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Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

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Using Ratio Test for a Series with odd and even indexed coefficients

The problem I have is the following: Suppose the radius of convergence of the series $\sum_{n=0}^\infty a_n z^n$ is equal to $2$. Find the radius of convergence of the series $$\sum_{n=0}^\infty 2^{\...
Hyperbolic Cake's user avatar
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Dilation of Open Ball to Unit Ball in Complex Plane

Suppose $S=\{z\in \mathbb{C} : |z-a|<r \}$ is an open unit ball in right half plane. Then can we get $c>0$ such that $c. S = \{cz : z\in S \}\subset \{z\in \mathbb{C}: |z-1|<1$? I tried the ...
VINI's user avatar
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Need Clarification on a problem about finding Laurent series

I am working on the following problem: Let $f(z) = \frac{1}{(1+z^2)(2-z)^2}$. Determine the principal part of $f$ at $z=2$ and determine the region where the Laurent series of $f$ at $z=2$ converges. ...
Koda's user avatar
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Given $f(z)=exp((1+z)/(z-1)$, prove $f(z) \in H^{\infty} (\mathbb{D})$.

Im a physics student trying to figure out some issues about complex analysis. I have been defined the group $H^{\infty}(\mathbb{D})$={ f is holomorphic in $\mathbb{D}$ / $||f||_{\infty}\leq \infty, \...
Iván's user avatar
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Uniqueness of Two Series in an Intersection

I've been working on some problems involving series and have found myself applying the identity principle to show that two representations of a series will lead them being unique under some conditions....
Hyperbolic Cake's user avatar
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20 views

Cauchy's theorem why emphasizing interior

I learned two typesof Cauchy's theorem in Stein and Shakarchi's book Complex analysis and quite confused about the descriptions of the second type. Suppose $\Omega$ is convex and open, $f$ is ...
Andrew_Ren's user avatar
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Simple parametrization of biholomorphism from a simple ellipse to the unit disk.

Let $D = \{ z \in \mathbb{C} \mbox{ with } |z|<1 \}$ be the open unit complex disk. Let $E_r= \{ z \in \mathbb{C} \mbox{ with } (\frac{\Re{z}}{r})^2 + (\Im{z})^2 < 1 \} $ be an open axis-...
ylvain's user avatar
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Do you know this special function?

I would like to know if anyone knows the name of this function which has this expression for $Re(s)>2$: $$\chi(s,z_1,z_2)=\sum_{m,n>0}\dfrac{1}{(mz_1+nz_2)^s}$$ It appears when giving the ...
BlueCharlie's user avatar
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11 views

Simple parametrization of automorphisme (Biholomorphism) over a simple annulus

Let $D = \{ z \in \mathbb{C} \mbox{ with } |z|<1 \}$ be the open unit complex disk. Let $A_r = \{ z \in \mathbb{C} \mbox{ with } 1<|z|<r \}$ be an open annulus, centered at the origin, of ...
ylvain's user avatar
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Principal Value Integral Using Contour Integration

The Principal Value of $\int^\infty_{-\infty}\frac 1{x(x^2+1)}dx=0$ since the integrand is an odd function. In the complex plane, the integrand has simple poles at 0 (residue 1) at $i$ (residue -1/2) ...
Drooga's user avatar
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Is there oriented scalar field line integral, or non-oriented vector field line integral?

I'm studying complex (integral) analysis and struggling with it. BTW, this is my first math stackexchange question, so please forgive any mistakes on my part. We have definition of complex line ...
Ryu's user avatar
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3 votes
1 answer
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Why is $2\sum_{k=a}^{b}ki^k=(1+i)i^a-i^a+i^b+(1-i)i^bb$

I am studying to skip into 10th grade, and on the curriculum, it has a section titled "Investigation of large sums and products of consecutive powers if $i$" (I kid you not) immediately ...
Lucien Jaccon's user avatar
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1 answer
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How to Find a Conformal Mapping from a Crescent-Shaped Region to an Annulus?

