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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2
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0answers
37 views

Is there any generalization of Riemann Mapping theorem?

Given any two regions in complex plane when can we say they are conformally equivalent? I mean does there exists some "complex-geometric" invariant which determines whether two regions are conformally ...
1
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1answer
32 views

Disc of convergence involving logs

Find the disc of convergence: $$\sum_{n=2}^\infty \frac{z^{n}}{n(log(n))^p};(p>0)$$ I have tried geometic series, ratio test, root test... What would be your thought on the best test to use?
6
votes
2answers
138 views

Computation of an iterated integral

I want to prove $$\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\frac{\pi^2}{2}.$$ Since the function $(x,y)\mapsto\sin(x^2+y^2)/(x^2+y^2)$ is not ...
0
votes
1answer
51 views

What function can turn $z=x+iy$ into something involving $xy$?

What function can turn $z=x+iy$ into something involving $xy$? What function takes the real parts of $z$ and then multiplies them? Or would I perhaps need to consider the point $(x,iy)$, rather than ...
0
votes
1answer
56 views

Why is it valid to set $r=e^t$ in $f(r)=\frac{r+r^{-1}}{2}$?

$f(r)=\frac{r+r^{-1}}{2}$ $f(re^{i \theta})=\frac{re^{i\theta}+r^{-1}e^{-i\theta}}{2}=\frac{r+r^{-1}}{2}\cos\theta+i\frac{r-r^{-1}}{2}\sin\theta$ Why is it then valid to set $r=e^t$, $-\infty≤t≤0$ ...
0
votes
2answers
43 views

Continuity of a function with complex variables

How could I show if or not the following piece-wise defined function is continuous at the point $z=-i$? $$f(z)=\left\{ \begin{matrix} \frac{z^2+2iz-1}{2z^2+iz+1}, & z \neq -i \\ 0, & z=-i ...
3
votes
2answers
47 views

Sketch the set of points satysfing an inequality $|z+1|+|z-1|\leq2$

The inequality is $$|z-1|+|z+1|\leq2$$ I used a triangle inequality to show that Since triangle inequality states: $$|z+w|\leq|z|+|w|$$ Then $$|z-1+z+1|\leq|z-1|+|z+1|\leq2$$ So $$|2z|\leq2$$ From ...
0
votes
1answer
22 views

Show that $\Sigma_{j=0}^n z^j=\frac{1-z^{n+1}}{1-z}$ [duplicate]

As in the question I have to show that $$\sum_{j=0}^n z^j=\frac{1-z^{n+1}}{1-z}$$ So if we suppose that the above is true then clearly ...
0
votes
1answer
39 views

Give the power series expansion of $\log z$ about $i$

I'm reading Conway's complex analysis book and I'm trying to solve the exercise 5 from page 74. In this exercise the author asks for the radius of convergence and power series expansion of $\log z$ ...
0
votes
0answers
30 views

Inequality on the unit circle-part 2

This is the follow up question to my earlier one Inequality on the unit circle . It seems to be $$\left|np(z)+(\alpha-1)zp'(z)\right| \\\leq\left|np(z)+(\alpha-z)p'(z)\right|$$ on $|z|=1,$ for ...
1
vote
1answer
33 views

Inverse of a Linear transformation in complex analysis

I am given $T z= \frac{z+2}{z+3}$ , where $T_1$ is a linear transformation on the complex number $z$. I need to find its inverse $T^{-1}z$. I considered a complex number $w =\frac{z+2}{z+3} $. This ...
2
votes
3answers
36 views

Compute the integrals using the residue theorem

Compute the following integrals: $I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$ $J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$ I do not know where to begin. I know I am supposed to use the substitution ...
0
votes
1answer
37 views

Closed sets and accumulation points

In complex analysis how to prove that if $S$ is closed in $\mathbb{C}$ then it contain all of its accumulation points. If $S$ is closed then $S$ contain all its boundary points.(If $z_{0} $ is a ...
0
votes
1answer
33 views

Finding an Entire function with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$

I am really stuck on a homework problem, which boils down to the following: We need to exhibit an entire function $f$ with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$. The only sorts of functions ...
1
vote
0answers
31 views

How to approach solving this Fourier series [closed]

$$f(x):=\frac{1}{e^{2+\cos x}-1}$$ Source. Hi. I need to find Fourier series for this function. This is even function so Fourier coefficient $b_n$ is 0. Basically I need to solve this integral ...
0
votes
0answers
21 views

Showing a function has a global primitive on the unit disk minus the origin

I've reached a dead end for a problem with proving there is a global primitive for a continuous function on the unit disk D minus the origin with the condition that $\lim_{z\rightarrow 0}{zf(z)}=0$. ...
0
votes
1answer
35 views

Finding $\sqrt{(1 + \sqrt{3i})}$

Find $\sqrt{(1 + \sqrt{3i})}$. I am trying to use the fact that $\sqrt{(1 + \sqrt{3i})} = re^{i \theta} = r(cos \theta + i sin\theta)$ but I am having trouble figuring out where to go from here. Any ...
0
votes
0answers
11 views

