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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What is the order of the pole of $\frac{\mathrm{Log}(z)}{(z-1)^3}$ at $z=1$?

I read somewhere that the series for the principal branch of $$\mathrm{Log}(z) = \sum_{n=1}^{\infty}\frac{(-1)^{(n+1)}(z-1)^n}{n}$$ If this is true does it means that the order of the pole is equal to ...
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1answer
14 views

Where does the sequence converge pointwise? Does it converge uniformly on this domain? (example)

I'm trying to learn about sequences of functions, which is a new concept to me, and I would like to have a simple example to go with what I already know from definitions. Unfortunately the notes that ...
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20 views

Proving that two branch cuts can cancel out

Define the following functions $\mathbb{C}\to\mathbb{C}:$ $$u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z}\quad \left[-\pi\leqslant\arg \left(z+\tfrac12\right)<\pi\right];\quad v(z)=\frac{\log ...
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1answer
27 views

What shapes, with boundary collapsed to a point, are homeomorphic to $S^n$?

Consider the following construction: Given a set $A\subseteq\Bbb R^n$, form the quotient space $A/\sim$ which identifies all the points on the boundary $\partial A$ (w.r.t $\Bbb R^n$). For which ...
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17 views

When can we take the integral limit from and to infinity?

I'm reading Conway's complex analysis book and on page 115 he writes this: I didn't understand why $\frac{x^2}{1+x^4}\ge 0$ implies this limit is true. What are the conditions to allow us to take ...
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13 views

Any bounded region $G \subseteq \mathbb{C}$ with transitive automorphism group and sufficiently “smooth” edges is biholomorphic to the unit ball

Let $G$ be a bounded region in $\mathbb{C}$ (i.e. we have $G ≠ \emptyset$, and $G$ is open and connected), and let $G$ have a transitive automorphism group (that is, for each two points $z_1, z_2 \in ...
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20 views

Tori conformally equivalent

Let $L_1,L_2$ be the lattices generated by $\{1,\tau_1\},\{1,\tau_2\}$ respectively and $X_1=\mathbb C / L_1, X_2= \mathbb C / L_2 $ the corresponding complex tori. We look to prove that $X_1,X_2$ ...
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28 views

Compute $\int_{0}^{\infty} \frac{x^{1/3}}{(x^2 + 1)^2} dx$

So I want to compute $\int_{0}^{\infty} \frac{x^{1/3}}{(x^2 + 1)^2} dx$ using complex analysis, Cauchys theorem and the residue theorem. What I did was the following: define $g(z) = e^{1/3(\ln|z| + ...
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20 views

ODE $Aw'' + iBw' + cw=0$ with complex coefficients, how to solve?

I have the ODE for $w:[0,\infty) \to \mathbb{R}$ : $$Aw'' + iBw' + cw=0$$ where $A, B \in \mathbb{R}$ and $c \in \mathbb{C}$ is a complex number. There is a boundary condition involving $w_y$. Also ...
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13 views

Composition with polynomial/ same type of singularity

Let $f\in O(D_1(0){}-\{0\})$ and $ p $ a non constant polynomial. Then $f$ and $p(f)$ have the same type of singularity at $z_o=0 $. I think its fairtly easy to Show that if $f$ has a ...
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2answers
40 views

Meromorphic, analytic, holomorphic and all that

I must have slept through something in my complex variables course, because all my life I have used the terms holomorphic, meromorphic, and analytic somewhat interchangeably. These are all also ...
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1answer
61 views

Proving that $\lim_{n \to \infty} (1+ \frac{i}{n})^n = e^i$

Prove that $\displaystyle \lim_{n \to \infty} (1+ \frac{i}{n})^n = e^i$ I was trying to proof in the same way of $\lim (1 + \frac{1}{n})^n = e$, but I couldn't proceed this way. Can someone give me a ...
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9 views

Any convex Reinhardt domain is logarithmically convex

I have the following question in Shabat p.59: Prove that any convex Reinhardt domain is logarithmically convex. I think I have a good idea about how to show this, but need to be clear on the ...
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1answer
24 views

classfiying singularities

\begin{equation} h(z)=\frac{z^2e^{\frac{1}{z^2+1}}}{\sin(z^2)} \end{equation} It seems to me the function has essential singularity at $z=\mp i$ It is clear that $e^{\frac{1}{z}}$ has essential ...
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17 views

Continuity of a function of two complex variables

Let $\Omega$ be a domain and a continuous function $f: \overline{\Omega}\rightarrow\mathbb{C}$ which is holomorphic on $\Omega$ and $f'$ extends continuously on $\overline{\Omega}$. Is the function ...
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16 views

Spherical Harmonics and $L_+$ and $L_-$ operators

I have the spherical harmonics $Y_{m}^{l}\left(\theta,\varphi\right)$ and I want to show that the operators $L^{\pm}$ act as "creation" and "annihilation" operators such that ...
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1answer
17 views

Residue of a non-identically zero function

Assume f(z)is analytic in the complex plane and let f be a complex function which is not identically zero.Then,show that Res(1/f(z^3),0)=0. I know that the residue is calculating for only ...
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1answer
23 views