I am trying to find a conformal mapping from $\{𝑧:∣𝑧∣<1\}\cap\{𝑧:|z−\frac{1}{2}∣>\frac{1}{2}\}$ onto an annulus. This question is the very last problem from an old complex analysis exam. I ...
Kadmos's user avatar
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Clarification on the definition of normal family

In Complex Function Theory, 2nd edition by Donald Sarason, the author defines normal families as follows: A family of holomorphic functions in a domain is called a normal family if every sequence from ...
Koda's user avatar
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Rudin's Construction of Lebesgue Measure 2

I am currently studying Rudin's RCA book and I have a question about Theorem 2.20, where the author constructs Lebesgue measure on $\mathbb{R}^k$. Here are the definitions and the notations I am ...
meh's user avatar
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3 votes
1 answer
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Strange substitution made in a paper to find asymptotics

In the quoted section from this paper, why is the author able to "substitute this result into Eq. (2.1)"? This should hold for $z$ large. But not everything on the contour is large. Why can ...
Sam Kirkiles's user avatar
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Equation in space of complex numbers

Giving an equation: $$z^2-(m+2)z+4(m-1)=0$$ I need to find the number of all the integer $m$ such that this equation has two complex solutions $z_1$, $z_2$ that satisfy: $$\vert z_1^2-m(z_1-4)\vert=\...
Lê Trung Kiên's user avatar
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How does one prove that a given function is localy summable? [closed]

For example the function ln(x+iy)
Mirindra Fandresena's user avatar
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Methods on finding Automorphism group on simply connected subset of $\mathbb{C}$.

Suppose $D$ is the unit disc and I want to find Aut$(D-\{0\})$ and Aut$(\mathbb{C}-\{0\})$. My approach: For the first automorphism group, I think immediately about Aut$(D)$ and if some how I can pick ...
Remu X's user avatar
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1 answer
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What is $\int_0^{\infty}\sin(ax)\cos(bx)dx$ ? where $a$ and $b$ are constant. [closed]

I am solving an E&M problem and I can't get the solution of integration above. I guess it is something to do with residue theorem, but I still can't solve it. Please help...
KingWangZZang's user avatar
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How to write $\sqrt{-1}$ in the base $\sqrt{-2}$ [closed]

I I’m trying to make a system that would be able to display all numbers (real and nonreal) with one digit string. In theory, you could do this in base $\sqrt{-2}$, so I tried to write some examples by ...
meeeeeeeeeeeee's user avatar
4 votes
1 answer
53 views

Finding $\int_\Gamma \frac{z f'(z)}{f(z)} \, dz$ over a given contour [duplicate]

Let $f(z)=z^4-2z^3+2z^2-3z+60$ and let $\Gamma$ be the circle $|z|=5$. I want to find $$\int_\Gamma \frac{z f'(z)}{f(z)} \, dz$$ Supposing we had $f'(z)$ in the numerator instead of $z f'(z)$, this ...
Grigor Hakobyan's user avatar
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27 views

For an analytic function $f$ in a domain $D$, if $\log|f|$ is harmonic in a neighborhood of $\partial D$ then $f\in C(\bar D)$.

I have stuck in one place while reading a paper. If $\phi$ is an analytic function on $D$, where $D$ is bounded multiple connected domains in $\mathbb{C}$. Now given that $\log\lvert \phi\rvert$ is ...
Ravi's user avatar
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1 vote
1 answer
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Understanding definition of degree of a holomorphic function at a point p

I am confused at the following notation: Let $f$ be non constant meromorphic function in a domain $U$ and $D$ a open set whose closure is a compact subset of $U$. Let $q\in \hat{\mathbb{C}}$, we set $$...
Remu X's user avatar
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1 answer
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Freitag-Busam proof of $\sum\limits_{\omega \in L \setminus \{0\}} |\omega|^{-s}$ converges when $s>2$.

One of the main results allowing the possibility of introducing the Weierstrass $\wp$ function is the following: $\sum\limits_{\omega \in L \setminus \{0\}} |\omega|^{-s}$ converges when $s>2$. I'...
TheWanderer's user avatar
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Inverse Laplace Transform of $e^{-as}/s^2$ for $a>0$

I am trying to compute the inverse Laplace transform of $$F(s) = \frac{e^{-as}}{s^2}$$ for $a > 0$. I computed it as follows: $$\mathcal{L}^{-1}_{s\to t} \left\{\frac{e^{-as}}{s^2}\right\} = \text{...
idk31909310's user avatar
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Find all elements of Automorphism on a simply connected domain

Find a conformal map from $U = \mathbb{D}_1(0)−[0, 1)$ to the upper half plane, and use it to find all elements of Aut$(U)$ that fix the point $(2\sqrt{2} − 3) \in U$ My approach: First we take a $\...
Remu X's user avatar
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-1 votes
0 answers
43 views

Can somebody please tell me about good resources on Hyperelliptic function and hyperbolic function? I need that for a project. [closed]

If possible please give me a pdf of the book or lecture notes on that topic...
Sayantika Bose's user avatar
1 vote
2 answers
48 views

Scalar complex holomorphic function - same derivative as scalar real function?

