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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7 views

Extension of biholomorphisms between planar domain to Möbius transformations

Let $\Omega, \Omega' \subset \mathbb{C}$ be two planar domains, i.e. open connected. For simplicity we can assume they are also bounded and with smooth boundary (but I don't know if this is actually ...
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1answer
39 views

The Origins of $Re \lbrace x \rbrace$

I am just starting to take a differentential equations course. We do not use the notation $Re \lbrace x \rbrace$ to describe the real part of a complex number, but I have come accross it in other ...
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1answer
63 views

$e^{2\pi i x} = (e^{2\pi i})^x$: What happens if x is rational? [duplicate]

I'm a bit embarrassed that I've had difficulty on getting around this one: $$e^{2\pi i x}$$ Solving it by itself, we can reduce it down to $(e^{2\pi i})^x = 1^x$ such that $e^{2\pi i x} = 1$ for all ...
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1answer
30 views

Exponential type of $\sin z$

An entire function $f$ is of exponential type if $\,\lvert\, f(z)\rvert\le C\mathrm{e}^{\tau\lvert z\rvert},\,$ for all sufficiently large values of $\lvert z\rvert$. The exponential type of $f$ is ...
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0answers
14 views

Distance from $f(0)$ to the boundary of $D$ if $f$ maps open unit disk to $D$ conformally

Let $f(z)$ be a conformal map from the open unit disk onto $D$, which is a domain. I would like to show that the distance from $f(0)$ to the boundary of $D$, denoted $\partial D$, is given by ...
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1answer
36 views

Show that $f(z)$ is constant

If $f(z)=u(x,y) + iv(x,y)$ is an entire function such that $u\cdot v$ is constant then $f(z)$ is constant. I know that I need to use the Cauchy-Riemann equations, but I don't know how to start. ...
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0answers
27 views

Let $f$ be a non-constant entire function. Prove that $f(z)=cz^n$ for some constant $c$ and positive integer $n$

Let $f$ be a non-constant entire function satisfying the following conditions: $$f(0)=0$$ for each $M \gt 0$ the set $$\{z| |f(z)| \lt M\}$$ is connected. Prove that $f(z)=cz^n$ for some constant ...
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19 views

Finding the harmonic function at a given point [on hold]

Draw concentric circles of radii $r_1 = |b|$ and $r_2 = |c|$, each centered at $z_0 = a + id$. Suppose $\theta(x,y)$ is a harmonic function inside the washer defined by these circles. The circle with ...
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3answers
59 views

Real Methods to Evaluate $2 \int_{-1}^{1}x^2 \sqrt{1-x^2}dx$

I was recently contacted by a friend to find the values of the two following integrals by any means. $$ I=2\int_{-1}^{1}x^2 \sqrt{1-x^2}dx$$ $$ J=\int_{-1}^{1}(1-x^2) \sqrt{1-x^2}dx$$ The first ...
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1answer
10 views

Derivatives of $(z-a)^kf(z)$ at $a$ knowing that $f\in H(D(a,r)\setminus \{a\})$

If $g(z)=(z-a)^nf(z)$ with $f\in H(D(a,r)\setminus \{a\})$. Can we said that $g^{(k)}(a)=0$ for all $k\in \{0,1,...,n-1\}$?
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1answer
18 views

Laurent series: how to join the 2 sums for $f(z)= \frac{1}{(z-1)(z+1)}$ about z = 1 for $0 < |z − 1| < 2$

We are to find the Laurent series for f(z) about $z = 1$ for $0 < |z − 1| < 2$: $f(z)= \frac{1}{(z-1)(z+1)}$ Assumptions: $|\frac{z−2}{1}| < 1 ⇔ |z − 1| < 2$ For $\frac{1}{(z-1)}$ we ...
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0answers
43 views

Homework: Awkward formulation of the exercise

The exercise: Let $U$ be open and connected, let $\bar D$ be a closed disk contained in $U$ and let $f : U \to \mathbb{C}$ be analytic. Denote by $γ$ the circle which is the boundary of $D$. Suppose ...
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1answer
82 views

