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
28 views

Integral with denominator raised to n-th power, residues

I don't know how to calculate this integral: $$\int_{-\infty}^{\infty} \frac{d x}{(1+x^2)^{n+1}}$$ If we denote by $\Gamma$ a curve = semicircle centered at $0$ with radius $R$ + segment $[\ R, R]$, ...
-1
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0answers
15 views

Complex Analysis-conformal Mapping [closed]

I studied complex analysis roudin book and i have a problem with conformal mapping section, in this section there is a theorem that i'll attache in follow, i couldn't find out how it was proved :i ...
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1answer
33 views

Chain of inequlities in Complex variables

I am having difficulty understanding the following inequalities which is part of a solution to a problem: Suppose $$|z-1|< 1/2$$ $$|z+1|< 5/2$$ and $$|z|> 1/2$$ Also Suppose ...
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0answers
20 views

Complex analysis proof of regions

how do i prove that the products of regions in C is also a region? Given that the intersection of these regions is not empty. I just know that a region is a non-empty open connected subset of C...
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0answers
57 views

Complete Proof of Lindelof Theorem [closed]

enter link description here![enter image description here][2]would you please solve this question for me: why (2) and (3) imply that (5)
4
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0answers
36 views

Obtaining a single-valued branch of $\ln \left( \frac{z-a}{z-b} \right)$ with a branch cut

It is rather easy to see that the function $$f(z) = \ln \left( \frac{z-a}{z-b} \right)$$ has branch points at $z=a$ and $z=b$, My question is why considering a branch cut "connecting" $a$ and $b$ ...
2
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1answer
42 views

Inverting complex cosine

I have been working out problem 3a in chapter 1 section 3 in Basic Complex Analysis by Marsden. He asks to solve $$ \cos z=\frac{3}{4}+\frac{i}{4} $$ After putting cosine in its exponential form and ...
7
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1answer
269 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
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1answer
25 views

Prove that space of Laurent series is not Hilbert

Let $z_0 \in \mathbb{C}, s>0,T(z_0,s):=\{z\in\mathbb{C}:|z-z_0|=s\}$ and let $V=V(z_0,s)$ be a vector space over field $\mathbb{C}$ of all Laurent series that are uniformly and absolutely ...
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1answer
22 views

How do I compute this Milnor number

I need to compute $\mu (x^5+y^5)=5$ on the point $p=(0,0)\in\mathbb{C}^2$. By definition, for $f\in\mathbb{C}[x,y]$, I have $$ \mu(f)=\dim\dfrac{\mathcal{O}_{(0,0)}}{<\dfrac{\partial f}{\partial ...
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1answer
49 views

The inverse of a bijective holomorphic

Let $U,V$ are open sets in $\mathbf{C}$, if $f:U\to V$ is holomorphic and bijective, then the inverse of $f$ $f^{-1}:V\to U$ is also holomorphic. How can I show that $f^{\prime}(z)\neq 0$ for all ...
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1answer
47 views

Let $w=1+3i$. Investigate whether $|iw+w|=|iw|+|w|$. [closed]

Let $w=1+3i$. Investigate whether $|iw+w|=|iw|+|w|$. This is a question on complex numbers.
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1answer
536 views

inverse Laplace transform of $e^\sqrt{as}$

I am trying to find the inverse Laplace transform of $e^\sqrt{as}$ for $a>0$. So we need to solve $\oint_B dz \: e^\sqrt{az} e^{z t}$ (Bromwich contour), but not sure how to start. How do we even ...
2
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1answer
45 views

Is a connected Reinhardt Domain which containg $0$ necessarely a polydisc?

I'm studying several complex variables basics. Roughly speaking: call $D\subseteq\Bbb C^n$ the set of points in which a given power series $$ \sum_{\alpha\in\Bbb N^n}a_{\alpha}(z-z_0)^{\alpha} $$ ...
3
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0answers
49 views

Real analytic function with radius of convergence 1 at non-negative integers

So, as the title states, the problem I was confronted with was to find a real-valued everywhere analytic function $$f:\mathbb{R}\to \mathbb{R}$$ s.t. at every non-negative integer, k ...
1
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1answer
36 views

Lagrange's identity in the complex form

I am trying to show Lagrange's identity in the complex form; that is, $$ \Bigl\lvert\sum_{i = 1}^na_ib_i\Bigr\rvert^2 = \sum_{i = 1}^n\lvert a_i\rvert^2\sum_{i = 1}^n\lvert b_i\rvert^2 - \sum_{1\leq ...
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1answer
37 views

Injective holomorphic function is conformal(i.e. nonzero derivative)

