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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32 views

A homeomorphism preserves irreducible components?

Let $f$ a homeomorphism between two Hausdorff topological spaces $X$ and $Y$. Assume that $X$ and $Y$ are reduced analytic spaces. Is true that $f$ takes an irreducible component of $X$ in an ...
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1answer
19 views

Under what conditions is the argument integral zero?

In my studies in complex analysis, I came across the usage of the argument principle to find the number of zeroes and poles of a meromorphic function within a given region. While trying to solve some ...
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1answer
33 views

Complex limit proving [duplicate]

For complex number $z$, how to prove the limit exits by definition $$\lim_{z\to 0} \frac{\sin z}{z} =1$$ By def I've tried, but I got a difference always greater than 1..
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2answers
5k views

Demonstration that 0 = 1

I have been proposed this enigma, but can't solve it. So here it is: $$\begin{align} e^{2 \pi i n} &= 1 \quad \forall n \in \mathbb{N} && (\times e) \tag{0} \\ e^{2 \pi i n + 1} &= e ...
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1answer
48 views

Maximum area of Triangle $\Delta PAB$

In Complex plane $A$ and $B$ are two points given by $z_1=5-2i$ and $z_2=1+i$ and if $P(z)$ is any Point such that $$|z-z_1|=2|z-z_2|$$ Find the Maximum area of Triangle $\Delta PAB$. I have done ...
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2answers
43 views

Concluding that a function is not analytic at a point?

I have the function $f$ defined on $\mathbb{R}$ by: $f(x)= \begin{cases} 0 & \text{if $x \le 0$} \\ e^{-1/x^2} & \text{if $x > 0$} \end{cases}$ I've inductively proved that $f$ is ...
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1answer
79 views

Why is Euler's formula valid for all $n$ but not De Moivre's formula?

The Wikipedia page on De Moivre's Formula says the formula doesn't hold for non-integer $n$, since non-integer powers of a complex number can have multiple values. It then goes on to say that this ...
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1answer
81 views

Useful device in complex analysis (Perron's formula)

I've come across the following useful device from complex analysis: $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}{\frac{y^z}{z}}{dz} = \left\{\begin{array}{lll} 0 & \text{if} & 0<y<1 ...
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1answer
28 views

Suppose $f$ is holomorphic in $D(0,r)$ and $|f(z)| \leq |z|, \forall z \in D(0,r).$ Is it true that $f^{(n)}(0)=0, \forall n =1,2,… \ ?$

Suppose $f$ is holomorphic in $D(0,r)$ and $|f(z)| \leq |z|, \forall z \in D(0,r).$ Is it true that $f^{(n)}(0)=0, \forall n =1,2,... \ ?$ May I verify if my proof is correct or wrong? Thank you. ...
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1answer
30 views

$1+e^{ix}+e^{i2x}+…+e^{inx}=\frac{(1-e^{i(n+1)x})(1-e^{-ix})}{(1-\cos(x))^2+\sin^2(x)}, n\in \mathbb{N}, x\neq 2k\pi , k\in \mathbb{Z}$

How to show $$1+e^{ix}+e^{i2x}+...+e^{inx}=\frac{(1-e^{i(n+1)x})(1-e^{-ix})}{(1-\cos(x))^2+\sin^2(x)}, n\in \mathbb{N}, x\neq 2k\pi , k\in \mathbb{Z}$$ I have tried to simplify the denominator, ...
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0answers
28 views

Complex Analysis (Analytic Function) [on hold]

Let, $f:\mathbb{C} \to \mathbb{C}$ be an entire function and $g:\mathbb{C} \to \mathbb{C}$ be defined by $g(z)=f(z)-f(z+1) \forall z \in \mathbb{C}$. If $f(\frac{1}{n})=f(\frac{1}{n} +1) \forall n \in ...
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29 views

Guidance for complex numbers/analysis problem needed [duplicate]

I'm looking at this one problem in a book of mine, but I can't even seem to start it. Let $z_1,z_2,...$ be a countable set of distinct complex numbers. If $|z_j-z_k|$ is an integer for every ...
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2answers
58 views

Complex Analysis problem and solution

Suppose $f(z)$ is holomorphic on $D(0,1)$ such that $|f(z)| \leq 1, \forall z \in D(0,1)$ and $f$ has zero of order $3$ at $z=0.$ Prove that $|f^{\prime\prime\prime}(0)| \leq 6$ and determine the ...
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1answer
17 views

How to bound the $L^\infty$ norm of a holomorphic function by its $L^2$ norm on a larger domain?

