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
34 views

Classifying the singularities of the a function

Classify all the singularities of the following functions $$ f(z) = \frac {z^2 - 1} {(z^3 + 1)(z-2)^3} $$ $$ g(z) = \frac {e^\frac{1}{z}} {\sin z} $$ I set the denominators equal to zero and solved ...
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1answer
26 views

$g: (U \times U - D) \to \mathbb{R}$ is continuous, $D$ diagonal? [on hold]

Do we have necessarily have that$$g: (U \times U - D) \to \mathbb{R},$$is continuous, where $D$ is the diagonal? Idea. Perhaps we want to apply the maximum-minimum principle to $G(z, z_0)$?
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14 views

Find $\frac{d}{dt}[\bar{f(\gamma(t))}]$ in the context of of finding $\frac{d}{dt}[|f(\gamma(t)|^2]$

I am trying to prove this exact problem, but more rigorously and without referencing analyticity: Ahlfors complex integration. I think the way to proceed is to try and find ...
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1answer
28 views

If $f$ has a pole of order $m$ at $z_0$ find the order of the pole of $g(z) = \frac{f'(z)}{f(z)}$ at $z_0$.

If $f$ has a pole of order $m$ at $z_0$ find the order of the pole of $g(z) = \frac{f'(z)}{f(z)}$ at $z_0$. What is the coefficient of $(z-z_o)^{-1}$ in the Laurent expansion for $g(z)$. M Since ...
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0answers
34 views

$f$ has pole of order $m$ and $g$ has a pole of order $n$ at $z_{0}$, show $f+g$ has isolated singular point there

I am faced with the following problem: Suppose $f(z)$ and $g(z)$ have poles of order $m$ and $n$ respectively, at a point $z_{0} \in \mathbb{C}$ with $m \neq n.$ Show that $z_{0}$ is an isolated ...
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1answer
25 views

Consider the complex-valued function $\frac{1}{(z^2+1)(z^2+z+1)(z^2-z+1)}$

I have found all the singularities of this function. They are $\frac{\sqrt 3}{2} + \frac{1}{2}i$, $i$, $-\frac{\sqrt 3}{2} + \frac{1}{2}i$, $-\frac{\sqrt 3}{2} - \frac{1}{2}i$, $-i$, $\frac{\sqrt ...
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1answer
41 views

Show the Cauchy-Riemann equations hold but f is not differentiable

Let $$f(z)={x^{4/3} y^{5/3}+i\,x^{5/3}y^{4/3}\over x^2+y^2}\text{ if }z\neq0 \text{, and }f(0)=0$$ Show that the Cauchy-Riemann equations hold at $z=0$ but $f$ is not differentiable at $z=0$ ...
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31 views

Let $P_1,P_2, \ldots,P_n$ be $n$ arbitrary points of the plane.

Let $P_1,P_2, \ldots,P_n$ be $n$ arbitrary points of the plane. If a variable point $P$ is confined to a closed bounded set $E$, show that the product $$\prod_{k=1}^n \overline{PP_k}$$ attains its ...
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0answers
19 views

Complex integral evaluation; I get the right answer, but one of my steps is a little fishy

The integral is $\int_{\gamma}\frac{1}{z^{2}-1}dz$ along the path $\gamma(t)=2e^{ti},\;t\in[0,2\pi]$ Which I attempt to do by parts: \begin{equation*} \int_{\gamma}\frac{1}{z^{2}-1}dz= ...
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0answers
52 views

Conformal map from the union of two disks onto half-plane

Let $U=D_2(-1)\cup D_2(1)$. Find a conformal equivalence from $U$ onto $\mathbb{H}$. We tried many things, like inversion thru one of the circles, and Möbius transformations, but none of that stuff ...
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1answer
30 views

Let $f: \mathbb D \to \mathbb D $ be holomorphic.Suppose for that $z_\circ\in \mathbb D$ we have $f(z_\circ)=z_\circ$ and $f'(z_\circ)=1$.

