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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Is a constant function considered to be an entire function?

Is a constant function considered to be an entire function? Constant function is differentiable everywhere. Liouville's theorem holds for them too.
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1answer
34 views

Basic Geometric Series Question

Calculation of $ \sum_{n=0}^{\infty}2^{2n} z^{2n} $ The answer is We note that the n-th summand has the form $(2z)^n$ Denoting w = 2z The sum is sigma of 0 to n summand being $(w)^n$ which can be ...
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1answer
107 views

Finding order of $f(z) = \cos\sqrt z$

What is the order of following entire function: $$f(z) = \cos\sqrt z$$
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3answers
33 views

Number of Complex satisfying a given condition

Suppose we have a complex number $z$ such that $|z|=1$ and $$|\frac{z}{z'} + \frac{z'}{z}|=1$$ where $z'$ is conjugate . How many complex number satisfy this? So I simplified second condition as ...
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1answer
34 views

Find the complex Fourier series

Find the complex Fourier series representation of the function $$ f(t) = \begin{cases} 1,\quad\text{if}\quad 0 < t < 2 \\ 0,\quad\text{if}\quad 2 < t < 4 \end{cases} $$ with the period ...
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1answer
84 views

Evaluating $\int_{-\infty}^{\infty} \frac{\cos x}{1+x^2} e^{-ixt} \,\mathrm dx$

$$\int_{-\infty}^{\infty} \frac{\cos x}{1+x^2} e^{-ixt} \,\mathrm dx \quad \quad \quad \text{for }t>0$$ Use residue formula, which contour should I try?
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1answer
215 views

Residue of $\frac{\cos(\frac{\pi}{z-1})}{z^2 \sin z}$ at $z=1$

Residue of $$\frac{1}{z^2 \sin z}\cos\left(\frac{\pi}{z-1}\right)$$ at $z=1$. More importantly, I don't even know whether it exists or not. The one who creates this question has made questions that ...
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2answers
62 views

Complex roots forming a equilateral triangle

Suppose we have relation $$z^2 + az + b=0 $$ where $a$ and $b$ are real and roots of this equation $z_1$ and $z_2$ form equilateral triangle with origin then what could be relation between $a$ and ...
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5answers
553 views

Geometrical Interpretation of Cauchy Riemann equations?

Differentiation has an obvious geometric interpretation, and the Cauchy Riemann equations are closely linked with differentiation. Do the Cauchy Riemann equations have a geometric interpretation?
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1answer
29 views

$Res[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$ -Proof

I need to prove that if $f$ and $g$ are analytic in $D_r(z_0)$ and $g$ has a simple zero at $z_0$ then, $$Res[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$$ When $f(z_0)\neq 0$ and since $1/g$ has a simple pole ...
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1answer
436 views

How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$. If $n=1$, the zeros ...
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1answer
44 views

Basic Question. What's the difference between a circumference and modulus of e^(iz)

We know Modulus of e^(iz) is 1 but circumference is 2*pi*r is modulus of e^(iz) multiplied with the angle it has rotated equal to circumference? modulus of e^(iz) calculates the distance of entire ...
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2answers
97 views

Polynomial growth, using the Cauchy Integral Formula,

Is this a true statement in Complex Analysis? If a function grows like a polynomial, then it is a polynomial. Or, is it really: if a function grows like a polynomial at infinity, then it is a ...
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2answers
63 views

Find cube roots of $-1$

If $-1$ and $\lambda$ are two cube roots of $-1$ find in terms of $\lambda$ the third cube root of $-1$. Am I right in saying that it is just $1-\lambda$?
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1answer
41 views

Solving a system of multivariable fifth degree polynomials

Suppose that $$x=u^5 - 10 u^3 v^2 + 5 u v^4$$ and $$y=5 u^4 v - 10 u^2 v^3 + v^5.$$ Given $x,y,u \in \mathbb{R}$, is it possible to find a $v \in \mathbb{R}$ that satisfies the above relations? I ...
3
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1answer
50 views

Explanation of the argument principle.

https://www.youtube.com/watch?v=RRDmCC8gKpY At 22:35 he's trying to explain how $ \arg h(z) $ changes. I am not understanding this though why is $ f(z_0 + Re^{it})$ it not a circle? And also why ...
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2answers
75 views

Finding a unique Mobius Transformation

Let $z_1, z_2, z_3$ be three distinct points in $\widetilde{\mathbb{C}}$. (1) show that there is a unique mobius transformation $g$ such that $g(z_1)=0, g(z_2)=1, g(z_3)=\infty$ (2) show ...
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1answer
65 views

Can someone please give me some practical application of liouville theorem

All I understand is liouville theorem states if f is entire on the domain specified, and modulus of f is bounded for all z on the domain then f is identically constant. This is all I know and ...
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0answers
54 views

