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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37 views

Radial Limits for Holomorphic Functions

Let $U$ be an open disc of center $0$ and radius $R > 0$ in the complex plane, and let $f:U \backslash \{0 \} \rightarrow \mathbb{C}$ be a holomorphic function, such that for some $a \in ...
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2answers
81 views

Residue theorem for Multi-valued functions

Im stuck with this problem show that: $$\int_0^\infty{\frac{x^a}{(x^2+1)^2}dx} = \frac{\pi (1-a)}{4cos(a \pi /2)}, \, -1<a<3, \, a \neq 1$$ I have the solution for it and everything but i ...
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0answers
15 views

Improper Integral in four dimensions

Considering the integral $I = \prod_{i=1}^4 (\int_\infty^\infty dk_i sinc (k_ia_i + b_i)sinc (k_iA_i + B_i) )\frac {k_1^n}{k_1^2+k_2^2+k_3^2+k_4^2}$ with parameters $n,a,b,A,B$ I tried to use ...
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39 views

What is the fundamental group of a modular curve $\mathcal{H}/\Gamma$?

Let $\Gamma$ be a finite index subgroup of $PSL_2(\mathbb{Z})$. What is the fundamental group of $\mathcal{H}/\Gamma$? By the Kurosh Subgroup theorem, $$\Gamma \cong F_n * C_2^{*r} * C_3^{*s}$$ ie, ...
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25 views

Glide reflections on the complex plane

Show that if a is real and non-zero then a). $z → \bar z + a$ is a glide reflection along the real axis, and b). $z → −\bar z + ia$ is a glide reflection along the imaginary axis. Let $z = x+ yi$ ...
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2answers
93 views

Contour integration of log over polynomial with fractional power

I've stumbled upon one integral which is rather challenging because of the fractional power of t: \begin{equation}\int_0^\infty \log(1+tx)t^{-p-1}dt, \end{equation} where $p\in(0,1)$ and $x>0$. ...
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1answer
33 views

Find the radius of convergence of $\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}z^{n(n+1)}$

Find the radius of convergence of $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}z^{n(n+1)}$$ And study what happens in the border. Ok so I calculated $$\limsup ...
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3answers
179 views

Example using Cauchy's Theorem and integrating over a contour

This is an example out of Stein and Shakarachi Complex Analysis. My question is, Why do they choose $f(z) = \frac{(1 - e^{iz})}{z^2}?$
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2answers
54 views

Converse of Cauchy Integral Formula

The Cauchy Integral Formula for a disk is stated as follows: Let $f$: D $\to \mathbb C$ and $ z_0\in D$ If $f$ is analytic, then for every $ r\gt0$ with $\overline{B_r(z_0)} \subset D$ we have: ...
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22 views

The left and right side of an orientation in complex

The definition of the right side of an orientation $(z_1,z_2,z_3)$ is $$\{z: Im(z,z_1,z_2,z_3)>0\}.$$ Why is it defined this way? Note: ...
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44 views

Using the Cauchy integral to construct holomorphic functions

I was going through David C. Ullrich's wonderful text Complex Made Simple, when I came to Theorem 10.3.1. The Theorem states If $f\in C(\partial\mathbb{D})$ then $C[f]\in H(\mathbb{D})$, where ...
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16 views

Showing that $e^{z}$ series is concentrated around indices close to $|\Re(z)|$

So, I believe (but am having trouble showing) that the power series for $e^{z}$ is concentrated around indices close to $|\Re(z)|$ (or at any rate is negligible beyond those indices). More precisely, ...
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24 views

Isometry identity map proof

Let $f : C → C$ be an isometry with equation $f(z) = e^{iθ} \bar z + b$ where $θ ∈ R$ and $b ∈ C.$ Prove that $f^2 = Id_C$ then $e^{iθ}\bar b + b = 0$. My Attempt: Given that this is an identity map, ...
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30 views

In complex analysis, need to prove corollary for schwarz's lemma

prove : Suppose f(z) is analytic for |z| < R with $f(0) = f ' (0)= … f^{(n-1)}(0) = 0$ If $|f(z)| \le M$ in |z| < R, then $|f(z)| \le \frac{r^nM}{R^n}$ (complex variables, silverman 270p) ...
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67 views

Approximating $(1+\frac{1}{z})^z$ where $|z|$ is large

I know that $$\lim_{x\rightarrow \infty}\left(1+\frac{1}{x}\right)^x=e$$ Is there an equivalent in complex analysis for $$\lim_{|z|\rightarrow \infty}\left(1+\frac{1}{z}\right)^z=?$$
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114 views

Where are the roots going?

