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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24 views

calculating curvilinear integral by residue theorem

Calculate the following integral by transposing to a curve integral and then using the residue theorem: $\displaystyle \int_{0}^{2\pi}{\frac{e^{int}}{C-e^{it}}dt}, \qquad |C|\ne1, n\in \mathbb N$. ...
3
votes
2answers
47 views

Non-constant holomorphic and bounded functions $f:\Omega_j\rightarrow\mathbb{C}$

Are there holomorphic, non-constant and bounded functions $$f:\Omega_j\rightarrow\mathbb{C}$$ with $\Omega_1=\mathbb{C}\setminus\{0\}$ $\Omega_2=\mathbb{C}\setminus[0,\infty)$? Since $\Omega_2$ ...
3
votes
1answer
70 views

Algebraic or Analytic Proof of a Polynomial Identity

Let $m$, $n$, and $r$ be integers with $0\leq r \leq \min\{m,n\}$. Define $$f_{m,n,r}(q):=\left(\prod_{j=1}^r\,\left(q^m-q^{j-1}\right)\right)\,\left(\sum_{\substack{{j_1,\ldots,j_r\in\mathbb{Z}_{\...
1
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1answer
37 views

Evaluating real integral by complex contour method

Please let me know where my mistake could be. I've verified the integral $$\int_{-\infty}^\infty \frac{dt}{(t^2+1)(t^2+4)}$$ to be equal to $\frac{\pi}{6}$ with a computer math system. However, I'm ...
1
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1answer
19 views

holomorphically convex hull

Let $U\subset\mathbb{C}$ be a domain. If $K=\{a_j:j\ge 1\}\cup \{a\}\subset U$ where $a_j\longrightarrow a$. How to prove that $$\widehat{K}_U=K$$ $\widehat{K}_U= \{z \in U: |f(z)| \leq \sup_K |f|, \...
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1answer
52 views

Analytic continuation of $\sum z^n/n$

I have the following function $$f(z)=\sum_{n=1}^\infty \frac{1}{n}z^n$$ It is easy to see that it converges for $|z|<1$ (root test, for example). How can it be analytically continuated beyond ...
2
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1answer
52 views

$f$ has no zeros in $\mathbb{D}$

Let $f$ be an holomorphic function, defined in a neighbourhood of $\overline{\mathbb{D}}$. Assume $f(\partial\mathbb{D})\in\mathbb{C}\setminus[0,\infty)$. Show that $f$ has no zeros in $\mathbb{D}$. ...
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1answer
19 views

How to remember Stolz Angle correctly

The Stolz angle is a condition used in Abel's Theorem: $$|1-z|\leq M(1-|z|)$$ Q1) How do I intuitively remember (and understand this)? Q2) In particular, is there a quick way to see that $$(1-|z|)...
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1answer
50 views

Is possible that an entire function satisfies this condition?

Let $f$ an entire function whose only zeros are all the negative integers. Is possible that $f$ satisfies $|f(z)|\leq C_1 e^{C_2 |z|^p},$ for some real constants $C_1, C_2,$ and $p<1$? Any help?
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0answers
27 views

Proof that if $z_0$ is a zero of order $k$ of $f$ then $\exists$ analytic $g$ saisfying this expression

Let $f:S\to \mathbb{C}$ be analytic, $z_0 \in S$ be of order $k$ of $f$. Prove that there exists an analytic function $g$ satisfying $$\frac{f'(z)}{f(z)}=\frac{k}{z-z_0}+g(z)$$ $\forall z$ in $\...
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2answers
23 views

holomorphic function $f:B(0,2)\rightarrow\mathbb{C}$ with $f(1/n)=f(-1/n)=1/n^3?$

Is there an holomorphic function $$f:B(0,2)\rightarrow\mathbb{C}$$ with $$f(1/n)=f(-1/n)=1/n^3?$$ I would guess no and I want to show this by using the identity theorem. How can I start?
4
votes
1answer
66 views

Laplace transforms of powers of cosine (solved!)