Complex integral $1/(z^2+1)$ along unit circle [on hold]

I want to compute the complex integral $$\int_{|z|=1}\frac{1}{z^2+1}dz$$ both i and -i lie on c , what can I do ? I tried caushy , series , i used def of contour integral along c = z(t)=exp(it) and ...
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1answer
21 views

Laurent series to converge in $0<|z-1|<R$

Question: Determine the largest number $R$ so that the Laurent series of $$f(z)=\frac2{z^2-1} + \frac3{2z-i}$$ about $z=1$ converges for $0<|z-1|<R$. Attempt: I really don't understand this ...
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2answers
40 views

What's wrong with this reasoning? (Cauchy integral theorem)

Asumme that $f$ is analytic and for $z\in \overline{B(x,r)}$: $$|f(z)|\leq d$$ Then for $z\in B(x,r)$: ...
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2answers
68 views

Determine $\int \limits_0^{\infty} \frac1{x^4+1}dx$

Let $$f(z)=\frac1{1+z^4}$$ (a) Find the sinularity of $f(z)$ in the first quadrant where $Re(z), Im(z) \ge 0$. (b) Find the residue of the singular point found in the first quadrant. (c) Let ...
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0answers
19 views

Help with $\int _{R_0<|z|<R_1}\frac{1}{z} dz$.

Consider the integral in $\mathbb{C}\simeq \mathbb{R}^2$ $$ \int_{R_0<|z|<R_1} \frac{1}{z}\; dx_1 dx_2 $$ where $0<R_0<R_1$ and $z=x_1+i x_2$ and $|z|=(x_1^2+x_2^2)^{\frac{1}{2}}$. So ...
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1answer
31 views

Finding $\lim \limits_{R \rightarrow \infty} \int _{\Gamma_R} \frac1{(z+i)^2 (z-i)^2}dz$

Let $\Gamma_R $ be the half circle centred at $0$ and radius $R>3$ with $Im(z) \geq 0$. Show that $$\lim \limits_{R \rightarrow \infty} \int _{\Gamma_R} \frac1{(z+i)^2 (z-i)^2}dz=0$$ Is this ...
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2answers
29 views

Integral on the real line between 0 and infinity using contour integration

For part (a) I have that the singularity is at $(1+i)/root2$ and it is a simple pole? For part (b) I have that the residue at $f(z)$ at that point is $-(1+i)/4root2$ For part (c) I used the ML ...
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2answers
25 views

Region of Convergence of power series

The power series $\sum_{n=0}^\infty 2^{-n} z^{2n} $ converges if a)$|z|\le 2$ b)$|z|\lt 2$ c)$|z|\le\sqrt2$ d)$|z|\lt\sqrt 2$ I tried this problem,my answer is d).I am not sure whether it is correct ...
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0answers
16 views

Proving max-mod principle by contradiction

This is a homework exercise I have to make which I am kind of stuck on. First let $U$ be open and connected, $\overline{D}$ be the closure of the disk $D$ contained in $U$ and ...
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1answer
13 views

Prove that $\sum_{n=0}^\infty e^{-nz}$ is analytic in the right half plane $\text{Re}(z)>0$

Consider$$\sum_{n=0}^\infty e^{-nz}$$ Using Weierstrass theorem, prove that the series is analytic in $\text{Re}(z)>0$. I know that $f$ is analytic if it satisfies Cauchy–Riemann equations. Could ...
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0answers
7 views

anlyalytic paths through convergent cauchy sequence II

Assume we have a Cauchy sequence $\{\vec{a_i}:i\in\mathbb{N}\}$ converging to $\vec{0}$ in $\mathbb{C}^n$ such that $|\vec{a_i}|<|\vec{a_j}|$ whenever $i>j$. Can we find an analytic path ...
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1answer
38 views

Evaluate the integral $\int _\gamma (e{^z}^{2} + \overline{z}) dz$

First part of the question asks me to state the path integral $\int_\gamma f$, which I defined as: \begin{equation} \int_\gamma f = \int^b_a f(\gamma(t))\gamma ' (t) dt \end{equation} And the second ...
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1answer
18 views