Each polynomial of degree $n$ has $n$ different roots. We know that $$e^z = \sum_{n=0}^\infty \frac {z^n}{n!}$$ has no roots. What is the behaviour then of the root of $S_N = \sum_{n=0}^N\frac ...
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17 views

Angle of the curves in complex analysis

Let $f(z) = U(x,y) + iV(x,y)$ be analytic in a region $G\subset \mathbb C$. Let $z_0= x_0+iy_0\in G$ be such that $f'(z_0)\neq 0$ a) Suppose that level curves $U(x,y)= C_1$ and $V(x,y)=C_2$ ...
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14 views

Maximum value of given expression

Find the maximum and minimum value of |z| , If |(z+(2/z))| = 2 Please explain in detail due to the fact that I'm new to complex numbers.. Thank you.
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130 views

Absolute Value of Cosine and Sine in $\mathbb{C}$

Is it generally true that $|\cos(z)|\leq1$, $|\sin(z)|\leq1$ $\forall z \in \mathbb{C}$? I think I'm missing something here (I think it does not hold, only if $z \in \mathbb{R}$). If this were not the ...
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1answer
40 views

Show there is no function from $\mathbb C \setminus \{0\}$ to $\mathbb C\setminus \{0\}$ with $f(zw)=f(z)f(w)$ and $f(z)^n=z$ [closed]

Let $n\geq 2$ be a natural number. There is no function $f : \mathbb{C}^*\rightarrow \mathbb{C}^*$ with the two properties $f(zw)=f(z)f(w)$ for all $z,w\in\mathbb{C}^*$, and $(f(z))^n=z$ for all ...
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41 views

Show that the function $f(z)=\frac{z}{z^{2}-1}-\frac{1}{z}$ has a primitive

Show that the function $$f(z)=\frac{z}{z^{2}-1}-\frac{1}{z}$$ has a primitive, i.e., there exists a function $F'(z)=f(z)$, on the region $\Omega=\{z\in\mathbb{C};\ \lvert z\rvert>3\}$. I know ...
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0answers
18 views

Does every nontrivial quotient $\mathcal{H}/\Gamma$ have an unramified cover

If $\Gamma$ is a proper finite index subgroup of $PSL_2(\mathbb{Z}) \cong C_2*C_3$, then must there exist a $\Gamma'$ finite index in $\Gamma$ such that ...
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3answers
174 views

Is there an analytic function with $f(z)=f(e^{iz})$?

Does there exist a non-constant analytic function $f$ satisfying $f(z)=f(e^{iz})$ ? I don't know where to start.
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31 views

Find all analytics functions in the domain $\Omega=\{z\in\mathbb{C}. |z|< R\}$

Find all analytics functions in the domain $\Omega=\{z\in\mathbb{C}; |z|< R\}$ that satisfy $f(0)=e^{i}$ and $|f(z)|\leq 1$ for each $z\in\Omega$. I try to write a series,i.e., ...
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2answers
37 views

Proof of a Complex Conjugate

I am told to prove the following: $$ \exp(\overline{z}) = \overline{\exp(z)}\,\! $$ for all $z$ in the complex plane. In what direction can I expand the equation?
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2answers
38 views

Proving complex conjugation

$f : C → C$ is an isometry such that $f(0) = 0$, $f(1) = 1$ and $f(i) = −i$. Prove that $f(z) = \bar z$ for all $z ∈ C$ I am having trouble figuring out where to go with this. I thought that since ...
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1answer
36 views

$\frac{d}{dz}$ and $\frac{d}{d\overline{z}}$ for $|z-a|^p$

I wish to find $\frac{d}{dz}$ and $\frac{d}{d\overline{z}}$ of $f(z)=|z|$ and $|z-a|^p$, $-\infty < p < \infty$. $\frac{d}{dz} = \frac{1}{2} \left ( \frac{du}{dx} + \frac{dv}{dy}\right ) + ...
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0answers
15 views