Regarding Smirnov domains

Suppose $G$ is a Smirnov domain in the complex plane and it contains infinity(we can think of it as the exterior to a closed Jordan curve) and $\phi$ is a conformal mapping from $\mathbb{D}$ onto $G$. ...
1
vote
2answers
47 views

If $f$ is analytic in $D$ and $|f(z)|<M$ everywhere on $|z|=1$, show for all $z:|z|<1$, $|f(z)| \leq M |\frac{z-a}{\bar a z - 1}|$

Suppose $f(a)=0$ for some $|a|<1$. If $f$ is analytic in $D$ containing the unit disk and $|f(z)|<M$ finite for all $z:|z|=1$, show for all $z:|z|<1$, that $$ |f(z)| \leq M ...
1
vote
0answers
27 views

Prove for $z$ in the unit disc (real/complex analysis)

Prove for $z$ in the unit disc $$1+\binom{k+1}{1}z+\binom{k+2}{2}z^2+\cdots+\binom{k+n}{n}z^n+\cdots=\frac{1}{(1-z)^{k+1}}\quad(k=0,1,2,\ldots)$$where the coefficients on the left are the binomial ...
0
votes
2answers
27 views

Show that the series is not absolutely convergent but is uniformly convergent in the whole complex plane

Show that the series is not absolutely convergent but is uniformly convergent in the whole complex plane. ...
0
votes
1answer
17 views

Help finding a second homogeneous polynomial of degree 5 that are also harmonic

Essentially I have to find 2 homogeneous polynomial of degree 5 that are also harmonic. Knowing z=(x+iy) is analytic I found my first polynomial to be f(z)=z^5 and that multiples of this would ...
0
votes
2answers
45 views

What's the relationship between hyperbola, hyperbolic functions and the exponential function?

The hyperbolic functions can be expressed using the exponential function. However how are these related to "hyperbolas"?
0
votes
0answers
17 views

If $b \in (-\infty, \infty)$ in $z=a+bi$, then how to mark the range of $z$?

Let $a$ be fixed. If $b \in (-\infty, \infty)$ in $z=a+bi$, then how to mark the range of $z$?
1
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0answers
33 views

How to classify harmonic functions on the punctured disk without the Schwartz reflection principle?

I am working through old qual problems at the University of Minnesota and am trying to find an alternate solution to the following problem. Determine all continuous functions on $\{ z : 0 < \left| ...
1
vote
0answers
19 views

Find the disc of convergence of the power series (real/complex analysis)

Find the disc of convergence: $$\sum_{n=1}^\infty \frac{(z-3)^{n}}{n(n+1}$$ I have tried geometic series, ratio test, root test... but I seem to get stuck each time. What would be your thought on ...
2
votes
0answers
104 views

Complex Root of Unity Analogue of Forward Difference Operator

In my studies I have come across a couple of operators; in particular; $$\Delta[f(x)]=f(x+1)-f(x)$$ $$\Delta^*[f(x)]=f(x+1)+f(x)$$ $\Delta$ has been called the Forward Difference Operator. I was ...
2
votes
5answers
80 views

How do I solve the following equation: $z^4+z^3+z+1=0$

Is there an existing method to solve the following equation: $z^4+z^3+z+1=0$?
1
vote
1answer
22 views

$\frac{1}{2}(z+\frac{1}{z})$ range is $[-1,1]$, when $|z|=1$?

The range of $$\frac{1}{2} \left(z+\frac{1}{z}\right), \quad z \in \mathbb{C}$$ should be $[-1,1]$, when $|z|=1$? Any idea how to see it? I tried de Moivre (since it has the $|z|$ term), but it ...
1
vote
2answers
34 views

Cantor Intersection Theorem extension [duplicate]

Task at hand: Show that in the Cantor Intersection Theorem, "compact" cannot be replased by "closed"; that is, find a nested sequence $\{F_n\}_{n=1}^\infty$ of nonempty closed sets in C such that ...
0
votes
0answers
30 views

Find the set of all accumulation points for the following set

Find the set of all accumulation points for the following set. $$\left\{\frac{1}{m}+\frac{i}{n}; m,n\in N\right\}$$ I am trying to partition the set in a way to find the points but I can not seem to ...
2
votes
3answers
46 views

Image of a family of circles under $w = 1/z$

Given the family of circles $x^{2}+y^{2} = ax$, where $a \in \mathbb{R}$, I need to find the image under the transformation $w = 1/z$. I was given the hint to rewrite the equation first in terms of ...
0
votes
0answers
30 views

Existence of a curve with index 1 around a compact set

Let $K \subset \mathbb{C}$ be compact. If $U$ is an open set containing $K$, I want to show that there exists a collection of (piecewise $C^1$) curves $\gamma_1...\gamma_n$ such that 1) For $ x \in ...
2
votes
1answer
52 views

If $(z_{n}) \in \overline{ \mathbb{C}}$, $z_{n} \to \infty$ as $n \to \infty$, what happens to $|z_{n}|$, $Re(z_{n})$, $Im(z_{n})$, $Arg(z_{n})$?

