue in first point over real axis? Firstly, we have to solve $1+z^n=0$. I think, that solution of this are $$z_k = e^{2 ...
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1answer
95 views

existence of analytic function related to identity theorm

Does there exist an analytic function $f$ in closed unit disk such that $f(z)\neq 0$ identically and $f\left(\frac{ni^n}{1+n}\right)=0$ for $n\in\Bbb N$?
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707 views

How to find PV $\int_0^\infty \frac{\log \cos^2 \alpha x}{\beta^2-x^2} \, \mathrm dx=\alpha \pi$

$$ I:=PV\int_0^\infty \frac{\log\left(\cos^2\left(\alpha x\right)\right)}{\beta^2-x^2} \, \mathrm dx=\alpha \pi,\qquad \alpha>0,\ \beta\in \mathbb{R}.$$ I am trying to solve this integral, I ...
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86 views

Closed form of $\int _{0}^{\infty }\!{\frac {x\cos \left( x \right) -\sin \left( x \right) }{{x}^{3} \left( {{\rm e}^{x}}+1 \right) }}{dx}$

Does it possibly have a closed form? $$\int _{0}^{\infty }\!{\frac {x\cos \left( x \right) -\sin \left( x \right) }{{x}^{3} \left( {{\rm e}^{x}}+1 \right) }}{dx}$$ Thank you! I found it. No more ...
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52 views

A question on Moebius transform and circles

Suppose $T : z \mapsto \frac{az+b}{cz+d}$ a Moebius transform, and $\mathcal{C}$ a disk of center $w$ and radius $r$. If $\mathcal{C} ' = T(\mathcal{C} )$ is a circle too, how to determine its center ...
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486 views

All roots of polynomial inside the open unit disc

I know from here that for a polynomial $p(z)=a_0+a_1z+...+a_nz^n$ with $0<a_0\leq a_1\leq...\leq a_n$ all roots are in the closed unit disk. What condition do we need to get that all roots are in ...
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1answer
50 views

A proof of a Proposition by Hurwitz

Here is a proof of a Theorem by Hurwitz. (Source: G. De Marco, Selected topic in Complex Analysis.) (In these notes, $B(c,r]$ means the closed ball in $\mathbb{C}$ centered in $c\in\mathbb{C}$ and ...
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226 views

Applications of Stein spaces in Algebraic Geometry

I want to know where are essential applications of the theory of Stein spaces in algebraic geometry. I heard Cartan's theorem A & B were used in Serre's GAGA, but are there any other applications? ...
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1answer
28 views

How to solve a limit of a complex integral over part of the real axis?

How do I solve: $$\lim_{\epsilon\to 0}\int_{|x|<1} {1 \over {x-i\epsilon}} dx$$ ? x is a real variable, and not complex.
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39 views

Maximum of $|(\sin (z)/z)-\cos (\alpha z)|$ on the disk $|z| \leq\pi/4$

I am trying to find the following maximum, whose existence is justified by the compactness of the close ball $\Delta$ of $\mathbb C$ and continuity of the function $$f(z)=\frac{\sin (z)}{z}-\cos ...
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1answer
1k views

Minimum Modulus Principle for a constant fuction in a simple closed curve

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
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2answers
107 views

Usage of Maximum Modulus Theorem?

I have a function $f(z)=ze^z$, I want to find the maximum value of $|f(z)|$ as $z$ varies over the region $D=\{x+iy: x^2+y^2\leq 4, x,y\geq 0\}$. I was thinking that this is case where I can use the ...
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1answer
23 views

Confused about a proposition about rational polynomial

I'm studing complex analysis.I saw that $R(z)$ is a rational function.Consider the function $R(1/z)$ which we can rewrite as a rational function $R_1(z)$,and set $R(\infty)=R_1(0).$ with the ...
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2answers
48 views

Series in $\mathbb{C}$

I'm having trouble to find the radius of convergence of the following series. $$\sum_{k=0}^\infty z_1^kz_2^k$$ where $z_1,z_2\in\mathbb{C}$ are arbitrary. Furthermore, how can I determine the domain ...
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1answer
92 views

Prove that $\int_{-\pi}^{\pi}$ $\frac{d\theta}{1+\sin^2\theta}$ = $\pi\sqrt{2}$ using the method of Residues

Prove that $$\int_{-\pi}^{\pi}\frac{d\theta}{1+\sin^2\theta} = \pi\sqrt{2}$$ using the method of Residues How do I do this? I know I need it from $0$ to $2\pi$ but I don't know how to modify it!! ...
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1answer
21 views

How to show that $f(A_1)=\{z: z(t)=x+i, x \in \mathbb{R} \}$?

Let $A_1=\{z: z(t)=\frac{1}{2}e^{it}- \frac{1}{2}i, t\in [\frac{\pi}{2}, \frac{3\pi}{2}] \}$ and $f(z)=\frac{1}{z}$. Show that $f(A_1)=\{z: z(t)=x+i, x \in \mathbb{R} \}$. My attempt: ...
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1answer
56 views

How to determine the equation of the circle which cover all the root and have least area

Gauss–Lucas theorem: If P is a (nonconstant) polynomial with complex coefficients, all zeros of P' belong to the convex hull of the set of zeros of P. so, for every polynomial $P=\sum a_i x^i$ ...
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1answer
56 views

Calculating the residues of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$

Calculating the poles of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$, where x is a fixed real number. I am trying to calculate the poles of this function at the trivial zeros of $\zeta$, ...
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3answers
86 views

Integrating $\int_{-\infty}^{\infty}\frac{e^{x/2}}{1+e^x}$

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{e^{x/2}}{1+e^x}dx$$ Show contours and discuss estimates needed to justify your method. I'm having trouble as to the region of integration which ...
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2answers
106 views

What's the behavior of $\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$ outside its radius of convergence?