More speficifically, what I need to prove is that for every positive numbers $r,s$ with $r<s$ there exists a $C_{r,s}$ such that for all holomorphic functions $f(z)$ on a region that contains $\bar ...
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2answers
29 views

$f$ is uniformly continuous $\Leftrightarrow$ $\mathrm{Re}f$ and $\mathrm{Im}f$ are uniformly continuous.

Let $f$ be a complex-valued function. In many complex analysis textbook, I have just found that $f$ is continuous $\Leftrightarrow$ $\mathrm{Re}f$ and $\mathrm{Im}f$ are continuous. I wonder that ...
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0answers
33 views

What is the Riemann surface of the exponential integral?

I have recently encountered a differential equation whose solution has a term \begin{equation} \frac{1}{2}e^{-\frac{1}{2 \varepsilon} e^{i \tau}} \int_{\tau_0}^\tau e^{\frac{1}{2 \varepsilon} e^{i ...
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1answer
36 views

Carleman's conditions

I compared Carleman's condition to Hadamard's radius of convergence for Taylor series. Given that the MGF can be re-expressed as a taylor series (that can be extended to a strip in the complex plane ...
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2answers
76 views

Proof of Taylor's Theorem with Wirtinger Derivatives (Complex coordinates)

Suppose that $f$, defined in $D_1(0)$, is infinitely differentiable. Show that for each $n \in \mathbb{N}$ we have \begin{equation*} f(z,\bar{z}) = \sum\limits_{0 \leq j + k \leq n} ...
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2answers
34 views

Finding the Roots of Cubic (Boas 2.14.25)

I'm working my way through Mathematical Methods in the Physical Sciences and came across the following problem: Use a computer to find the three solutions of the equation $x^3−3x−1=0$. Find away ...
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33 views

weighted Laplacian

My aim is to study a weighted Laplacian defined as $$ L_1=\partial_z \left(\lvert z \rvert^{-2} \bar \partial_z\right), \quad z\in \mathbb{D} $$ Here $\partial_z$ and $\bar \partial_z$ denote the two ...
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0answers
36 views

Where to find current literature, especially dissertations, on complex analysis?

Is there any public website or any other source which classify written master or doctoral thesis classify with respect to their content? Especially, I am going to make some research about complex ...
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3answers
126 views

Proving $\arg(zw)=\arg(z)+\arg(w)$

This is my attempt I know this is incomplete or may even be wrong. Let $θ_1 \in \arg(z)$ and $θ_2 \in \arg(w)$. Then, $θ_1+θ_2 \in \arg(z)+\arg(w)$. Also, $θ_1+θ_2 \in \arg(zw)$. Is this sufficient ...
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2answers
44 views

Are the Cauchy-Riemann equations and the continuity of partials enough for analyticity?

I know that if the partial derivative of real and imaginary parts of a complex function satisfy the cauchy-riemann equations and the partial derivatives are continuous at a point then the function ...
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1answer
34 views

Wirtinger derivatives and conjugate

I haven't found anywhere in the literature (that's available to me, at least) a proper explanation of the following relations for a function $f \in \mathcal{C}(\Omega)$, $\Omega$ domain of ...
7
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0answers
98 views

Another way of expressing $\sum_{k=0}^{n} (-1)^k\frac{H_{k+1}}{n-k+1}$

In this post Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$ I asked for a solution of the non-alternating series. How about the alternating series? Can we find a nice way of ...
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38 views

Understanding branch cuts by manually choosing the branch cuts of a function

Below I will explain what I have done in order to illustrate my confusion with branch cuts of a typical function. If I say something wrong at any point please do not hesitate to correct me! In order ...
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1answer
34 views

Appolonius circle with $\infty$ as one limit point.

