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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21 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
33 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
32 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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votes
2answers
26 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
8 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
20 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
26 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 ...
-2
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0answers
11 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
35 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
34 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 ...
3
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1answer
51 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
59 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
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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
74 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
42 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
52 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
28 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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votes
1answer
33 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
25 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
45 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
28 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 ...
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1answer
47 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
39 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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1answer
65 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 ...
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1answer
27 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 ...
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1answer
27 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 ...
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votes
0answers
41 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 ...
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0answers
22 views

Integrating $\operatorname{Log}(z+2)$ along the unit circle [duplicate]

For the function $f(z) = \operatorname{Log}(z + 2)$, where we choose the principal branch of logarithm (namely, $−\pi < \operatorname{Arg}(z) < \pi$), and the contour $C := \{z \in ...
2
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4answers
249 views

Cauchy integral formula

Can someone please help me answer this question as I cannot seem to get to the answer. Please note that the Cauchy integral formula must be used in order to solve it. Many thanks in advance! ...
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0answers
38 views

There exist a sequence $Z_n$ with $Z_n \to Z_0$ such that $\lim_{n \to \infty} |f(z)| = \infty$

Suppose $f$ has an Essential Singularity at $Z_0$. Then there exist a sequence $Z_n$ with $Z_n \to Z_0$ such that $\lim_{n \to \infty} |f(Z_n)| = \infty$ Here two cases arise If there exist a nbd ...
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0answers
22 views

Interior of a closed curve

I'm working through a proof that contains this particular argument which I think is highly non-trivial but no justification is given - the context is complex analysis and the proof is of Lindelof's ...
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1answer
21 views

A question about zeroes and poles of complex functions. [on hold]

Let $f (z)=\frac {z}{z} $ be a complex function. Is 0 a zero, a pole, or neither of these?
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1answer
29 views

How can I find these partial derivatives?

I'm reading a book which gives this function $f(x,y)=x^2y/(x^2+y^2)$ if $(x,y)\neq (0,0)$ and $f(0,0)=0$ as a $C^1$ function in $\mathbb R^2-\{(0,0)\}$, continuous in $(0,0)$ and it has the partial ...
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1answer
68 views

Differences between real and complex analysis?

To start with, real analysis deals with numbers along the (one dimensional) number line, while complex analysis deals with numbers along two dimensions, real and imaginary, Cartesian style. Could this ...
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2answers
41 views

Type of singularity of $\sin(z)/z^3$ at $0$

I would have thought that this is a pole of order $3$ but on the answers it says it is of order $2$. I don't see why...
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0answers
38 views

Complex function

Can anyone give me a hint to approach this question? I haven't done anything like this before so I'm bit confused about what this question is asking. Thank you very much for all your help.
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2answers
32 views

Finding Laurent series with imaginary numbers

$$f(z)=\frac{2z}{z^2+1}=\frac1{z-i} +\frac1{z+i}$$ Find Laurent series in powers of $z$ in the domain $|z|<1$. So I got to find two Taylor series of the two terms in the function but how do you do ...
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0answers
21 views

How can I get the region of Convergence for zcos1/z? [on hold]

Find laurent Series and the region of convergence for ZCos(1/Z), I can find the series but I can't get the region of convergenc
2
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0answers
21 views

An extension of Kato's Selection Theorem?

One formulation of the well-known Kato Selection Theorem states that, given an analytic family of $n \times n$ complex, symmetric matrices $M(t)$, one can choose an orthonormal basis $\{e_i(t)\}_{i = ...
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1answer
33 views

Analytic paths through converging sequences in the complex space.

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 ...
-2
votes
1answer
19 views

How can I find Laurent Series and the region of convergence for $z/((z+1)(z+2))$ for$ z= -1$? [on hold]

How can I find Laurent Series and the region of convergence for $z = -1$ of $$ \frac{z}{(z+1)(z+2)} $$
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votes
1answer
32 views

Properties of a non-constant analytic map from the annulus $A(0,1,2)$ to the unit disk such that $\lvert f(z)\rvert = 1$ on $\partial A(0,1,2)$

Let $f$ be an analytic map that sends the annulus $A(0,1,2)$ to the unit disk such that $|z|=1,|z|=2$ get mapped to the points $|f(z)| = 1$. Furthermore f is not constant. Prove: 1) $f$ has at ...
1
vote
1answer
24 views

Let $f\in H(\mathbb{C})$. Prove that: $\exists_{M\in\mathbb{R}^+} \forall_{z\in\mathbb{C}}\ \ \ \ |f(z)|> M \Rightarrow f$ is constant

Let $f\in H(\mathbb{C})$. Prove that: $\exists_{M\in\mathbb{R}^+} \forall_{z\in\mathbb{C}}\ \ \ \ |f(z)|> M \Rightarrow |f(z)|> M \Rightarrow f$ is constant Completely don't know how to bite ...
0
votes
0answers
39 views

Laurent series confusion

I've split it up into partial fractions and got $1/z$ - $2/(z-1)$ + $1/(z-2)$ but I'm unsure sure what to do now. I think I have done part $(i)$. I get $$z^{-1} + \sum_{n=0}^\infty ...