Reflections represented in complex form

An isometry $f$ of the plane $R^2$ can be written in complex form either as $f(z) = az + b$ or as $f(z) = az + b$ where $a, b ∈ C$ and $|a| = 1$. Find specific formulas of one of these forms for each ...
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2answers
52 views

Determine the image of the set $G = \{ z \mid |z|<1,\; Im(z)>0\}$ under $f(z) = \frac{z+\frac{1}{z}}{2}$

I wish to describe the image of $G = \{ z \mid |z|<1,\; Im(z)>0\}$ under $f(z) = \frac{z+\frac{1}{z}}{2}$. $G$ is the open upper unit=semicircular region. $f(z) = \frac{z^2 + 1}{2z}$. I thought ...
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2answers
44 views

$\sum_{n=1}^{\infty}\frac{z^n}{n}$

Find the radius of convergence of this series and study what happens in the border. $\sum_{n=1}^{\infty}\frac{z^n}{n}$ ($z\in \Bbb{C}$) I easily found that the radius of convergence is $\rho =1$, ...
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1answer
25 views

Growth rate of power series coefficients for function holomorphic on a strip

Suppose $f$ is holomorphic on the strip $\{ z \in \mathbb{C}: |\text{Im} \, z| < \delta \}$. Consider the expansion $f(z) = \sum_{k=0}^\infty a_k z^k$. Is there a bound on the growth rate of the ...
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4answers
81 views

Proving that if $|z_1|=|z_2|=|z_3|=|z_4|$ and $z_1+z_2+z_3+z_4=0.$ then $z_1,z_2,z_3,z_4$ are points of some rectangle

...or two pairs of points are identical. The assignment is done as follows: $$z_i=|z|e^{i \varphi_i}=re^{i \varphi_i}, i=1,\ldots,4$$ the assignment goes on to prove that ...
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0answers
40 views

Question about the first step inside the derivation of the Abel-Plana formula.

The derivation of the Abel-Plana formula is found here. At the first step for How does $$\lim_{h\to\infty}\int^{{b}\pm{ih}}_{{a}\pm{ih}}\left[g(z) \pm f(z)\right]dz = 0$$ What two functions for ...
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1answer
39 views

To show there exists a unique function $u \in C^{1}(\mathbb{C^n})$ that satisfies $(\bar{\partial u})=f$

Assume $n \gt 1$. Let $f$ be a $(0,1)$ form in $\mathbb{C^n}$, with $C^1$-coefficients and compact support $K$, such that $\bar{\partial} f=0$. Let $\Omega_{0}$ be the unbounded component of ...
3
votes
2answers
67 views

If $z_1,z_2,z_3 \in \mathbb{C}$ and $|z_1|=|z_2|=|z_3|$ and $z_1+z_2+z_3=0$. Prove that $z_1,z_2,z_3$ are points of a equilateral triangle [duplicate]

If $z_1,z_2,z_3 \in \mathbb{C}$ and $|z_1|=|z_2|=|z_3|$ and $z_1+z_2+z_3=0$. Prove that $z_1,z_2,z_3$ are points of a isosceles triangle that is on a unit circle with the center in the coordinate ...
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10 views

A complex subseries $\sum_{k=1}^{\infty}|z_{n_k}| $ if a complex serie $ \sum_{n=1}^{\infty}|z_n|$ converges absolutely

A complex subseries $\sum_{k=1}^{\infty}|z_{n_k}| $ converges absolutely if a complex serie $ \sum_{n=1}^{\infty}|z_n|$ converges absolutely This is a bidirectional proof, but I can't do this part$
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1answer
23 views

If $|z_1|=|z_2|=|z_3|$ and $\arg z_1\leq \arg z_2 \leq \arg z_3$ prove that $\arg{\frac{z_3-z_2}{z_3-z_1}}=\frac{1}{2}\arg \frac{z_2}{z_1}$

If $z_1,z_2,z_2 \in \mathbb{C}$ and $|z_1|=|z_2|=|z_3|$ and $\arg z_1\leq \arg z_2 \leq \arg z_3$ prove that $$\arg{\frac{z_3-z_2}{z_3-z_1}}=\frac{1}{2}\arg \frac{z_2}{z_1}$$ Answer:$\arg ...
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4answers
197 views

