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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23 views

$\frac{1}{z}-\frac{1}{\sin z}$ at the origin- Classify singularities

I tried for a while to classify the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked A way to do this it's to consider a hint of a colleague : ...
-2
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1answer
20 views

Classify singularities - Hint [duplicate]

I tried for a while to classifiy the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked. Is there someone who is able to help me at this point?
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1answer
12 views

How do the coefficients in the linear combination of cosines impact the number of local minima of the sum?

Consider the following function: $$f(\theta) = r_0 + r_1 \cos(\theta + \phi_1) + r_2 \cos(2\theta + \phi_2)$$ where $\theta$ is an angle between 0 and $2\pi$. For all $0\leq k\leq 2$ we have $r_k\geq ...
-1
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1answer
42 views

$\frac{1}{z}-\frac{1}{\sin z}$ at the origin - Classify singularities [on hold]

I tried for a while to classifiy the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked. Is there someone who is able to help me at this point?
3
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1answer
48 views

Prove that $ζ(4)=π^4/90$ knowing that $\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right)$

The question Knowing that: $$\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right) \tag{1}$$ obtain the Taylor series expansion of $\frac{\sin(πz)}{πz}$ to deduce: $$ \sum_{1 ≤ n_1 < n_2 ...
3
votes
0answers
27 views

Find and classify singular points of $\cot\left(\frac{1}{z}\right)$

I need to find and classify singular points (i.e., decide whether the point is removable, a pole of order $N$, essential, or not an isolated singular point), including infinity, of ...
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0answers
24 views

What is the Laurent series of $\frac{2}{z-1} - z$ in $1<|z|<2$?

What is the Laurent series of $\frac{2}{z-1} - z$ in $1<|z|<2$? I can factor the first term to get $$ f(z) = \frac{2}{z} \sum_{n=0}^\infty \frac{1}{z^n} - z $$ where the series converges ...
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1answer
21 views

Holomorphic function with given real part on unit circle

From de Branges' book Hilbert spaces of Entire Functions (page 2): If $h(\theta)$ is a continuous real-valued $2\pi$-periodic function, define $$g(z) := \frac{1}{2\pi} \int_0^{2\pi} \frac{e^{i ...
1
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1answer
26 views

$f(z)=\frac{1}{z^2-2z+2}$ - Maximum modulus principle

Let the function $f(z)=\frac{1}{z^2-2z+2}$. I have to find $\max_{z \in D(0,1)} |f(z)|$, but I already know that the maxixum would be on $\bar{D}-interior(D)$ by the maximum modulus principle. Is ...
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0answers
32 views

Contour integral of continuous but not holomorphic functions

This question was transferred here following Mathoverflow suggestions. Let us consider two functions $f(z)$ and $g(z)$, both holomorphic on a domain $U$ (a simply connected subset of $\mathbb{C}$). ...
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0answers
20 views

$\int_C \frac{dz}{z+1}$ - Specific question

To compute the integral $\int_C \frac{dz}{z+1}$, where $C=C(0;1)$, could it possible to use the Cauchy theorem or I have to compute normally this integral? I know that the integrand is not define at ...
0
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1answer
14 views

Help in the demonstration of the theorem 1.2 chapter V from Conway's complex analysis book

I'm reading Conway's complex analysis book and I'm stuck in this little detail in the demonstration of this theorem: Why does $|\int _{T_1}g|\le \epsilon$?
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0answers
19 views

Convergence behaviour of Eichler integral

Consiger $g : \mathbb H \to \mathbb C$ a modular form of weight $2-k, k \in \frac{1}{2}\mathbb Z$. Let $z \in \mathbb H$ and consider the following integral: ...
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0answers
18 views

Integration with branch cuts.

This may be a silly question, but when integrating over closed contours in the cut complex plane (a complex plane with a branch cut) do we need to integrate along the branch cut? For instance, if we ...
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0answers
5 views

Proof of decomposition of homotopy into elementary decompositions?

In my Complex Analysis notes, the following lemma is stated without proof: If $G$ is an open connected domain, and $C$ and $C'$ are homotopic in $G$, then the homotopy can be decomposed into a finite ...
6
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3answers
86 views

Calculate $\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$ using residues

I'm supposed to calculate $$\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$$ using residues. The typical procedure on a problem like this would be to integrate a contour going around an upper-half ...
3
votes
2answers
65 views

an analytic function being zero

Let $f$ be an analytic function defined on the unit disc $D=\{z:|z|<1\}$. If $|f(z)|\leq 1-|z|$ for all $z\in D$ then show that $f$ is a zero function on $D$. Please give only hints.
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0answers
11 views

Sum of Roots of Unity With Weighted Exponents

I have the following conjecture that I want to believe has some sort of classical result associated to it, but have yet to find any such evidence. Let $\ell,r\in\mathbb{Z}^+$, and fix ...
0
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0answers
14 views

Uniqueness of a solution to a functional equation

I have two complex-valued functions, $f$ and $g$, that satisfy the following properties. $\overline{x}$ denotes the complex conjugate of $x$ below. $$g(t)\overline{g(t+h)} = f(h) \quad ...
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3answers
60 views

how can I find solvability condition of $aZ+b\overline{Z}=c$ equation? [on hold]

I must find the solvability condition for $aZ+b\overline{Z}=c$ linear algebraic equation, where a,b and c ar complex numbers. Are there anybody for being help me?
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1answer
32 views

Does tanz assumes all complex numbers? [on hold]

$f(z)=tanz$,$z$ $\in$ $C$ a)assumes all complex numbers b)assumes none of complex numbers c)assumes all complex numbers except $i$ d)assumes all complex numbers except $i$ and $-i$ I think ...
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3answers
63 views

Contour Integral of $\int\limits_0^{2\pi}\frac{d\theta}{1+a\cos\theta}$ for $a^2<1$ (textbook wrong?)

My book is telling me that the answer is $\frac{2\pi}{\sqrt{1-a^2}}$. I'm getting an extra a on the numerator. Could somebody verify if I'm wrong, or if it's my book (it has been wrong numerous ...
1
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1answer
29 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 ...
1
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1answer
23 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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0answers
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 ...
1
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1answer
27 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 ...
2
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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 ...
0
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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 ...
1
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1answer
40 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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0answers
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= ...
0
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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 ...
2
votes
1answer
29 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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votes
1answer
34 views

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

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.
0
votes
1answer
39 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 ...
1
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2answers
33 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 ...
1
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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 ...
3
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2answers
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 ...
2
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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 ...
16
votes
2answers
2k views

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, ...
0
votes
0answers
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 ...
0
votes
1answer
32 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.
0
votes
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 ...
2
votes
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 ...
0
votes
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 $g=h^{-1} \circ f \circ h$ ...