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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1answer
36 views

Proving $f(z)$ entire function in complex analysis

If $f\in C(\Bbb C)\cap H(\Bbb C\backslash \delta B_1(0))$ then $f\in H(\Bbb C)$ [C means continuous, $\Bbb C$ means complex plane, H means analytic and $\delta$ means boundary] I don't even know ...
0
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3answers
229 views

Cauchy's argument principle, trouble working simple contour integral

I'm trying to teach myself Cauchy's argument principle by doing a simple example, but apparently I'm missing something, because every time I try to do the contour integral I get 0. Cauchy's argument ...
5
votes
1answer
154 views

Mean value property implies harmonicity

It is fairly easy to show that harmonic functions satisfy the mean value property, but it seems harder to show the converse. I've seen the following theorem without proof: If $u \in C(\Omega)$ ...
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0answers
17 views

Logarithm in Complex Plane and Open Set

I am still kind of confused about logarithms in complex plane. If you cut off the one of the line such as positive x-axis. I understand every set excluding positive x-axis will be holomorphic and ...
2
votes
1answer
35 views

When I can use Cauchy's Int. Formulae

Let $C$ be the circle $|z| = R$ traversed anticlockwise. Let z be any point in $\mathbb{C}$. I'm asked to calculate: $$\frac{1}{2\pi i} \int_C \frac{e^w}{w^3}(w^2+wz+z^2) \ dw$$ I'm wondering here ...
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1answer
45 views

Problem in complex analysis

Let $G$ be a bounded region and $f\in H(G)\cap C(\bar{G})$ [$H$ means analyticity and $C$ means continuity]. If $|f(z)|$ is constant on the boundary of $G$. prove either $f(z)$ has a zero in $G$ or ...
1
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2answers
71 views

complex analysis problems related to nalyticity… [on hold]

I don't even know how to start for the following problem. It would be highlt appreciated if you could help me. Let $z_n=\frac{1}{n},\forall n\in\Bbb N$. If possible find $f\in H(\bar{B}_1(0))$ [H ...
0
votes
2answers
48 views

Need help with complex analysis

I am struggling with following problem: Let f be an entire function, If $Im(f)>|z|$ for all $|z|>2$, show that f is constant. I think I have to use Liouville's theorem to do this problem. That ...
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2answers
64 views

If $f(z)g(z)=0$ then $f(z)=0$ or $g(z)=0$ [duplicate]

Let $D$ be a domain and let $f,g$ be analytic in $D$. I need to prove that if $f(z)g(z)=0$ for all $z\in D$, then $f(z)=0$ for all $z\in D$ or $g(z)=0$ for all $z\in D$.This is my answer. Assume ...
2
votes
1answer
58 views

let $f$ be analytic and bounded above, can I prove f is constant?

I've read up on Lioville's theorem and I was wondering if this could also be proved using the theorem: let $f$ be analytic on $\mathbb{C}$ and let $K>0$ be s.t. $|f| \geq K$, could I prove using ...
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0answers
25 views

Conformal Map for Circle to Circle

I am trying to find a conformal map that maps a circle in the $\zeta$ plane to a circle in the $z$ plane. As far as I know, a Mobius transformation is appropriate for this. These are the conditions ...
0
votes
1answer
21 views

Statement of maximum modulus principle and question

Question 1 My book (Complex Variables by Churchill) states If a function $f$ is analytic and not constant in a given domain $D$, the $|f(z)|$ has no maximum value in $D$. that is, there is no ...
3
votes
0answers
40 views

Construction of an explicit series of meromorphic functions.

I have to construct an explicit series of meromorphic functions that converges locally uniformly on the unit disk, and such that it has poles of first order at the points $a_k:=\frac{k-1}{k}$ with ...
1
vote
3answers
20 views

Taylor Series of $\frac{\log(z+1)}{z+1}, z_{0}=0 $

Finding the Taylor Series of $\frac{\log(z+1)}{z+1}, z_{0}=0 $ Where Log is the complex logarithm. At first, I tried to find the series for $\log(z+1)$ and $\frac{1}{z+1}$ and multiply them. ...
6
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3answers
64 views

Video Lessons in Complex Analysis

Does anybody have some link for good video lessons of a complete course in Complex Analysis? Grateful.
0
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1answer
53 views

Minimal surface and Weierstraß parametrization

If I have $f(z) = 1$ and $g(z) = \frac{1}{z}$ and I am looking for a minimal surface on $\mathbb{C} \backslash \{0\}$ using the Weierstraß-Enneper representation of minimal surfaces. Now I was ...
1
vote
1answer
36 views

definition of derivative for complex analysis

How can I use the definition of derivative to find the derivative of $\dfrac{\bar{z}^2}{z}$. My attempt, $\dfrac{\dfrac{\overline{z+\Delta z}^2}{z+\Delta z}-\dfrac{\bar{z}}{z}^2}{\Delta z}= ...
2
votes
1answer
33 views

When does this complex series converge?