During the past several hours, while studying the Laplace transform, I've started wondering what \begin{equation} \mathcal{L} \{ \cos^n(at)\}(s) \end{equation} would look like – since it won't ...
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1answer
15 views

Stolz Angle $M$ range

I refer to Stolz angle (https://en.wikipedia.org/wiki/Abel%27s_theorem), the region of the open unit disk where $|1-z|\leq M(1-|z|)$ for some $M$. My question is, are there any conditions on $M$? Can ...
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0answers
16 views

The Klein-Gordon equation Green's function.

How would I go about solving the following integral? \begin{equation} G_\text{ret}(x-x')=i\theta(t-t')\int\frac{d^3\mathbf{p} }{(2\pi)^32E}\left[e^{-iE(t-t')+i\mathbf{p}\cdot(\mathbf{r}-\mathbf{r}')}-...
1
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1answer
55 views

Riemann Zeta function in polar form

The Riemann Zeta function is expressed in complex rectangular form generally. When it is expressed in polar form as seen in plot of Riemann Zeta function $ \zeta( \frac12 +i \,t) $ along critical ...
6
votes
6answers
504 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log z)^...
3
votes
4answers
421 views

Geometry with complex numbers.

Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle, such that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$. Find a simple expression for $d$ in ...
2
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1answer
517 views

Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
2
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1answer
57 views

Laurent series with $e^z$

I'm trying to find the Laurent series Expansion for $$ f(x) = \frac{e^z-(z-1)}{z-1} $$ on the annulus $0<|z|<\infty$. I'm aware that I am supposed to use substitution of known series. I am ...
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1answer
35 views

Cauchy-Riemann equations on $f(z)=\begin{cases}(z\overline{z}^{-1})^2&z\neq 0\\1&z=0\end{cases}$

Let $$f(z)=\begin{cases}(z\overline{z}^{-1})^2&z\neq 0\\1&z=0\end{cases}.$$ I need to show that the Cauchy-Riemann equations hold for $f$ in $0$ but $f$ is not (complex) differentiable in $0$....
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0answers
17 views

Convergence for $\sum_{\text{m composite}}\frac{e^{2\pi\sigma(m)i/(m+1)}}{m^s}$, where $\sigma(n)$ is the sum of divisors function

Let $\sigma(n)=\sum_{d\mid n}d$ the sum of divisors function. When one writes informally the identity $$\sum_{n=1}^\infty \frac{e^{\frac{2\pi\sigma(n)i}{n+1}}}{n^s}=1+\mathcal{P}(s)+\sum_{\text{m ...
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2answers
44 views

Evaluating $\int_0^{2\pi} e^{2i\theta} f(e^{i\theta})\,d\theta$ using Cauchy's Theorem

If $f(z)$ is analytic on the disk $|z| \leq 2$, evaluate $$\int_0^{2\pi} e^{2i\theta} f(e^{i\theta})\,d\theta.$$ This problem comes from Mathematical Methods in the Physical Sciences 2nd edition by ...
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1answer
21 views

It's possible to generalize the Ml inequality (also call Estimation Lemma)?

The ML inequality property in complex integral says $|\int_{c}f(z)dz| \leq ML$. If I have two function in the integral, I can write the inequality: $|\int_{c}f(z)g(z)dz| \leq ML|\int_{c}g(z)dz| $ ?. ...
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0answers
50 views

Residues and Cauchy Residue Theorem [closed]

I am working on the Cauchy Residue Formula. This is my function (I apologize I do not have the tools to write using latex right now) Find the integral of 1/(z^2*(z-1)(z+pii) on the path |z|=2 I ...
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1answer
70 views

What is the fundamental group of a modular curve $\mathcal{H}/\Gamma$?

Let $\Gamma$ be a finite index subgroup of $PSL_2(\mathbb{Z})$. What is the fundamental group of $\mathcal{H}/\Gamma$? By the Kurosh Subgroup theorem, $$\Gamma \cong F_n * C_2^{*r} * C_3^{*s}$$ ie, $...
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1answer
39 views

Condition on $(a_n)_{n \in \mathbf{N}}$ for Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ to converge for $Re(s) > 1$.