STATEMENT: If $f:U\rightarrow V$, where $U,V$ are open subsets of $\mathbb{C}$, is holomorphic and injective, then $f'(z)\neq 0$ for all $z\in U$. Proof: We argue by contradiction, and suppose that ...
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2answers
130 views

Are there explicit forms of these integrals

Let $a>0$, and $z_0=r_0e^{i\theta_0}$, where $0<r_0<a$, $0<\theta<\gamma<\frac{\pi}{2}$, do we have a closed form of each of the following integrals $$ ...
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2answers
33 views

Flipping sign of $i$s

Why do we flip the signs of all $i$ s in a complex number when we want to take the conjugate of it? I mean, conjugating means making $x + iy$ into $x - iy$, but given a number of the form: $$\frac ...
2
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1answer
14 views

Determining uniform convergence of complex power series

I'm working on some practice problems for my complex analysis course, and I'm having trouble with uniform convergence. The question asks whether the following series converges uniformly for ...
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4answers
35 views

Proving arg(z/w)=arg(z)-arg(w)

I need to prove that $$arg\left(\frac{z}{w}\right)=arg(z)-arg(w)$$ However, I am a little stuck as to how to go about this. I know the proof for $arg(zw)=arg(z)+arg(w)$ happens by letting ...
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0answers
51 views

Complex Analysis (Complex Mapping) stuck on professor's method of simplification in math notes

I'm having an exam this university semester and need some help with my math notes. I've come accross some problems with the section "Complex Mapping." Link to Image of my Notes: Click Me (see first ...
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0answers
27 views

Ordering in the complex plane [closed]

Explain why there is no ordering ≺ on the complex numbers that satisfies the usual properties of the relation < on the real numbers I just need help starting this problem. I'm not sure what ...
2
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0answers
21 views

Claim about the z-transform of a discrete function

Claim: $\lim_{k\to\infty} x[k]$ exist and if finite is $X(z)$ the Z-transform of $x[k]$ has no pole in $|z|>1$ and at most 1 pole at $z = 1$ Attempt: \begin{align*} X(z) &= ...
3
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0answers
42 views

Finding Laurent Series of a function

I've been assigned to write a computer program which then calculates the Laurent series of a function. Of course I'm familiar with the concept, but I've always calculated the Laurent series in an ad ...
1
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1answer
17 views

Harmonics conditions for a plucked string

Given a plucked string which is taken on the interval $[0,\pi]$, and it satisfies the wave equation with $c=1$. The initial position of the string is: $\ f(x) = \frac{xh}{p}$ ($0\leq x\leq p$), and $\ ...
6
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1answer
128 views

Can $f(g(x))$ be a polynomial?

Let $f(x)$ and $g(x)$ be nonpolynomial real-entire functions. Is it possible that $f(g(x))$ is equal to a polynomial ? edit Some comments : I was thinking about iterations. So for instance ...
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0answers
133 views

How many complex functions reduce to a given x-y function?

A 2D or x-y coordinate function has a complex analog, which is formed by replacing x with with the complex variable z. That function can then be separated into real and imaginary parts. Graphing the ...
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0answers
34 views

problem in complex analysis about the modulus [closed]

Let $z$ and $w$ be any two complex numbers such that $|z|\leq 1, |w| \leq 1$ and $\bar w z$ $\not= 1$. Prove that $$\frac{|(w-z)|}{|(1-\bar w z)|} \leq 1,$$ with equality if and only if $|z| = 1$ or ...
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0answers
28 views

Does uniform convergence on $D\subseteq \mathbb{C}$ imply uniform convergence on all subsets of $D$?

Let $f_n:D\rightarrow \mathbb{C}~\forall n\in \mathbb{N}$. If $(f_n)_{n\in\mathbb{N}}$ converges uniformly on $D\subseteq \mathbb{C}$ against $f:D\rightarrow \mathbb{C}$, does $(f_n)_{n\in\mathbb{N}}$ ...
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1answer
62 views

Find roots of $ω^x+(ω^x)^2+1=x$ [closed]

We have to solve this equation at complex numbers group $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right
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0answers
41 views

Solve equation $ω^x+ω^{2x}+1=x$ [closed]

We have that to solve $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right.
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2answers
153 views

Evaluate $\oint_C |z|^2 dz$ around the square with vertices at $(0,0), (1,0), (1,1), (0,1)$

I don't think I quite understand how to go about this. My solution so far: $\oint_C |z|^2 dz = \oint_C (x^2 + y^2)dz = \oint_C (x^2 + y^2) d(x+iy) = \oint_C x^2 + y^2 dx + i\oint_Cx^2+y^2dy$. ...
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0answers
14 views

Find the equation of an ellipse in a complex plane.