I want to check the behavior of $$\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$$ outside its radius of convergence. I've tried to use the ratio test as follows: ...
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130 views

What is the order of this pole at $z=0$?

What is the order of the pole: $$f(z) = \frac{1}{(6\sin (z)-6z+z^3)^3}\,\, \mathrm{at}\, z=0$$ I thought about doing the expansion for sin and then pulling out a $z$ from there to get an order of ...
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1answer
277 views

A problem related to Schwarz reflection principle

Let $\Omega$ be a bounded domain in $\mathbb{C}$. Suppose there is a function $f$ which is analytic in $\Omega$ except a simple pole at $a\in\Omega$, such that $(z-a)f(z)$ is continuous on ...
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1answer
36 views

How can I prove that $_{max}|Az^{n}+b|$ =$ |A|$ + $|B|$ when |Z| $\leq$ 1?

How can I prove that $_{max}$$|Az^{n}+B|$ = $|A| + |B|$ when $|Z|$ $\leq$ 1? Remember that Z is a complex number which is why I had to include the magnitude.
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2answers
99 views

sequence and series (complex analysis)

Let $N_0\in \mathbb{N}.$ If a sequence of complex numbers $\{F_N\}_{N \in \mathbb{N}}$ has the following properties: $$\lim_{N \rightarrow \infty} |F_N|^{1/N}=0$$ and for all $N \geq N_0$, $$|F_N|\leq ...
2
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1answer
88 views

Prove that $|f(z)|\leq |z|^2$ where $f$ is analytic on the unit disc and has a zero of order 2 at zero.

$f(z)$ is analytic on the unit disk $|z|<1$. If $f(z)$ has a zero of order $2$ at the origin and $|f(z)|\leq 1$ on the disk. Prove that $|f(z)|\leq |z|^2$ when $|z|<1$. What I did: Since ...
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2answers
93 views

Finding singularities of a projective curve

For $w \in \mathbb{C}$ we define the projective curve $$p(x,y,z):= x^3+y^3+z^3+wxyz.$$ Now I have to find all $w \in \mathbb{C}$ for which the projective curve $p(x,y,z)$ is singular and show that ...
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4answers
284 views

Integral$\int_0^{\pi/4} \log \tan \left(\frac{\pi}{4}\pm x\right)\frac{dx}{\tan 2x}=\pm\frac{\pi^2}{16}$

Hi I am trying to prove $$ \int_0^{\pi/4} \log \tan \left(\frac{\pi}{4}\pm x\right)\frac{dx}{\tan 2x}=\pm\frac{\pi^2}{16}. $$ What an amazing result and a clever one this is. I tried writing $$ ...
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1answer
151 views

Integral $\int_0^\infty \ln x\,\exp\left(-\frac{1+x^4}{2\alpha x^2}\right) \frac{x^4+3\alpha x^2- 1}{x^6}dx$

$$I:=\int_0^\infty \ln x\,\exp\left(-\frac{1+x^4}{2\alpha x^2}\right) \frac{x^4+3\alpha x^2- 1}{x^6}dx=\frac{(1+\alpha)\sqrt{2\alpha^3 \pi}}{2\sqrt[\alpha]e},\qquad \alpha>0.$$ This one looks very ...
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2answers
133 views

Cauchy Integral Formula for a continuous function defined on a circle

Let $D$ be the open unit disc and $C$ be the unit circle. Suppose $$f:C\to\mathbb{C}$$ is continuous. Show that $$g(w)=\dfrac{1}{2\pi i}\int_{C} \dfrac{f(z)}{z-w} \rm{d}z$$ is an analytic function of ...
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1answer
44 views

Prove statement about complex series

The problem statement Let $(a_n)_{n\geq o}$, $(Z_n)_{n\geq 0}$ sequences of complex numbers such that $(a_nZ_n)_{n\geq 0}$ converges. Show that $\sum_{n=0}^{\infty} (a_n-a_{n+1})Z_n$ converges if ...
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2answers
83 views

why $\infty$ +$\infty$, $\infty$ - $\infty$ and 0⋅$\infty$ are left undefined.

I'm reading http://en.wikipedia.org/wiki/Riemann_sphere, and having the following question. 1. What's the mean of symbol $\infty$?Is it a surreal number? 2. they write note that ∞ + ∞, ∞ - ...
1
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1answer
158 views

How to show $\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$

For $z\in\mathbb{R}$ it's very easy to show that it holds $$\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$$ But how do we show the same thing for $z\in\mathbb{C}$
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2answers
95 views

$\frac{\sin(z)}{z}$ Bounded on $\mathbb{C}-\{0\}$?

Is $\frac{\sin(z)}{z}$ bounded on $\mathbb{C}-\{0\}$? Based on real analysis intuition it is, and even though complex $\sin$ is not bounded the $\frac{1}{z}$ makes me think it is.
2
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1answer
76 views

Find maximum of a complex function $f(z)$

I am trying to find the following maximum, whose existence is justified by the compactness of the close ball $\Delta$ of $\mathbb C$ and continuity of the function $$f(z)=\sum_{k=1}^\infty ...