On $\zeta(s)=h^2(s)$ when we presume that the Riemann zeta function has no zeros for $\Re s>\frac{1}{2}$

By specialization of a theorem from complex analysis, one has that on assumption that the Riemann zeta function $\zeta(s)$ has no zeros with $\Re s>\frac{1}{2}$, then there exists an analytic ...
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26 views

Laurent Series, How it is done

Suppose that a series $$\sum_{n=-\infty}^{\infty}x[n]z^{-n}$$ converges to analytic function $X(z)$ in some annulus $R_1<|z|<R_2$. That sum $X(z)$ is called the z-transform of $x[n]$ $(n=0,\mp ...
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1answer
31 views

How can I prove that it is an Entire Function

Prove that if $$ f(z)=\left\{ \begin{array}{ll} \frac{\cos z}{z^2-(\pi /2)^2} & \hbox{when} \; z\neq \mp \pi/2\\ -\frac{1}{\pi}, & \hbox{when} \;z= \pi/2. \end{array} \right. $$ ...
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33 views

Challanging problems on [Grade-12]Complex Number [on hold]

recently we are introduced to interesting world of complex number but except for 3-5 problems in the my books,all the problems are just plug-and chug,expression manipulation,etc.. which bores me out ...
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26 views

Power series representation with Gamma function

This is taken from Stein and Shakarchi's Complex Analysis (Chapter 6, Exercise 4): Prove that if we take $$f(z) = \frac{1}{(1-z)^\alpha}$$ for $|z|<1$ (defined in terms of the principal branch ...
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27 views

Non-trivial inverse Laplace transform

I'm trying to compute the inverse Laplace transform of $f(s) = s^c/(N + s^{ir} )$ where $c,N \in \mathbb{C}$ and $r \in \mathbb{R}^+$ using the Bromwich integral $$ F(t) = \frac{1}{2 \pi i} \int_{- ...
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4answers
57 views

How do i write the analytic function $f(z)$ in terms of $z$?

I have an entire function, consider the function : $f(z)= (3x^2 + 2x - 3y^2 - 1) + i(6xy + 2y)$ I want to write $f(z)$ in terms of $z$.
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2answers
37 views

Contour integration with logarithms

I'm having trouble calculating the below integral to get the right answer: $$\frac{1}{2\pi i}\int_\gamma \frac{3}{z-2}\; dz$$ where $\gamma$ is parametrised by $\gamma(t)=3e^{it}, t\in [0,2\pi]$. So ...
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34 views

$|p(z)|=1$ contains no circles [duplicate]

Help with the following problem would be appreciated: Let $p(z)$ be a polynomial over $\mathbb{C}$ with at least two distinct roots. Prove that no connected component of the set $\{z \in \mathbb{C} : ...
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20 views

About the proof of a corollary of Arzela-Ascoli Theorem.

This is from Scheidemann, Complex Analysis. Theorem (Arzela-Ascoli): Let $K$ be a compact separable metric space, $E$ a finite-dimensional Banach space and $(f_j)_{j\in\mathbb{N}}\subseteq C(K,E)$ ...
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46 views

Is this a legitimate way to compute a contour integral?

I wish to calculate $$\int_{\Gamma}\cos(z)\sin(z)~\text{d}z$$ where $\Gamma$ is the line segment given by $\gamma(t)=\pi t+(1-t)i$ for $0\leq t \leq 1$. Here is what I did: We have that $$\int ...
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1answer
62 views

Laurent series of $\frac{e^z}{z^2+1}$

I cant figure out the laurent series of the following function. Let $f(z)= \frac{e^z}{z^2+1} $ and $|z|\gt 1$ $$\frac{1}{z^2+1}=\sum_{n=0}^{\infty}(-1)^nz^{-2n-2}$$ and $$e^z = ...
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1answer
60 views

Entire function $f$ such that $\lim\limits_{z\rightarrow \infty}f(z)=0$ and $f(0)=1$?

The question is this: Does there exist an entire function $f$ such that $\lim_{z\rightarrow \infty}f(z)=0$ and $f(0)=1$. I immediately would point to $f(z)=e^{-z}$. It is entire and satisfies the ...
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1answer
24 views

How to show $f(z)=x^2+y^2+i2xy$ is differentiable at $z_0=x_0+i0$?

How to show $f(z)=x^2+y^2+i2xy$ is differentiable at $z=x_0+i0$? Here is what I have done we know by the Cauchy Riemann (its it very easy to verify) that these can only hold for $z_0=x_0+i0$ that is ...
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32 views

Radius of convergence of the series $\sum\limits_{-\infty}^{\infty}(2^{-n}+4^{-n}) z^n$

I'm trying to find for what values of $z\in\mathbb{C}$ the series $$\sum_{n=-\infty}^{\infty}(2^{-n}+4^{-n})z^n$$ converges. My main methods are the nth root test and the ratio test. I believe it can ...
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0answers
27 views

When finding Laurent series when to use partial fractions?