Quintic Roots of 1

z^5=1 has five roots. How does z^5=32 relate to those roots? Its basically those roots, but multiplied by 32 right?
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2answers
84 views

Integral of exponential

I have been reviewing complex analysis since it's been awhile since I dove into it I am stumped of the following integral, $$\int^{\infty}_{-\infty}e^{iax-bx^2}dx.$$ By setting $z=x-i\frac{a}{2b}$, I ...
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2answers
3k views

Singularity at infinity of a function entire

How to prove that every non-constant entire function $\,\,f:\mathbb{C}\rightarrow\mathbb{C}\,\,$ has a singularity at infinity? What type of singularity must this be?
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1answer
487 views

Can one give me some concrete examples explaining Picard's Great Theorem

Picard's Great Theorem Every non-constant entire function attains every complex value with at most one exception. Furthermore, every analytic function assumes every complex value, with possibly one ...
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1answer
132 views

A tough question on meromorphic functions- Conway

This is a question I came across in JB Conways book. Let $f$ be meromorphic function in the punctured disk $D_r(z_0)$ \ {$z_0$}. Suppose there is a sequence {$p_n$} of poles of $f$ in ...
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2answers
173 views

Geometric interpretation of Cauchy-Goursat Theorem?

This theorem seems almost magical. The algebraic derivation doesn't really provide any insight into why it works. So could someone give me a geometric interpretation of it? This: Geometrical ...
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1answer
42 views

analysis of complex vector space

null vector in complex space let is vector scalar product of which to itself is zero, for example let us take vector scalar product to itself $(1,i)*(1,i)=1-1=0$ let us consider all null vector ...
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1answer
47 views

How can I calculate this complex integral?

The integral is the following: $$\int_{|z|=r} \frac{z+1}{z(z^2+4)} dz , r>0, r \neq 2 $$ I'm a little bit lost, I know that its partial fraction expansion is $$ \frac{z+1}{z(z^2+4)} = ...
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1answer
56 views

If $f$ is analytic in some punctured disk and if $|Re f(z)|$ is bounded in a sub disk then it has a removable singularity

Suppose $f$ is analytic in in $D_r(z_0)$ \ {$z_0$}. By Riemman's theorem we know that if $f$ is bounded in some punctured sub disk centered at $z_0$ then $f$ has a removable singularity at $z_0$. Now ...
3
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1answer
66 views

If $f$ has a zero of order m then $f'/f$ has a simple pole

Let $f$ be analytic in $D_r(z_0)$ and has a zero of order $m$ at $z_0$. I need to show that $f'/f$ has a simple pole at $z_0$. This is part of attempt. Since f has a zero of order $m$ there exists ...
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0answers
29 views

find ALL conformal map with a certain property

I'm working on this problem from a qualifying exam of some years ago. It reads: find all conformal transformations sending the upper half disc of radius 2, $\{z: |z|<2\}$ to the unit disc, with the ...
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1answer
68 views

complex plane questions

Find where the points of the complex plane are if, a) $|\pi - \arg z| < \pi/4$ b) $|\Re z| < 1$ c) $\Im \left(\frac{z+1}{z+i}\right) = 0$ d) $z = z_1 + t(\cos x + i\sin x), 0\leq ...
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0answers
78 views

roots of complex polynomials with real coefficients in conjugate pairs?

I used the Argument Principle and applied Rouche's Theorem to show that a polynomial with real coefficients had 4 zeroes inside the unit disk. I then argued that, since these roots must come in ...
3
votes
1answer
184 views

Geometric Interpretation of Liouville's Theorem?

The only bounded entire functions in $\mathbb{C}$ are constants. Could someone please give me a geometric interpretation of the theorem above? I don't intuitively understand why it's true. Also, ...
4
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2answers
145 views

Complex integral using cauchy residue formula

I want to compute $ \displaystyle \int_{0}^{+\infty} \frac{dx}{x^n-1} $ I've proved that $ \displaystyle \int_{0}^{+\infty} \frac{dx}{x^n+1} = \frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}$ in a ...
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3answers
393 views

Prove that $\arcsin z = \frac{\pi}{2} - \arccos z$

I have $\arccos (z) = -i\ln (z + \sqrt{z^2-1})$ and $\arcsin (z)=-i \ln(iz +\sqrt{1-z^2}).$ Now I must prove, that $\arcsin (z) = \frac{\pi}{2} - \arccos (z)$. I get: $$\arcsin ...
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0answers
31 views

$\wp$ via Jacobi triple product

$$\wp(z;\tau) = -(\log \vartheta_{11}(z;\tau))'' + c $$ $$\vartheta_{11}(z|q) = -2 q^{1/4}\sin(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right)$$ Then ...
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0answers
31 views

Homotopies and simply connectedness

QUESTION: If we are given a space $X=\mathbb{C}$ that is simply connected and two points, call them $a,b \in X$, and two curves $f,g:[0,1]\rightarrow X$ such that $f(0)=g(0)=a$ and $f(1)=g(1)=b$, can ...
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1answer
199 views