Singularity of Product of two complex function $f$ and $g$

Suppose $f$ has an essential Singularity at $z = a$ and $g$ has a pole at $z = a$. Then the product $fg$ has an essential Singularity at $z = a $. Is this hold if $g$ has removable ...
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1answer
25 views

Some doubts on singularity of the complex function $f(z) = 1/ \sin (1/z)$ at $z=0$

$$f(z) = \frac{1}{ \sin (1/z)}$$ has a non isolated singularity at $z =0$. Since by definition of isolated singularity, every nbd of $0$, $S_{1/n}$ , $\exists $ $\frac{1}{n \pi}$ which is a zero of ...
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0answers
9 views

x,y plane, divide up the x-axis by placing marks at x=c, x=b, and x = a [on hold]

In the x,y plane, divide up the x-axis by placing marks at x=c, x=b, and x = a. Suppose theta is harmonic in the upper half plane, and on the segments of the x-axis defined by your marks, theta takes ...
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0answers
34 views

Open sets in $\mathbb C$ and open sets in $\hat{\mathbb C}$

I usually have a lot of trouble with complex variable when it comes to the geometric representation of $\hat{\mathbb C}$ and what happens in there. I have the next exercise and from quick look at it I ...
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1answer
30 views

Classify singularities of $\frac{e^z \sin(3z)}{(z-\sqrt2)(z+\sqrt2)z^2}$

They are $0, \pm \sqrt2$. With the zero, $f(0)$ makes the numerator vanish and I have no idea how you would expand the whole function at $0$ because of the denominator. So what do you do to classify ...
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1answer
49 views

Prove that the range of the entire function $z^2+\cos(z)$ is all of $\mathbb{C}$.

Prove that the range of the entire function $z^2+\cos(z)$ is all of $\mathbb{C}$. I'm aware this question has been asked already, but the explanations were a little shakey and referenced a google ...
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3answers
58 views

Complex numbers? [duplicate]

There are plenty of questions out there asking what complex numbers mean and I never seem to get any of them. I have a few specific questions i want to ask about complex numbers. 1) what is the ...
2
votes
1answer
22 views

Find $f,g$ such that $f \equiv g \mod 2i\pi $ has finitely many solutions

I'm interested by two holomorphics functions $f,g : \mathbb C \to \mathbb C$ such that the set $$ E := \{z \in \mathbb C \mid e^{f(z)} = e^{g(z)} \}$$ is finite and non-empty. For example : $f,g$ ...
3
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1answer
53 views

Complex stationary point of $\frac{z}{1-e^{-z}}+z$?

I apply the method of steepest descents I need to know the stationary points $z_0$ of the function $$ p(z)=\frac{z}{1-e^{-z}}+z, $$ such that, $ 0 <\mathrm {Im} (z)<2 \pi$. That is, I want $z_0$ ...
3
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2answers
71 views

Integrals on the real line using contour integration

I know I am supposed to split it up like this and gamma(R) tends to zero and the other tends to my integral as R tends to infinity? I compute the residue at $2i$ which I think is $sin(2i)/2$ ? ...
2
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2answers
40 views

Troubles working with Residue Theorem

I try to compute the integral on the positively oriented circle $$\int_{\partial D(1,2)} \frac{z dz}{(z+2)(z^2 -2z + 2)}$$ So I try working with the Residum Theorem. First I compute the singularities ...
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1answer
51 views

Calculate $\int _{\Gamma} \frac1{z^4 +16}dz$

Where $\Gamma $ is $|z-i|=1/2$ positively orientated. I have thought of every method to do this but still cant. It wont factor such that it would be in the form of Cauchy's integral formula. It ...
2
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1answer
58 views

Why doesn't the derivative of a holomorphic function vanish in a border maximum?