Let $$f(z)=\sum_{n=0}^{\infty}\frac{z^{n [\mathbb{Re} z]}}{n}$$ For which $z\in\mathbb{C}\setminus\{0\}$ does this series converge? I have trouble with this example. When I use the ratio test, I ...
0
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0answers
12 views

An inequality in Zygmund space

How to derive (3) from (2)? Thanks for help.
2
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0answers
22 views

How to integrate $\int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}$?

In Lancaster & Blundell's QFT book they show that \begin{equation}A:= \int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}\end{equation} returns a nonzero value for $x$, $t$ and $m$ ...
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1answer
29 views

A question about the properties of the pseudospectrum

Assume that $A\in \mathbb{C}^{n\times n}$. The $\epsilon-$pseudospectrum of $A$ is defined by $$\sigma_{\epsilon}(A)=\{z\in C \quad | \quad \Arrowvert (zI-A)^{-1} \Arrowvert>\frac{1}{\epsilon}\}.$$ ...
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3answers
49 views

easier way to decompose fraction into partial fraction

It is a question in a test, and I couldn't manage to complete it. Given a complex fraction $\frac{1}{(z-1)^3(z+1)^3}$, we know that it can be decompose into ...
0
votes
1answer
9 views

Sum over the branches of a composition of an entire function with the branches of an algebraic function is entire.

Let $l(t)$ be the solution of the polynomial equation $g(l,t)=\det(lE-(A-tB))=0$, where E is the identity and A, and B are $n\times n$ matrices. The natural domain of definition is a Riemann surface ...
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0answers
12 views

Determine all of Möbius transformations $S$ such that $S(D(0,1))=D(0,1)$. [duplicate]

Determine all of Möbius transformations $S$ such that $S(D(0,1))=D(0,1)$. I thank you for your help.
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0answers
33 views

Quadratic formula and complex numbers

Let $az^2+bz+c=0$ be a complex quadratic equation. We know that it has $2$ roots: $z_1=\frac{-b-\sqrt{b^2-4ac}}{2a}$ and $z_2=\frac{-b+\sqrt{b^2-4ac}}{2a}$ If $b^2-4ac=1+i$ for example we have to ...
2
votes
2answers
29 views

Finding the Laurent series of $\frac{1}{z^{2}(1-z)}, 1<|z|<\infty $

Finding the Laurent Series of $\frac{1}{z^{2}(1-z)}, 1<|z|<\infty $ I tried to divide both numerator and denominator by $z^2$, so that there is a term of $\frac{1}{(1-z)}$. I am tempted to ...
2
votes
2answers
60 views

Using Residue theorem

I read Book by Egorychev in Russian. I don't understand the following identity $$S_m=\frac{1}{2\pi i m}\int_{|w|=\frac12}(2+w)(1+w)^m(1+w+w^2)^{-1}w^{-m-1}dw=-\frac1m ...
1
vote
1answer
30 views

Laurent Series of $\frac{e^{z}}{(z+1)^{2}}, 0<|z+1|<\infty $

$\frac{e^{z}}{(z+1)^{2}}, 0<|z+1|<\infty $ I am utterly unable to solve this problem. I have tried to write it as $e^(z-2\ln(z+1))$, but the resulting series is completely hideous. I have ...
0
votes
1answer
17 views

Show that $H(\mathbb{C})$ is a Frechet space.

Let $H(\mathbb{C})=\{f: \mathbb{C}\rightarrow \mathbb{C}; f \text{ holomorphic}\}$. For each $n$ let the seminorm $p_n$ be $p_n(f)=\sup_{|z|\leq n}|f(z)|$, and let $d(f,g)=\sum_{n=1}^\infty ...
1
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1answer
72 views

$\{x^nf(x)\}_{n\in\mathbb{N}}\subset L_2(a,b)$ as a complete system

I read in Kolmogorov-Fomin's (p. 430 here) the statement, sadly left without a proof, that if function $f:(a,b)\to\mathbb{C}$, measurable almost everywhere on $(a,b)$, where $-\infty\leq ...
3
votes
1answer
36 views

Determining the value of an integral using complex methods

I need to find the value of the following integral using complex analysis: $$\int_{-\infty}^{\infty}\frac{\sin(k_1\ x)+\sin(k_2\ x)}{x^2-a^2}\ dx$$ where $k_1, k_2, a$ are real coefficients. The ...
1
vote
2answers
38 views

Bounding a real integral involving complex constant

Is this integral finite $$|\int_{-\infty}^{\infty} e^{-i\pi x^2}\ dx|$$ can we use the fact that $e^{-\pi x^2}$ have compact support to estimate the above integral?
1
vote
1answer
36 views

Laurent series for $f(z)= \frac{1}{ (z-i)(z+2i)}$

I'm struggling with this question. I tried to break $f(z)$ using partial fractions and modify each equation so it looks like $\dfrac{1}{1-z}$ series but that's where I get stuck. Any help would be ...
2
votes
0answers
53 views

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent. I use a root test: $\lim_{n\rightarrow\infty} |\frac{e^{nz^2}}{n}|^{1/n}=\lim_{n\rightarrow\infty} ...
-3
votes
1answer
57 views