Let $f: \mathbb D \to \mathbb D $ be holomorphic.Suppose for that $z_\circ\in \mathbb D$ we have $f(z_\circ)=z_\circ$ and $f'(z_\circ)=1$. What can you say about $f$? My intuition says $f$ should be ...
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1answer
34 views

Integrate $\int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x$ [closed]

I'm having a trouble with this integral expression: $$\int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x$$ I want to solve to using residue but it seems hard.
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1answer
40 views

Schwarz's Lemma application

Need help with this problem. Let $f$ be an entire function such that $|f'(z)| \leq |z|$ for all $z$. Show that $f(z) = A+Bz^2$, with $|B| \leq \frac{1}{2}$. My attempt: What I think is the way ...
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0answers
18 views

Without using the Residue Theorem, evaluate the integral

Evaluate $$\int _C\frac{ e^{iz}\, dz}{(z+2i)^7}$$ where C is given by the square with vertices $(\pm3,\ 3)$, $(\pm3, -3)$ I know that there is one singularity at $z=-2i$, which lies ...
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0answers
15 views

Branch dependence on Laurent Series?

Consider the function: $$f(z)=\ln(1-z)$$ in the principal branch (i.e. with $arg(1-z)\in [-\pi, \pi]$) the Laurent series of this function is: $$\ln(1-z)=-z-\frac{1}{2}z^2-\frac{1}{3}z^3...$$ My ...
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2answers
39 views

Prove a doubly periodic entire analytic function in complex plane is a constant [duplicate]

I got stuck on this problem. So I really appreciate if anyone can give me some hint to move on. Thanks a lot. Prove that an entire analytic function $f:\mathbb{C} \rightarrow \mathbb{C}$ is a ...
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2answers
53 views

Complex Analysis: How isolated singular points behave

I am working on the following question: Suppose $z_0 \in \mathbb{C}$ is an isolated singular point of the function f of a given type (removable, pole of order N, essential). Show that $z_0$ is an ...
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1answer
29 views

Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$.

Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$. I'm thinking about a contradiction proof. Assuming ...
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40 views

Suppose $\triangle ABC$ is an equilateral triangle inscribed in the unit circle C(0,1).

Suppose $\triangle ABC$ is an equilateral triangle inscribed in the unit circle C(0,1). Find the maximum value of $$\overline{PA}\cdot\overline{PB}\cdot\overline{PC}$$ where $P$ is a variable point in ...
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1answer
62 views

Show that Riemann Surface is connected?

I was reading Artin's Alegbra when this question came into my mind. Consider $f(t,x)=x^{2}-t$ , The locus X of zeros in $\mathbb C^{2}$ of a polynomial is called Riemann surface of f. I understood ...
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2answers
41 views

Show that the function $\phi(z)=\sum_{j=1}^n |f_j(z)|^2 $ has no local max

Suppose $f_j(z) \in H(\Omega)(j=1,2, \ldots,n) $. Show that the function $$\phi(z)=\sum_{j=1}^n |f_j(z)|^2 $$ has no local maximum in the region $\Omega$ unless all the functions $f_j(z)(j=1,2, ...
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0answers
24 views

bIf $f$ is differentiable at $a$ then $f$ is differentiable on $D_{\delta}(a)$ for some $\delta$

So I have that $f:D_r(\alpha)\rightarrow \mathbb{C}$ is differentiable at $a$. I would like to know if there is anything I can say about differentiability or even continuity on some delta ball at ...
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Does there exist a complex function which is differentiable at one point and nowhere else continuous?

Let $f\colon\mathbb{C}\to\mathbb{C}$. We know that if $f^{\prime}(a)$ exists for some $a\in\mathbb{C}$ then $f$ is continuous at $a$. This is because, from the definition of the derivative, ...
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12 views

What is the definition domain of the function $(z^2-4)^i$, that is defined by the analytic branch of $Log(z)$, which is $L_{\frac{\pi}{2}}$.