Let $\lambda>1$ and show the equation $\lambda - z -e^{-z} = 0$ has exacly one solution in the half plane $\{z:Re(z)>0\}$ [duplicate]

Let $\lambda>1$ and show the equation $\lambda - z -e^{-z} = 0$ has exacly one solution in the half plane $\{z:Re(z)>0\}$. Show that this solution must be real. What happens to the solution as ...
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2answers
413 views

Integral via complex analysis. Integral via hypercomplex analysis

If I remember rightly there are some integrals of real functions which are easier to compute by using complex analysis. Is this because of properties of the particular function or because of a lack ...
3
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3answers
112 views

Theres a small detail in this proof on why $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^2} = \frac{\pi^2}{6}$ that I cant figure out

http://www.maa.org/sites/default/files/pdf/upload_library/2/Kalman-2013.pdf Here is a link to the article I have been reading. Its really interesting and easy to follow. What bothers me is a result ...
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2answers
75 views

Prove or disprove the existence of polynomials $p$ and $q$ for which $pe^p+qe^q=1$.

Prove or disprove that there exist non-constant polynomials $p$ and $q$ such that $p(z)e^{p(z)}+q(z)e^{q(z)}=1$ for all $z\in \mathbb{C}$. This question was first asked here Prove or disprove the ...
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1answer
32 views

If $f$ is an entire dunction such that $|f(z)|\le k|z| , \forall z \in \mathbb C$ ; $f(1)=i$ ; then how to find $f(i)$?

If $f$ is an entire dunction such that $|f(z)|\le k|z| , \forall z \in \mathbb C$ ; $f(1)=i$ ; then how to find $f(i)$ ? I can do it if I know whether $\dfrac {f(z)}z$ is constant or not for $z \in ...
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1answer
35 views

Bounding a $C^k$ function on the unit disk

In my reading there is the following "simple" claim: If $u \in C^k(\overline{\mathbb{D}})$ and if all derivatives of $u$ up to order $k$ vanish on $\partial \mathbb{D}$, then for some $C>0$ it ...
5
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2answers
156 views

Dogbone countor integral (evalutate $\int_0^1 \frac{x^n}{x^a(1-x)^{1-a}}dx$)

I'm confronted with the following problem which I really don't seem to find a way to solve properly: Let $n\in \mathbb{Z}$ be fixed. Determine for what values of the parameter $a\in\mathbb{C}$ the ...
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1answer
29 views

BigOh Complexity: $\frac{x^{3} + 2x}{2x + 1}$ is $O(x^2)$?

Show $\frac{x^{3} + 2x}{2x + 1}$ is $O(x^2)$ Can I do it like this? Since exponent rules/laws allow this: $\frac{x^{3} + 2x}{2x + 1}$ $=$ $\frac{1}{2}x^{2} + 2x$ Must show a constant c>0 and k ...
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2answers
67 views

Cauchy's Integral Formula?

Say you have a function with a single discontinuity (a simple pole). Now say you have want to compute the line integral of a simple closed curve around the discontinuity. You can deform the curve into ...
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2answers
50 views

complex differentiation, alternative way?

$\partial/\partial\bar{z}$ is defined as $1/2[\partial/\partial x+i\partial/\partial y]$. So lets say you have a function $f(z,\bar{z})$ in order to find $\partial f/\partial \bar{z}$ I have to write ...
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1answer
21 views

Inequality extension to the boundary of domain

Let $f$ be holomorphic on the unit disk $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$. Then I know that the function $|f|^3$ is subharmonic on $\mathbb{D}$. So for every $r<1$ I have by ...
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1answer
22 views

is $(5-2z-2/z)^{1/2}$ regular in this sector?

Is $f(z)=(5-2z-2/z)^{1/2}$ regular in the sector A? $$A=\{z\in \mathbb{C};-\pi/2\leq \arg z\leq \pi/2; z\not = 0; z\not = \infty \}$$ Is $f(z)$ regular in $B$? Where $B$ is $A$ with extra restriction ...
1
vote
1answer
46 views

If $f(z_0)=0$ then can we say the $\lim_{z\rightarrow z_0}|1/f(z)| \rightarrow \infty$?

Suppose f is analytic on $D_r(z_0)$. If $f(z_0)=0$ then can we say the $\lim_{z\rightarrow z_0}|1/f(z)| \rightarrow \infty $? I suppose this should be correct but I am not 100% confident. Any help ...
2
votes
1answer
51 views

$\int_0^1 g(x) \ dx = \int_{|w|=1} g(w)\log(w) \ dw$

$$\int_0^1 g(x) \ dx = \int_{|w|=1} g(w)\log(w) \ dw$$, where $g(x)$ is a holomorphic function on the closed disc centered at the origin of radius 1. A hint suggests to use $\log w = \log |w| + i ...