For two complex numbers $a,b$ circle of appollonius with limit point $a$ and $b$ are given by $$\left\lvert \frac{z-a}{z-b}\right\rvert=r.$$ I can only see these circles when limit points are finite ...
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1answer
33 views

Why is the set $E$ measurable?

I am having trouble understanding a proof presented in Rudin's Real and Complex Analysis. The theorem states, if $f$ is a complex measurable function on $X$, there is a complex measurable function ...
2
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1answer
46 views

Solve the equation:$ \bar{z}=z^{n-1}$

Solve the following equation: $\bar{z}=z^{n-1}$ Where $\bar{z}$ is the complex conjugate of z, and n is a natural number such that $n\neq 2$. I have tried to write z in rectangular form and polar ...
3
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1answer
64 views

The periods of the Weierstrass function $\wp(z)$

Is it true that the periods $\omega_1$, $\omega_2$ of $\wp(z)$ are $\omega_1 = 4K$ and $\omega_2 = 4iK'$, respectively? Here, $K = K(k)$ is the complete elliptic integral of the first kind, and $K' = ...
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1answer
23 views

Is $\sum c_n z^n$ analytic when $c_n$ is Banach-valued?

I'm trying to define "Analytic function". I want a definition that covers all interesting cases. To be specific, let me explain what exactly I want Here is the definition of analytic function in ...
5
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2answers
72 views

Prove that $|e^{i\theta_1}-e^{i\theta_2}| \leq |\theta_1 - \theta_2|$

I'm trying to prove the inequality $$|e^{i\theta_1}-e^{i\theta_2}| \leq |\theta_1 - \theta_2|$$ I have tried to use Taylor's formula and got this $$|e^{i\theta_1}-e^{i\theta_2}| = |(1+i\theta_1 - ...
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1answer
13 views

Convergence of Taylor series about centre of open disc for analytic function.

I define a function on an open set of the complex plane to be analytic if about any point $z_0$ in that set it can be expanded as a power series in $(z - z_0)$ that converges in some neighbourhood of ...
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1answer
55 views

Problem identifying branch cuts of a square root function

Just when I thought I understood the basics of branch cuts, I started to plot some standard functions to see how they were handled on a computer. I used python 2.7 I plotted the function ...
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1answer
36 views

Why does these complex sequences converge uniformly?

I have one complex series and one sequence. It is used in complex analysis in a part of my book where they are integrated. However, as you know in order to change limit and integration order it has to ...
2
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1answer
38 views

Find loci of the points in complex plane such that $\mathrm{Im}(\frac{z-z_1}{z-z_2})^n=0$

Find loci of the points in complex plane such that $$\mathrm{Im} \left (\frac{z-z_1}{z-z_2}\right )^n=0,$$ where $n\in\mathbb{N}$, $z_1, z_2$ are the given points in $\mathbb{C}$. When $n=1$, it ...
5
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4answers
219 views

Wolfram alpha says that $\int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$

Wolfram alpha says that $$ \int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$$ holds. But it has two different values ($\sqrt{i}$). How should I understand this?
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0answers
45 views

Complex Gaussian Integral - $\int_{-\infty}^\infty e^{-p(t+c)^2}dt = \sqrt{\frac{\pi}{p}}$?