I'm curious about the conditions on a sequence $(a_n)_{n \in \mathbf{N}}$ of real numbers such that the Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ converges absolutely for $Re(s) > 1$. ...
0
votes
2answers
49 views

Laurent series of $z\mapsto\frac{z^4}{z^2-1}$ in $\infty$

I want to calculate the Laurent series and the main part in $\infty$ of $$f:z\mapsto\frac{z^4}{z^2-1}.$$ The Taylor series of $f(\frac{1}{z})$ in $0$ is $$\sum_{n=0}^{\infty}z^{2(n-1)}=1/z^2+1+z^2+z^...
4
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1answer
588 views

extension of Cauchy's Integral formula

This question is from Brown and Churchill's Complex Variables and Applications, 8ed., Section 52, Question 6. Let $f(s)$ denote a continuous function taken along a simple contour, $C$ enclosing a ...
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votes
0answers
34 views

limit of the argument function [closed]

Prove that: $$ \lim _{z\to i} Arg(z) = \frac{\pi}{2} $$ I tryed to prove it by definition but I did not succeed. Any suggestion? Thanks for helpers!
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1answer
76 views

Prove that $u$ is upper semicontinuous on $\Delta(0,\rho)$.

Let $u:\Delta(0,\rho)\rightarrow \mathbb{R}$ be a function such that $u(x+iy)$ is convex in $x$ for each fixed $y$, and convex in $y$ for each fixed $x$. Prove that $u$ is subharmonic on $\Delta(0,\...
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0answers
15 views

What's about $\sum_{n=1}^\infty\frac{\mu(n)}{n}f(X^{\frac{1}{n}})$, where $\mu(n)$ is the Möbius function?

Let $X=\sigma+it$ the complex variable, and $\mu(n)$ the Möbius function. Inspired in Riemann function $R(X)$ I would like to ask you Question. What conditions are required to be satisfied by a ...
3
votes
2answers
72 views

Approaching a contour integral with singularities on each axis

How do I solve an integral like this using complex methods? $$ \int_{0}^{\infty} \frac{\ln(x)}{\left(x^2 + 2\right)\left(x^2 + 1\right)}dx.$$ I tried using two semi circles in the upper half plane ...
6
votes
2answers
236 views

Complex numbers as exponents [duplicate]

Is there any formula to calculate $2^i$ for example? What about $x^z$? I was surfing through different pages and I couldn't seem to find a formula like de Moivre's with $z^x$.
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1answer
40 views

Antiderivative of holomorphic and bounded function

Let $f:D(0,1) \to \Bbb C$ holomorphic and bounded. Do the antiderivatives belong in the disc algebra? Disc algebra $= \{f| f: $holomorphic on $D(0,1), $continuous on the closed disc $\}$
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1answer
43 views

Which vectors are obtainable by my function?

Imagine a disc with $N$ radially displaceable masses $m_g$. A total imbalance with respect to the center of the disc can be calculated as follows (using the respective radiuses $r_1,...,r_N$): $$\...
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1answer
472 views

Show $ \coth $ is a conformal mapping of the horizontal strip

I want to show that $\coth=\frac{e^{2z}+1}{e^{2z}-1}$ is a conformal mapping of the horizontal strip $S=\{z\in C: \pi/4<\text{Im}(z)<3\pi/4\}$ onto the unit disc U, but I can't seem to get the ...
2
votes
3answers
72 views

$f:\mathbb{C}\rightarrow\mathbb{C}\setminus\{z\in\mathbb{C}:\Im(z)=0\text{ and }\Re(z)\ge 0\}$ is constant

I want to show that if $$f:\mathbb{C}\rightarrow\mathbb{C}\setminus\{z\in\mathbb{C}:\Im(z)=0\text{ and }\Re(z)\ge 0\}$$ is holomorphic, then f is constant. I tried to show that $f$ is bounded, then I ...
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1answer
61 views