My question is about the derivative of holomorphic complex functions. Assume there is a function $f(x) := \mathbb{R} \rightarrow \mathbb{R}$ , and a function $f(z) := \mathbb{C} \rightarrow \mathbb{C}$...
Bastian's user avatar
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0 votes
1 answer
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How can I show that $f:\mathbb{C} \to \mathbb{C}$ is bounded? [duplicate]

at the moment I am working on the following exercises: Let $v,w \in \mathbb{R}^2 = \mathbb{C}$ be two linear independent $\mathbb{R}$-vector. Now let be $f:\mathbb{C} \to \mathbb{C}$ holomorphic and ...
WomBud's user avatar
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1 vote
1 answer
74 views

Showing the identity log($z^n$) = n log (z) for a particular value of n and z

I was asked to show $\log(z^n) = n \log (z)$ where $z = 1 + i$ and $n = 5$. The worked solutions state that they are not equal for those values but I do not understand why given that we find the ...
Oofy2000's user avatar
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-3 votes
1 answer
32 views

Is $|1-e^w| \leq c|w|$ true? [closed]

Let $w$ be a complex number. Is it true that if $|w| \leq 1$ , $|1-e^w| \leq c|w|$ for some constant $c$?
SunnyMath's user avatar
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2 votes
0 answers
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Application of open mapping theorem in complex analysis?

I recently came along the following question. Let $\Omega\subseteq\mathbb{C}$ be a open connected set and $f\in\mathrm{H}(\Omega)$, i.e., $f$ is holomorphic in $\Omega$. Assume that $f^2(z)=\overline{...
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0 answers
23 views

How to show a complex-valued function defined by a series converges on complex plane [closed]

I studied about Weierstrass $P$-function (a.k.a Weierstrass elliptic function). In order to prove the fact that this function converges to a meromorphic function on the whole complex plane, my ...
PKS's user avatar
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2 votes
0 answers
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A limit related to Poisson kernel in unit disc

I am trying to show $$\lim_{r\to1^-}\int_0^{2\pi}\exp\left(\frac{1-r^2}{1-2r\cos\theta+r^2}\right)d\theta=+\infty.$$ An approach is to apply the theory of Hardy space. Indeed, the function $g(z)=\exp\...
FFGG's user avatar
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1 vote
1 answer
78 views

Fourier transform of incomplete gamma function

Ultimately I am interested in the Fourier transform of $$ e^{-i\zeta}(-i\zeta)^{-2\epsilon}\Gamma(2\epsilon,-i\zeta) $$ in a series expansion around $\epsilon=0$, so to first order in $$ \lim_{\...
Tobias's user avatar
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-4 votes
0 answers
71 views

How does the Schrödinger equation on compact Riemann surfaces under SL(2, R) action impact? [closed]

In our study, we delve into the closure of orbits and the uniform distribution of points within the context of the SL(2,R) action on the moduli space of compact Riemann surfaces, akin to unipotent ...
amir mohammadi's user avatar
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0 answers
12 views

Lower&upper bound $\alpha_1\|x_1\|_{\infty}+\alpha_2\|x_2\|_{\infty} \leq \|x_1+x_2\|_{\infty}\leq\beta_1\| x_1\|_{\infty}+\beta_2\| x_2\|_{\infty}$?