Solution to Complex Equation $\cos z = 2i$

I tried solving the equation $\cos{z}=2i$ where $z$ is a complex number. The solution $i$ have ended up with is $z$, $= (4k+1)\frac{\pi}{2} - i\ln{(2+\sqrt{5})}$. However the text book solution is ...
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62 views

Understanding Complex Differentials (forms)

In the study of Riemann surfaces, many books bring in their discussions, the complex differentials or differential forms, and there my understanding gets stopped. I personally interacted with many ...
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1answer
60 views

Why does $\sum\limits_{n=0}^{\infty}c_nz^n=0 \implies c_n=0, \forall n$?

In so many arguments for solutions to ordinary differential equations via the Frobenius Method do I see this argument - that if an infinite polynomial with constant coefficients is identically zero ...
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1answer
40 views

Absolute value of product is less than product of absolute values: $|(1+a_1)(1+a_2)\dots (1+a_n)-1|\leq (1+|a_1|)(1+|a_2|)\dots (1+|a_n|)-1$

For a sequence $a_n\in\mathbb{C}$ I want to show that $$|(1+a_1)(1+a_2)\dots (1+a_n)-1|\leq (1+|a_1|)(1+|a_2|)\dots (1+|a_n|)-1$$ I think I should show this by induction on $n$. For the base case I'm ...
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1answer
63 views

Proving a criterion for connectedness

The English translation of Markushevich's Theory of functions of a complex variable contains the following sufficient condition for the connectedness of a compact set on the complex plane. Theorem ...
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0answers
25 views

Describe the image of the set $G = \{ z \mid |z| >2, 0 < arg(z) \leq \pi\}$

Describe the image of the set $G = \{ z \mid |z| >2, 0 < arg(z) \leq \pi\}$ where $f(z) = z^3$ So $f(z) = r^3 ( \cos(3\theta) + i\sin(3\theta))$ and $r^3 > 2$. So is it all points in the ...
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29 views

Riemann Sphere Complex Analysis

I've been posed with the task of explaining, if $\{x_n\}$ is a sequence of points on the Riemann Sphere which converges to $(0,0,1)$ why does the corresponding sequence $\{z_n\}$ in the complex plane ...
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1answer
72 views

Continuous extension of $\int_\mathbb{R} dt\, e^{-t^2}/(t-z)$ from $\operatorname{Im} z < 0$ onto $\mathbb R$

I am asked to show that the continuous extension of $$ F(z) = \int_{-\infty}^{\infty} dt\, \frac{e^{-t^2}}{t-z}, \quad \operatorname{Im} z < 0 $$ onto $\mathbb R$ is given by $$ ...
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1answer
57 views

how to check if a singularity is isolated?

I have a function $1/(\sin(1/z))$ and I must show if the singularities are isolated or not. Is taking the limit of the number a little to the right and a little to the left enough? If not, how can ...
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2answers
14 views

Find the set of points where equality holds.

Suppose $z\in \mathbb{C}$, $|z|=R,R>1,m\geq 1, n\geq 0$. a) Prove that $\displaystyle \Bigg|\frac{z^n}{z^m-1}\Bigg|\leq \frac{R^n}{R^m-1}$. b) Find the set of points where equality holds. ...
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2answers
36 views

Describing the image of the complex set $G = \{ z = x+iy \mid x^2 + y^2 + 2x + 2y + 1 = 0\}$

Let $G = \{ z = x+iy \mid x^2 + y^2 + 2x + 2y + 1 = 0\}$, $f(z) = \frac{2z+3}{z+i}$ I wish to describe the image of the set $G$. However, I am not sure how to apply the restriction to $f$. Does ...
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votes
1answer
30 views

Path connectedness of the complement of countable set

Let $G$ be an open (path) connected subset of $\mathbb{C}$. Let $f:G\rightarrow \mathbb{C}$ be a nonconstant anlytic function I proved that the subspace topology on $f^{-1}(0)$ is discrete and ...
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1answer
16 views

Proving $|z-1|\leq ||z|-1|+|z||argz|$ where $z$ is a complex number..

The following substitution is written: $$z=|z|e^{i \varphi}$$... and the assignment goes on to do some transformations, operations that are trivial with this and then comes to the conclusion, which is ...