Find an equation for the ellipse that passes through $3+7i$ if its foci at $i$ and $-1$. This is what i have so far. I know that the equation of an ellipse is $|z-p| + |z-q| = c$. so $|z-p| + |z-q| = ...
2
votes
3answers
74 views

Can a non-periodic function have a Fourier series?

Consider two periodic functions. Assume their sum is not periodic. The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents ...
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1answer
32 views

Composition of harmonic and holomorphic function

Simmiliar to this question my problem is as following: If $u$ is harmonic, and $f$ is holomorphic function, are $u \circ f$ and $f \circ u$ harmonic? I tried to do it like this: $$\Delta (u \circ f)= ...
269
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25answers
24k views

Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I ...
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1answer
76 views

Prove $\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$

I have to prove that for $f(x,y)=e^{-xy^2}\sin(x)$ and $\forall b>0$ we have $$\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$$ I've tried to ...
0
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1answer
46 views

Limit involving branch point of a complex function

I am having trouble with the following problem : If we restrict ourselves to that branch of $f(z)= \sqrt{z^2+3}$ for which $f(0)=\sqrt 3$ , prove that $$\lim_ {z\to 1}\frac{\sqrt{z^2+3}-2}{z-1} = ...
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1answer
30 views

Complex Analysis (Limits)

Let $a, b$ be complex numbers. Use the definition of a limit directly (not just the properties of limits) to prove that $$ \lim_{z \to z_0}az + b = az_0 + b. $$ Sorry for the wrong notation, I do ...
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1answer
215 views

Problems in interpreting an integral that should be solved with residue method

Usually, when I solve an integral using residue method, I find real functions as integrands. I am not able to provide an interpretation for the following complex integral $$ \int_{-\infty}^{\infty} ...
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1answer
45 views

Calculation of an integral via residue.

$$\int_{-\infty}^{\infty}{{\rm d}x \over 1 + x^{2n}}$$ How to calculate this integral? I guess I need to use residue. But I looked at its solution. But it seems too complicated to me. Thus, I asked ...
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1answer
19 views

Schwarz Reflection Principle vs. Analytic Continuation

Analytic continuations are unique on simply connected domains: $$F,F':\Omega\to\mathbb{C}:\quad F\restriction=F'\restriction\implies F=F'$$ Schwarz reflection principle offers analytic continuations ...
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2answers
38 views

complex numbers equality question

Let $a$ be given a complex number. Show that $$\left|\frac{z-a}{1-a^*z}\right|=1$$ for $z$ with $|z|=1$ and $a^*z\neq 1$. If $|z|=1$, that means $z$ can be equal to $i$, $-i$, $1$ or $-1$ right? ...
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1answer
33 views

How many analytic functions are there on a given set

Consider the set $S=\{0\} \cup \bigl\{\frac{1}{4n+7}:n=1,2,...\bigr\}.$ Then the number of analytic functions which vanishes only on $S$ is (a) infinity (b) 0 (c) 1 (d) 2 I think, the answer is ...
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1answer
65 views

Mean Value Theorem for $f: \mathbb{R} \rightarrow \mathbb{C}$

Let $f: \mathbb{R} \rightarrow \mathbb{C}$ be a continuous and differentiable function on $[a, b]$. Then does there exists a $c \in (a,b)$ such that $$\frac{|f(b) -f(a)|}{b - a} \leq |f'(c)|?$$
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1answer
18 views

Express $\sin(z)$ and $\cos(z)$ in Rectangular Form

"Express $\sin(z)$ and $\cos(z)$ in rectangular form." For $z \in \mathbb{C}$ (complex numbers), we have defined \begin{equation} \sin (z)=\frac{e^{iz}-e^{-iz}}{2i} \end{equation} and ...
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2answers
16 views

Defining domain in complex plane

I am asked to define the domain for the following given that $z=x+iy$: $a) \quad f(z) = \dfrac 1 {z^2 + 1}$ $b) \quad f(z) = \dfrac 1 {1 - |z|^2}$ How would this be different from a normal domain ...
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2answers
35 views

Plotting on a complex plane

I'm very confused how you would plot the relationship $|z-4| \leq |z|$. I tried to change it in form which could become $-|z|\leq|z-4|\leq|z|$ and I guess the same can be done for z-4. But I don't ...
0
votes
1answer
16 views

Computing Principal Logarithm on Different Intervals

Compute the principal logarithm of a complex number $z=\sqrt{3}+i$ using $\mathrm{Arg}(z) \in [0,2\pi)$ and $\mathrm{Arg}(z) \in (-\pi,\pi]$. Wikipedia shows how the answer can be different for the ...