What is the definition domain of the function $(z^2-4)^i$, that is defined by the analytic branch of $Log(z)$, which is $L_{\frac{\pi}{2}}$. Note: the way I understood it, $L_{\frac{\pi}{2}}$. means ...
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1answer
37 views

the closure of the set $\{ e^{in\theta}:n\;\text{non-negative integer numbers} \}$

May I ask a question about the closure of the set $\{ e^{in\theta}:n\;\text{non-negative integer numbers} \}$, where $\theta\in\mathbb R$. Many thanks.
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1answer
26 views

Prove the integral is always imaginary

Show that if f is analytic on D and γ is a closed curve in the region then the integral $$\int \overline{f(z)}f'(z)$$ is purely imaginary. I think this problem would use some extension of cauchy ...
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0answers
52 views

Sum of Gamma Function Residues

I was exploring Cauchy's residue theorem with the gamma function and came across an interesting identity. Consider $$\int_{C_R} \Gamma(z) \, dz $$ Over the complex plane where $C_R$ is the curve ...
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2answers
26 views

Conjugate Mobius Transformations Are Invariant

This is for homework and I was hoping to get some help with clarifying some of the concepts. The problem is as follows: Let $f,g$ be conjugate Mobius transformations, say $g=h^{-1} \circ f \circ h$ ...
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48 views

Evaluate the integral without using the Residue Theorem

Without using the Residue Theorem, evaluate $$\int _C\frac{ z\, \cos(z-1) \, dz}{z^4+z^2-20} $$ where $C$ is given by $x^2 + y^2 = 3$. Assume that C has positive orientation. I know that $C$ is a ...
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2answers
15 views

Calculating Taylor series of complex function

I'm going through a past exam paper and found a question I can't do. The question is to write down the Taylor expansion of $\frac{z^2}{z-2}, z \in C$ \ {2}, on the disc $|z| < 2$ I've been ...
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1answer
43 views

Residue theorem integral calculation

Use the residue theorem to calculate $$\int_{0}^{2\pi} \frac {27} {(5+4\sin\theta)^2} d\theta $$ I know $$ \operatorname{Res}_{z_0} f = \frac 1 {2\pi i} \int_\gamma d\theta f(\theta) $$ My question ...
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0answers
34 views

$\frac{\omega}{2\pi}\int_0^{\frac{2\pi}{\omega}}\frac{\sin^2\theta \cos^2\omega t}{(1+\beta\cos\theta \sin\omega t)^5} dt$

I'm going to write out the whole problem as it is given to me (bad grammar and all) even though some of the info may be irrelevant to finding a solution. A charge $e$ moving along a straight line ...
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1answer
16 views

Fourier transform of $\cos(x) f(2x+3) $

So when trying to compute the Fourier transform I believe I can use the convolution theorem to evaluate this as a whole. $$ \widehat{g*h}(x) = 2\pi \hat{g}(k) \cdot \hat{h}(k) $$ If it let $ g(x) = ...
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1answer
38 views

Complex supremum function is strictly monotone

I'm having great troubles to solve the following exercise: Let $f$ be a holomorphic function on the unit disc. For $0\leq r < 1$ is $$M(r):=\sup\limits_{|z|=r}|f(z)|$$ Show that ...
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3answers
47 views

How to write the following polynomial in $(1-\frac{x}{a}) (1-\frac{x}{b}) (1-\frac{x}{c}) (1-\frac{x}{d})$?

I was given the following problem: Write the polynomial $f(x) = \frac{1}{24} \displaystyle \prod_{i \mathop = 1}^4 (x-i)$ in the form $(1-\frac{x}{a}) (1-\frac{x}{b}) (1-\frac{x}{c}) (1-\frac{x}{d})$ ...
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1answer
30 views

Cauchy integral formula with singularities

I am stuck on this question. $$\int \frac{e^{sin z^{2}}}{(z^{2}+1)(z-2i)^{3}}dz $$ along the path γ where γ is a circle centered at the origin of radius different from 1 or 2. I initially thought of ...
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3answers
62 views

How do I compute the following complex number? [on hold]

This was the problem I was given: Compute the complex number for $\frac{(18-7i)}{(12-5i)}$. I was told to write this in the form of $a+bi$. So please give me a hint of how to do this. :)
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1answer
53 views

Knowing the representation of $\frac{π^2}{\sin^2(πz)}$, deduce the Taylor series of $\frac{z^2}{\sin^2{z}}$

The question Consider the representation: $$ \frac{π^2}{\sin^2(πz)} = \sum_{n∈ℤ}\frac{1}{(z+n)^2} \tag{0}$$ valid for all $z ∈ ℂ \setminus ℤ$. Deduce that the Taylor series of $z^2/\sin^2(z)$ for ...
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13 views

Determine the singular part of $f$ at these poles.