Let $K$ be a compact disc in the domain $G \subseteq \mathbb{C}$ and $a \in \partial K$ be a point where the absolute value of the nonconstant holomorphic function $f \colon G \to \mathbb{C}$ attains ...
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1answer
27 views

Find $\int _{\Gamma} \frac{\cos(2z)}{(z-\pi/4)^2}dz$

Where $\Gamma$ consists of the sides of a triangle with vertices $i$, $-1-i$ and $\pi -i$. I think we use Cauchy's integral formula but I cant get it in the standard form of it. I don't think partial ...
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1answer
31 views

Largest $R$ value in domian $0<|z-1|<R$

Determine the largest real number $R>0$ such that the Laurent series of $$f(z)=\frac1{z-1} +\frac2{z-i}$$ about $z=1$ converges for $0<|z-1|<R$. The singularities are $1$ and $i$. But in the ...
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2answers
33 views

Is there such a thing as complex rational numbers and does it have the same properties as the usual complex numbers as extension of the real numbers?

I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$). I immediately see a problem with ...
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1answer
23 views

Laurent series in domain $|z|>0$

Find Laurent series, in powers of $z$, of $$f(z)=\frac{\sin(2z)}{z}$$ valid in the region $|z|>0$. The singularity is $0$ but $0$ isn't inside the region of the domain so what do you exactly ...
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0answers
43 views

what would be the formula of $\phi$ in this question?

Suppose $\phi:\mathbb{C}^2\longrightarrow\mathbb{C}^2$ be an entire map (i.e, the components of $\phi$ are entire in each variable separately) with $\phi_1$,$\phi_2$ as its components satisfying ...
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1answer
26 views

Calculating residues of multiple poles?

How would I calculate $$\mathrm{Res}\left(\frac{\pi}{\sin(\pi z)(2z+1)^3}\right)?$$ I understand it has singularities at $z=n$ and $z=-1/2$, I'm interested in the residue when $z=-1/2$. I know that ...
0
votes
1answer
45 views

From $\mathbb{H}$ to Poincaré disc? [on hold]

What is the mapping that takes one from the Poincaré upper half plane $\mathbb{H} = \{ z\in \mathbb{C} \mid \operatorname{Im}(z)>0 \}$ to the Poincaré disc? Here $z=x+i y$.
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2answers
38 views

Contour integral, f(z)=$ze^{z^2}$

For part $(a)$ is the answer just $0$? Using Cauchy-Goursat theorem? For part $(b)$ I am confused. Do I use ? It seems very complicated. Am I missing a trick?
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1answer
22 views

Continuous Choice of Argument

Since $\arg(z)$ is a set, if we define it with a specific branch, there will be discontinuity at the branch line. However, suppose $z:[a,b]\to \mathbb C\backslash\{0\}$ is continuous (it is a curve ...
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0answers
46 views

Let $F(r)=\sum_{k=1}^m{|P(rz_k)|^2}$ for $r>0$. Prove that the function $F(r)$ is increasing if $m>n>0$.

Let $P(z)$ be a polynomial of degree $n$ with complex coefficients. Further, let $$z_k=e^{\frac{2 \pi i k}{m}}$$ for some $m$ and $k=1,2,...,m$. In other words, $z_1,\cdots z_m$ are the $m$th roots of ...
1
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1answer
23 views

If $\forall f \in \mathcal{H}(\Omega)$ such that $f(z)\neq 0$ exists a square root then $\Omega$ is simply connected

If $\forall f \in \mathcal{H}(\Omega)$ such that $f(z)\neq 0$ for all $z\in \Omega$ $\exists$ $\varphi \in \mathcal{H}(\Omega)$ such that $\varphi^2=f$ $\implies$ $\Omega$ is simply connected. Is ...
0
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1answer
26 views

Proof of the Lindelöf theorem related to the radial limit of an analytic function in the unit disc

Hi I am looking for the proof of this theorem here by Lindelöf: "Suppose $\Gamma$ is a curve with parameter interval $[0,1]$, such that $|\Gamma(t)| < 1$ if $t < 1$ and $\Gamma(1)=1$. If $g \in ...
-1
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0answers
40 views

The Coin-Exchange Problem (Application of the Residue Theorem) [on hold]

These day, I have met a problem about application of the Residue Theorem, see section 10.4 of enter link description here.Could anybody help me solve it? (The Coin-Exchange Problem) Suppose $a$ and ...