Let $f(z)=z+\frac 1 z$ for $z\in \mathbb C$ with $z\neq 0$ [closed]

Let $f(z)=z+\frac 1 z$ for $z\in \mathbb C$ with $z\neq 0$. Which of the following are true? $f$ is analytic function on $\mathbb C\setminus \{0\}$. $f$ is a conformal map on $\mathbb C\setminus ...
3
votes
2answers
49 views

$\int_{\mathbb{R}}f(x)e^{-ixz}d\mu_x$ analytic for $f\in L_1$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$ and let $x\mapsto e^{\delta|x|}f(x)$ also be Lebesgue-summable on all the real line. I have added the condition that ...
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votes
0answers
33 views

Question about Cauchy-inequality

Let $f:\mathbb{C} \to \mathbb{C}$ a holomorphic function and assume that there exist $M > 0$ and $r>0$ such that $$ |f(z)| \leq M |z|\ln |z| $$ $\forall z \in \mathbb{C}$ with $z \geq r$. I ...
3
votes
1answer
74 views

Riemann-Roch Theorem

Could somebody give a simple plain English explanation as to what the Riemann-Roch theorem is about to somebody who knows only standard one-variable complex analysis. Thanks.
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2answers
55 views

What does it mean for a function to be holomorphic?

I am trying to wrap my head around the definition of holomorphic. Wikipedia tells me that: A holomorphic function is a complex-valued function of one or more complex variables that is complex ...
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0answers
6 views

Time series of Complex numbers and Lyapunav exponent calculation

I have a time series of length $n$ of complex numbers that is $\{(x_i,y_i)\}$. $i=1,2,\dots,n$ where $(x_i, y_i)$ is a complex number. I would like to calculate the Lyapunav exponent of the series. ...
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64 views
+100

Approximation of holomorphic functions and topological properties

So, in the last couple of lectures of my complex analysis class we've proved some approximation theorems of holomorphic functions. Eventually, we showed the following propositions: Theorem 1. Let ...
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1answer
45 views

Singularities of $\overline{z}$ (complex conjugate) [closed]

How do I approach this in order to use the usual methods of identifying singularities?
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0answers
35 views

What are the possible expansions for $f(z)$ at $0$ for disks and annuli?

For the expression $f(z)$ what are its all possible expansions (I am considering disks and annuli) around the origin and where do they converge? $$ f(z) = z + 2z^2 + 3z^3 + \ldots + nz^n + \ldots = ...
0
votes
1answer
26 views

Laurent Series - when do singularities on the boundary of an annulus require a Laurent series instead of Taylor?

I need to find the Laurent Expansion of $F(z) = \dfrac{1}{(z-1)^2(z+2)}$ in the regions $A_1 = D(0,1)$ and $A_2 = \{z: 1 < |z| < 2 \}$. After doing partial fractions on $F(z)$, how do I know ...
2
votes
1answer
26 views

Asymptotic behaviour of Fourier transform: $|F[f]|=|\lambda^{-k}F[f^{(k)}]|$ for absolutely continuous $f$

I read in Kolmogorov-Fomin's (p. 429 here) that if function $f:\mathbb{R}\to\mathbb{C}$ is such that $f^{(k-1)}$ [the $(k-1)$-th order derivative] on any finite interval and if $f,...,f^{(k)}\in ...
3
votes
1answer
22 views

maximal injective neighborhoods centered at the zero of a polynomial

I was working on a particular problem involving the injectivity of a certain polynomial, $p(z) = z^5 + z -1$, $z \in \mathbb{C}$, in which I needed to find a neighborhood around it's real root so that ...
5
votes
2answers
62 views

Solve $e^{z-1}=z$ with $|z| \leq 1$

I'm looking for solutions to $$e^{z-1}=z$$ when $z \in \mathbb{C}$ with $|z| \leq 1$. The obvious solution is $z=1$, but I don't know how to show that there aren't any others. This question is ...
0
votes
1answer
30 views

Showing $1$ is not a branch point for $f(z) = z^2$?

I can see geometrically why $1$ is not a branch point for $f(z) = z^\frac{1}{2}$ as if we take a a point $p$ on the Riemann surface for $z$, $\epsilon$ distance away from $1$ are able to rotate that ...
1
vote
1answer
17 views

Finding an example of an analytic function, 0 on set of points (1/n)

As per the title, I want a function $f(z)$ which is analytic on $\mathbb{C}$ on the set of points $\{1/n\}$ for $n \in \mathbb{Z}^+$ and with $f(z) \neq 0$. What would this look like if the function ...
0
votes
1answer
12 views

Searching for a bound on a Mobius functions defined on $B(0,r)$.

Let $\alpha \in \mathbb C$ such that $|\alpha| \in (0,1)$. Prove that if $z\in \mathbb C$ is such that $|z| \le r < 1$ then : $$ \frac{\alpha+|\alpha| z}{\alpha(1-\overline{\alpha} z)}\le ...