Show that $f(z)=\tan(z)$ is analytic in $\mathbb{C}$ excpet for simple poles at $z=(n+\frac{1}{2})\pi$ for $n \in \mathbb{Z}$. Determine the singular part of $f$ at these poles. My thought is to ...
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0answers
18 views

Having problem solving this integral

In a book I'm reading, the author says that $$\int_0^t dt'e^{i(w-w_0)(t'-t)} \rightarrow \pi \delta(w-w_0) - i\textrm{P} \frac{1}{w-w_0}$$ where P indicates that the Cauchy principal part is to be ...
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1answer
58 views

Suppose $f$ is analytic and $f(a) = f(b) = 0$. Show that $|f(z)| ≤ |{z − a \over 1 − z\bar{a}}| · |{z − b \over 1 − z\bar{b}}|$.

Suppose $f$ is analytic from $D(0, 1)$ to $D(0, 1)$ and $f(a) = f(b) = 0$ for two different numbers $a, b$ in $D(0, 1)$. Show that $\left\vert f(z) \right\vert ≤ \left\vert{z − a \over 1 − z\bar{a} ...
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1answer
32 views

Proving that a function is real-analytic

I try to solve the following exercise: Let $f:\mathbb{R}\to\mathbb{R}$ with $f(x):=\frac{1}{1+x^4}$. Prove that $f(x)$ is real analytic and compute the radius of convergence of it's Taylor series at ...
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0answers
23 views

Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to $f^{ (k)}$ uniformly on any compact subset of $G$.

Suppose $f_n$ is analytic in some region $G$ and suppose $f_n$ converges to $f$ uniformly on any compact subset of $G$. Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to ...
3
votes
1answer
79 views

Why does $z^{-1}$ not have an anti derivative?

I had been given the question as shown in the following image, with the answer also given. Surely however the anti derivative of $z^{-1}$ would be $\log(z)$ ? I have seen a similar question asked, ...
0
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2answers
39 views

Fiding imaginary part of a complex number [closed]

What is the imaginary part of $i^i$ ? I've tried multiple approaches, including using log. I can't seem to understand how to work with complex numbers as logarithmic functions. Also, it would ...
2
votes
0answers
21 views

how to prove $\left( {\frac{1}{\sin z}} \right)^{2}=\sum_{-\infty}^{+\infty}\frac{1}{\left( {z-\pi n} \right)^2}$ [duplicate]

How can I show that $$\left( {\frac{1}{\sin z}} \right)^{2}=\sum_{-\infty}^{+\infty}\frac{1}{\left( {z-\pi n} \right)^2}.$$ I don't know how to do this. Any help will be appreciated.
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0answers
16 views

Finding number of zeros of analytic function. [closed]

Find number of zeros of $e^z - 2z = 0$ around the curve $\mid z \mid $ = 3, where z is a complex number of the form $z = a + ib$.
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0answers
31 views

Holomorphic, entire functions and the Cauchy-Riemann equations

Where is $f\colon\mathbb C\to\mathbb C,~z\mapsto |z|^2$ $\mathbb C$-differentiable? Is there a restriction of $f$ that is holomorphic? Is there an entire function $f$ with ...
0
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0answers
12 views

Equality involving holomorphic function and its series coefficients [duplicate]

Function $f(z)=a_0 + a_1z +a_2z^2+...$ convergences on $\left\{z:|z|<R\right\}$. Prove that for any $0<r<R$ $$\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^{2}dt= ...
2
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1answer
57 views

Regarding the weight of the given modular form

I am following this course. A question regarding the modular form of weight two that is constructed in lecture 33 emerged. Let me briefly tell you what's going on there. Let $\mathbb{H}$ denote the ...