When finding the Laurent series of $$f(z):=\frac{1}{z(z-1)(z-2)}$$ valid in the region $1<|z-2|<2$ for example do we just use partial fractions to break $f(z)$ up and the just find the Laurent ...
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1answer
20 views

Evaluating an integral using Cauchy Integral Formula and a further application

Question: $i)$ Evaluate $$\int_{\gamma}\frac{e^{2z}}{z}dz$$ Where $\gamma=${$z\in \Bbb{C}: \lvert z\rvert$=1} $ii)$ Hence find $$\int_{0}^{2\pi}{e^{2\cos(t)}}.\cos(2\sin(t) dt$$ My attempt: $i)$ ...
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48 views

Holomorphic function with $f(z)^2=z$

Is there an holomorphic function $f:B_1(0)\setminus\{0\}\rightarrow\mathbb{C}$ with $f(z)^2=z$?
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1answer
30 views

Help in this inequality in Conway's complex analysis book

I'm reading Conway's complex analysis book and on page 118 he write the following inequality: Why is this inequality true?
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1answer
23 views

Why $|\alpha|\lt 1$ and $|\beta| \gt 1$?

I'm reading Conway's complex analysis book and on page 117 he writes: I didn't understand why $|\alpha|\lt 1$ and $|\beta| \gt 1$. I could only prove $\beta\lt -1$.
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1answer
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disk of convergence for complex-valued series

Find the disk of convergence of $\displaystyle \sum_{k=0}^{\infty} \frac{(z+2)^k}{(k+2)^3 4^{k+1}}$, where $z \in \mathbb{C}$. I tried applying the ratio test: $\lim_{k \to \infty} \left| ...
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1answer
54 views

Can $\int_0^1 \frac{1}{x} e^{-x} dx$ be integrated?

I have an integral with a singularity at $x = 0$. $$\int_0^1 \frac{1}{x} e^{-x} dx$$ It's not a removable singularity so is it possible to perform the integration? For example could some complex ...
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2answers
28 views

Geometrical Description of $ \arg\left(\frac{z+1+i}{z-1-i} \right) = \pm \frac{\pi}{2} $

The question is in an Argand Diagram, $P$ is a point represented by the complex number. Give a geometrical description of the locus of $P$ as $z$ satisfies the equation: $$ ...
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1answer
19 views

Recommendations for tutorials specifically devoted to real integration using contour integral techniques.

Complex analysis, and in particular contour integrals and the residue theory have proved a very powerful tool in computing a large class of real function integrals which would be quite troublesome to ...
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22 views

Series expansion of complex exponential

Prove that $e^z= \displaystyle \sum_{k=0}^{\infty} \frac{z^k}{k}$. I took the taylor series $f(z)=\displaystyle \sum_k \frac{f^{(k)}(z_0)}{k!}(z-z_0)^k$ centered at $z_0=0$ and obtained $$e^z= ...
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2answers
96 views

Integral of $e^{\cos t}$

I’d like help with computing the following integral: $$\int_0^\pi e^{\cos t}\,dt.$$ (This is a problem in complex analysis [supposedly].)
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1answer
29 views

Absolute converge of real and complex parts of a series

If the real and imaginary parts of a complex series converge absolutely, then the complex series converges absolutely. Is this true? If we write our complex series $\sum_{k=0}^{\infty} b_k = ...
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1answer
40 views

How to find the Laurent series of $\frac{1}{z^4(1-z)^2}$ for |z|>1?

A hint is given that $$\frac{1}{(1-\frac{1}{z})^2} = \frac{z^2}{(1-z)^2}$$ and we know that $$\frac{1}{1-w} = \sum_{n=0}^{\infty} w^n$$ for $|w|<1$. I don't know how to make ...
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26 views

How to calculate this integral using Residues?

$$\int_{-\infty}^{\infty}\frac{\sin(x)}{x^2-2x+5}dx$$ I have calculated the Residue in the upper half plane to be $1/4i$ which is correct according to wolfram alpha but I am unsure on how to proceed I ...
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19 views

Question regarding the Cauchy Residue Theorem

Sorry about the vague title, I'm not quite sure how to word it. Any edits would be very helpful! Question: Let $f(z)$ be analytic with $f'(z_0)\ne0$ where $z_0$ is a complex number Define $$g(w)= ...
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15 views

calculating $\int_0^{\pi} \frac {d\theta} {(a+b\cos \theta)^2}$ using Residual Theorem [duplicate]

Could anyone help me provide a way to calculate $$ \int_0^{\pi} \frac {d\theta} {(a+b\cos \theta)^2} $$ using the Residue theorem in complex analysis? Many thanks
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2answers
23 views

First two terms of the Taylor series of the $n$-th iterated of a holomorpic function

Let $G$ be a region in $\mathbb{C}$ (i.e. $G ≠ \emptyset$ is simply connected and open), with $0 \in G$. Let $f: G \to G$ be a holomorphic function that's Taylor series (around $0$) has the shape $z + ...
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22 views

Name or reference about a inequality with integrals?

I have wrote down some class notes and I think I copied something wrong. It is an integral inequality; $$\iiint_{B^n}|\nabla\psi|^2\frac{1}{|x|^{n-2}}dV\leq C\iint_{\partial B^n}|\psi|^2dA$$ where ...