I found some formulas on books, especially the complex gaussian integral formula: $$ \int_{-\infty}^\infty e^{-p(t+c)^2}dt = \sqrt{\frac{\pi}{p}} $$ for $p,c\in\mathbb C$. Then if $p=i=\sqrt{-1}$, the ...
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1answer
14 views

Removable singularity of exp(f(z))

I have already classified the case when $f(z)$ has a non removable singularity then what about the nature of the singularity of $e^{f(z)}$ ?? [ - as in the exercise of Gamelin's book] Now my question ...
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28 views

Homogeneous function in Complex plane and its Periodicities

Consider A function $f:\mathbf{C}^2\rightarrow \mathbf{C}$ defined as $$f_{\alpha, \beta}(z,w)=\frac{\alpha}{z}+\frac{\beta}{w}$$ where $\alpha$ and $\beta$ both are complex number. It is easy to ...
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0answers
30 views

Maximum modulus principle $\sup_{z\in\overline{U}}|f(z)|=\sup_{z\in \partial U}|f(z)|$

Is the following proposition true or false? Let $f:\overline{U}\subset \mathbb{C}\longrightarrow \mathbb{C}$ be a bounded continuous function and holomorphic on $U$ (unbounded connected open) then ...
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3answers
78 views

Every line or circle in $\mathbb{C}$ are solution sets to the equation…

Here is a complex analysis homework problem I can't quite figure out: Prove that every line or circle in $\mathbb{C}$ is the solution set of an equation of the form $a|z|^2+\bar{w}z+w\bar{z}+b=0$, ...
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2answers
91 views

Does the equality $1+2+3+… = -\frac{1}{12}$ lead to a contradiction? [duplicate]

Is $1+2+3+4+5.... = -\frac{1}{12}$ self-contradictory ? I've heared much that $1+2+3+.... = -\frac{1}{12}$, although the fact that this series is diverging. I saw a proof of it by a physicist. In ...
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1answer
20 views

Find the positive integer $n$ such that $p(z)=z^5-30z^2+1$ has exactly $3$ zeros(counting multiplicity) in $\{z \in \mathbb{C}:n<|z|<n+1\} $

Could anyone advise me on how to find the positive integer $n$ such that $p(z)=z^5-30z^2+1$ has exactly $3$ zeros(counting multiplicity) in $\{z \in \mathbb{C}:n<|z|<n+1\} \ ?$ Hints will ...
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1answer
22 views

laurent series for $\frac{z}{(z-1)(z+4)}$

$f(z)=\frac{z}{(z-1)(z+4)}$ and I need the Laurent series for $0<|z-1|<5$ and $5<|z-1|$. I think I have $0<|z-1|<5$ figured by decomposing $f$ into $\frac{1}{5(z-1)}+\frac{4}{5(z+4)}$ ...
3
votes
2answers
45 views

A problem on fixed point

Let $f$ be holomorphic function defined on a domain which contains the closed unit disk $\overline {D(0,1)}.$ Suppose $f$ maps $\overline {D(0,1)}$ into open unit disk $D(0,1).$ Could anyone advise me ...
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0answers
19 views

Injectivity of Theta functions

Let $\vartheta_{00}$ and $\vartheta_{01}$ be Jacobian Theta functions (notations like on wikipedia). $F:=\left\{ \tau \in \mathbb{C}: Im(\tau)>0, \left| Re(\tau)\right|<1, ...
-1
votes
1answer
31 views

laurent series expansion ${z+2\over z-1}$ on two annuli

i am trying to find the laurent series expansion for $$f(z)={z+2\over z-1}$$ on both $0<|z|<1$ and $|z|>1$. For $0<|z|<1$, my thought has been to rewrite ${z+2\over z-1}$ as ${3\over ...
0
votes
1answer
31 views

If $P$ is a polynomial of degree $n>0$, then there exists circle $C$ of radius $R$ such that $\int_{C} \frac{P^{\prime}(z)}{P(z)} dz=2n\pi i $

Let $P$ be a polynomial of degree $n>0.$ Could anyone advise me on how to show there exists $R>0$ such that if $C$ is the circle $|z|=R$ anticlockwise oriented, then $\begin{align}\int_{C} ...
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0answers
42 views

complex limits, how to show they go to 0?

In complex integration my book uses that some limits go to zero as R goes to infinity. However I do not now how to show this, these two limits are: $e^{-\pi(R^2+2iRy-y^2)}$, where y is a real number ...