Suppose the sequence $(z_{n}) \in \overline{\mathbb{C}}$ (where $\overline{\mathbb{C}}$ is the extended complex plane) converges to infinity as $n \to \infty$. I need to determine what this implies ...
1
vote
0answers
31 views

$(2+z^2) f(z) + 3 z + 4 = f(z+1)$?

Consider the equation $(2+z^2) f(z) + 3 z + 4 = f(z+1)$ Such that $f(z)$ is analytic near the positive real axis and the functional equation holds for real $z>0$. Can we express Some solutions ...
0
votes
0answers
16 views

Fourier transform of windowed complex exponential

I have a function on the form $$f(x) = g'(x)*e^{i\pi g(x)}.$$ Where $g'(x)$ is a window function with support in the range $-R \ldots R$. I want to find the fourier transform $\mathcal F(\omega)$ ...
0
votes
0answers
56 views

Existence of an analytic function with nonvanishing derivative mapping the punctured unit disk to the unit disk

Here is an exercise on the page 164 of Conway's book(in the section of Riemann mapping theorem): Show that there exists an analytic function $f$ defined on $G=ann(0;0,1)$ such that $f'$ never ...
0
votes
0answers
15 views

Univariate residue in multivariate contour integration?

Consider a multivariate contour integral on the complex polydisc: $$\oint \frac{dz_1dz_2...dz_n}{f_1(z_1,z_2,...,z_n)f_2(z_1,z_2,...,z_n)...f_n(z_1,z_2,...,z_n)}F(z_1,z_2,...,z_n)$$ The $f_i$ in ...
0
votes
1answer
28 views

Show $\prod_{n=1}^\infty 1 + \frac{-\left( 1 + z \right)}{n^2 + \left( 1 - n^2 \right) z}$ has no analytic extension past the unit disk

I'm studying for a qualifying exam (tomorrow) and I was hoping someone could show me how to finish solving this problem. Let \begin{align} a_n = 1 - \frac{1}{n^2}, && f(z) = ...
5
votes
0answers
62 views

Finding an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$

I encountered the following problem in the lecture note in my complex analysis class: Problem. Find an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$ for $n = 1, 2, \cdots$. ...
1
vote
1answer
25 views

Cauchy integral formula12

$\int 1/(z^2 + 2i)dz$ I've tried writting it as $1/(z-1+i)*(z+1-i)$ but then it's impossible to find solution. Any help would be great, thanks in advance.
2
votes
0answers
19 views

Artin approximation vs implicit function theorem in the class of analytic functions

I am not an algebraist so my question might be stupid. I am doing mainly complex analysis and recently I was informed about the existence of Artin's theorem and it sounded like it could be of interest ...
1
vote
0answers
16 views

sketch the set of points in a complex plane

I have two questions in here and absoultely no idea how to approach them. $$0<arg(z-1-i)<\frac{\pi}{3}$$ and $$log|z|=-2arg(z)$$ My approach: In first case since we want the argument to be ...
1
vote
1answer
15 views

What to do when there is only one valid value to be used in the Cauchy-Riemann equations

I just did 2 problems where the $u$ part of the C-R equation was $0$. I'll give one as an example. I'm confused as to what conclusions I can correctly arrive at. $$f(z)=Im(z)$$ So I can say that ...
1
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0answers
14 views

A clarification on an answer on residues and Polya fields

In this very informative and interesting answer about the relation between residues and representation of complex functions as vector fields the author states that the function $$f(z) = \frac{1}{z}$$ ...
1
vote
2answers
42 views

Does the inverse of $f(x)=x^3$ have a non-negative domain to have a real output?

I'm not familiar with complex analysis. While playing with Mathematica (a mathematics software), I found that it keeps spitting out unexpected results, and the reason was that it considers differently ...
0
votes
1answer
19 views

Where are the following functions differentiable? Where are they holomorphic? Determine their derivatives at points where they are differentiable.

$$ f(z) = e^{−x}e^{−iy}$$ I used the Cauchy Riemann equations to determine that $x=iy-\ln(i)$, but I'm not sure what I'm supposed to conclude. Could I say that the function is differentiable wherever ...
1
vote
2answers
38 views

Is it a removable singularity?

In the function: $$ f(z)=2iz\frac{(1-z^{2})^{\frac{1}{2}}}{1-2z^{2}} \qquad \qquad (z \in \mathbb{Z}) \,\, , $$ There is a singularity at the point $z=\pm \sqrt{1/2}$. Is that a removable ...
4
votes
2answers
35 views

zeros of $p(z)=z^4+2$

I want to find all zeros of $p(z)=z^4+2$ and I'm not sure if I've done everything correctly. Can you correct this if something is wrong? $$x^4+2=0 \iff x^4=-2=2\cdot(-1)$$ $$\Rightarrow x_k= ...
1
vote
0answers
23 views

Let $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ then is $f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$?

I am a little stuck here, suppose we have some function $$f(z)=f(x+iy)=u(x,y)+iv(x,y)$$ then is $$f'(z)=\frac{\partial u}{\partial x}(x,y)+i\frac{\partial v}{\partial x}(x,y)$$ assuming $f$ is ...