Show that integral is analytic

Let $h:[0,\infty)$ be an integrable function. Prove that the function $$g(z)=\int_0^\infty h(t)e^{tz}\,dt$$ is analytic on $\{z=x+yi:x<0,y\in\mathbb{R}\}$. How do I start for this question? I ...
0
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1answer
17 views

Laurent Series Expansion

I need to find all possible series expansions with center at $c=0$ for $f(z) = \frac{1}{z^2(z-1)(z-2)} $ I know that I need to use the fact that $\frac{1}{1-z} = z^n$ for substitution and that the ...
-3
votes
1answer
33 views

Expansion of $\frac{\cos(z^2)-1}{z^2}$ about the origin [closed]

Expand the following functions about the origin and find the region of convergence in each case. $\frac{\cos(z^2)-1}{z^2}$ $\frac{e^z-z-1}{z^2}$ Please help with these functions, I'm ...
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0answers
32 views

complex analysis singularites

These are the directions that I have for my three functions below. Please help. Determine the type of each singularity of the given function, if the singularity is removable, define the function ...
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0answers
45 views

Proof that Harmonic Implies Conformal

How do I show that for some function $u$ that $$\Delta u = 0 \implies u \> \> \text{is analytic}$$ and assuming $u$ has non-vanishing derivative everywhere, how do I show $u$ is conformal? ...
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0answers
35 views

Surjectivity on the image of a annulus

I'm trying to prove the Fundamental Theorem of Algebra as it is done in Birkhoff and MacLane. Unfortunately, I don't have access to the book, only to a sketch. Therefore, I'm filling the gaps myself. ...
2
votes
1answer
71 views

Primitive of $\frac{1}{z-z_1}-\frac{1}{z-z_2}$ on an open set

Let $\Omega\subset\mathbb{C}$ be open and assume $z_1$ and $z_2$ belong to the same connected component of the complement of $\Omega$. First, prove that there exists a holomorphic function $f$ on $\...
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1answer
30 views

Determinant of complex matrix from real and complex parts

Is is possible to find the absolute value of the determinant of a complex matrix M, given two real matrices A & B of the same dimension of M; where for N = 2, M would be: $\begin{matrix} a_{00}...
0
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3answers
67 views

Laurent series of $\frac{1}{(z-1)(z-2)}$

How can I compute the Laurent series of $$f(z)=\frac{1}{(z-1)(z-2)}$$ on the circular ring $A(0;1,2)=B(0,2)\setminus \overline{B(0,1)}$? I tried to take the Cauchy product of geometric series but ...
3
votes
2answers
31 views

Singularities of $\frac{\cos(z)}{(z-\frac{i}{2})^2}$ in $\mathbb{C}\cup\{\infty\}$

What are the (types of) singularities of $$f(z)=\frac{\cos(z)}{(z-\frac{i}{2})^2}$$ defined on $\mathbb{C}\cup\{\infty\}$? I know that $i/2$ is a pole of order $2$. There are no other singularities ...
0
votes
0answers
9 views

Relationship between complex normal and bivariate normal distributions

Suppose I have a complex random variable $X$ which follows a complex normal distribution (with $0$ mean). I've been trying to represent the complex normal in a simpler way, but I'm not sure how. Is ...
1
vote
2answers
46 views

Calculate $\int_{\gamma}zdz$ by using Cauchy's integral formula

How can I calculate $$\int_{\gamma}zdz~\text{with }\gamma:[0,1]\rightarrow\mathbb{C},t\mapsto te^{2\pi i t}$$ by using Cauchy's integral formula? The line $\gamma$ isn't even closed. Has anyone a hint?...
-6
votes
2answers
71 views

Complex Analysis - Write in x+iy form

Write ($\frac{\sqrt{2}}{(i-1)})^{10}$ in $x+iy$ form. I'm completely stuck and cannot figure it out. Can someone please walk me through the solution?