If $x_1, x_2 \in \mathbb{C}^n$, then is there any lower and upper bound for $\|x_1 + x_2\|_{\infty}$, where $\| x\|_{\infty} := \max_{i=1,\ldots,n} |x_i| $? More specifically, I am wondering, is there ...
learning's user avatar
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0 answers
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Singularities and their nature

What are the singularities of $f(z) = e^{\frac{\sin z}{z}}$ ? It clearly has a removable singularity at $0$. A textbook says that it has essential singularities at $kπ$ for $k \in \mathbb{Z}$. How and ...
Anonymous's user avatar
0 votes
1 answer
52 views

A question of complex numbers involving inequality

$\begin {aligned}|a(z_2-z_3)+(z_3-z_1)|&\ge |a|\, |z_2-z_3|+|z_3-z_1|\text{(by triangle inequality)}\\ &\ge 2\sqrt a\cdot \sqrt{|z_2-z_3|\cdot|z_3-z_1|} \text{(by AM-GM inequality)} \end{...
user1318878's user avatar
0 votes
1 answer
69 views

How to show $\log(z) = \log(r) + i \theta$ without implicitly assuming $z = r \exp (i \theta)$ - from Penrose Road to Reality

In Roger Penrose's book Road to Reality - Chapter 5 - he goes to great lengths to arrive at the standard polar expression for a complex number $w = r e^{i \theta}$ via a discussion of complex ...
a_former_scientist's user avatar
3 votes
1 answer
58 views

Radon Nikodym derivative and distribution function

Let $\mathfrak{B}$ be the Borel $\sigma$-algebra over $\mathbb{R}$ and $\beta$ the Borel-Lebesgue measure over $\mathfrak{B}$. Let $\mu$ be a Borel measure over $\mathbb{R}$ s.t. the distribution ...
Kham Bodrogi's user avatar
4 votes
1 answer
97 views

Analytic $f: \mathbb{D} \to \mathbb{D}$, $f(0)=0$, and $f$ has five zeros in $\overline{\frac{1}{2}\mathbb{D}}$

Suppose $f: \mathbb{D} \to \mathbb{D}$ is a holomorphic function and $f(0)=0$. The function $f$ has a total of five zeros (counting multiplicities) in the closed half-disc $\overline{\frac{1}{2}\...
Grigor Hakobyan's user avatar
0 votes
1 answer
45 views

Integral of a multivalued function

I need to calculate this integral using complex analysis methods $$ \int_{0}^{1}\frac{\,\sqrt[4]{\,{x^{3}\left(1 - x\right)}\,}\,}{x + 1}\,{\rm d}x $$ I encountered a problem: I received an inadequate ...
  Alina Gabriel's user avatar
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0 answers
7 views

Which of the following function does not have a Laurent series around the point z=0 with a non-zero region of convergence around z=0. [closed]

(i) Which of the following function does not have a Laurent series around the point z=0 with a non-zero region of convergence around z=0. $tan(1z)$ $e^{(-iz)^{-2}}$ $1z(z-1)$ $log(1+z)$ Contour
Miss M's user avatar
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1 vote
0 answers
23 views

Construction of a function subject to conditions

Is it possible to construct $f \in H(B(0,1))$ such that $f(1/n) = z_n$, where: $z_n = (-1)^n$ $z_n = \frac{n}{n+1}$ $z_n = 0$ if $n$ is even and $z_n = \frac{1}{n}$ when $n$ is odd? Is the following ...
A. Random's user avatar
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0 votes
0 answers
16 views

Arithmetic Mean of a Harmonic Function over concentric circles

I'm going through Ahlfors Complex Analysis. I'm trying to understand theorem 20 in the book. Basically, given that in spherical coordinates the conjugate differential is $^*du=r(\partial u/\partial ...
Redcrazyguy's user avatar
1 vote
0 answers
41 views

Singularities of a complex function with exponential components

I'm working on trying to classify all singularities and find their residues in the following function in $\overline{\mathbb{C}}$: \begin{equation} f(z)= \frac{1}{e^{z^2}-e^{4z-4}} \end{equation} I ...
Febrero's user avatar
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6 votes
0 answers
55 views

Anything interesting known about this generalization of even and odd functions?

Let $n \in \mathbb N$. Let's say a complex function $f: U \rightarrow \mathbb C$ is "of type $k \pmod n$" if for one (and hence every) primitive $n$-th root of unity $\omega$, $$f(\omega z) =...
Torsten Schoeneberg's user avatar
1 vote
0 answers
22 views

Using the Singular Inverstion Theorem to count the number of $r$-ary trees

For $r \ge 2$ let $\mathcal{C}_r$ denote the class of $r$-ary trees, i.e. Cayley trees in which every vertex has at most $r$ children. We denote the EGF of $\mathcal{C}_r$ by $C_r(z)$. Show that the ...
3nondatur's user avatar
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