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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22 views

Homotopy between circle and semi-circle.

It is easy to see that a circle and a semi-circle (as in the image), of the same radius, are homotopic. I was trying to exhibit a homotopy, but it seems that I'll have to use analytic geometry, solve ...
2
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0answers
25 views

Finding the number of zeros in the right half plane of $\mathbb{C}$. [duplicate]

I am attempting to learn an exercise in the chapter on Rouche's Theorem in Ahlfors Complex Analysis. It comes from page 154, number 3: Find the number of zeros of $$f(z) = z^4 + 8z^3 + 3z^2 + 8z + ...
2
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1answer
30 views

Evaluate the complex integral $\int_{C_R}\frac{z^3}{(z-1)(z-4)^2}$

Let $C_R$ be the positively oriented circle with centre $3i$ and radius $R > 0$. Use the Cauchy Residue Theorem to evaluate the integral $$\int_{C_R}\frac{z^3}{(z-1)(z-4)^2}$$ Your answer ...
2
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1answer
54 views

Weird (?): Use Cauchy's Integral formula to calculate $\int _{|z|=3} \frac{z}{z^2-\pi^2}dz$

Weird (?): Use Cauchy's Integral formula to calculate $\large \int _{|z|=3} \frac{z}{z^2-\pi^2}dz$. But the function $\frac{z}{z^2-\pi^2}$ is holomorphic on all of $|z|=3$. Am I missing something? ...
2
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1answer
30 views

Harmonic Function with linear growth

We want to find harmonic functions $w$ in (say) $\mathbb{R}^2$ that are zero on $\{y=0\}$ with the linear growth bound \begin{equation} \sup_{\mathbb{B}_R} |w| \leq C(1+R) \end{equation} where ...
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2answers
15 views

Solution verification: Integral of the conjugate equals the conjugate of the integral

For all $f(z)$ holomorphic in all $\mathbb C$ and every smooth curve $\gamma$, is it correct that: $\overline {\int_\gamma f(z)dz}=\int_\gamma \overline { f(z)}dz$ My solution: If $f$ is holomorphic ...
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1answer
24 views

Taking the complex conjugate of some complicated composite function

I'm aware of the rule where to take the complex conjugate of anything, you simply replace any $i$'s with $-i$, and to conjugate any composed functions (i.e. $f*(g(z))=f(g*(z)))$ What is the ...
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1answer
34 views

Calculating integral using Cauchy integral formula in two variables

I want to compute the integral: $\iint_{\partial_0P}\frac{1}{1-4zw}dzdw$ (or any similar integral) using Cauchy integral formula for two complex variables over polydiscs. The distinguished boundary is ...
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1answer
33 views

Given a complex function in $x$ and $y$ change into $z$, where $z=x+iy$

I have $$f(x,y)=\sinh{x}\sin{y}+e^y\sin{x}+i(e^y\cos{x}-\cosh{x}\cos{y}+C)$$ for $C$ a constant. Now if I want to write this in terms of the complex variable $z$ I've heard that I can just set $x=z$ ...
2
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1answer
27 views

Basic question about Beltrami differentials

my question must be terribly naive, but I'm stuck right now and don't know how to proceed.. Let $X,Y$ be Riemann surfaces and $f:X\rightarrow Y$ an orientation preserving diffeomorphism. It is known ...
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2answers
39 views

why we can't find function satisfying this conditions?

why we can't find function satisfying this conditions? In fact >>I don't know how to start solving
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0answers
20 views

Where does $\cos z$ conjugate is holomorphic

I solved this question using Cauchy-Riemann equations and got a contradiction,meaning not analytic everywhere, but I am not sure I am right. Got from 1 equation $x=\pi k$ from the other $x=\pi/2+\pi ...
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1answer
30 views

Prove that $f(z)$ is of the $Ce^{z}$

Let $f(z)$ be an entire function, such that : $|f(z)|\geq e^{\operatorname{Re}(z)}$. Prove that there is a constant $C$ such that $f(z)=Ce^{z}$. I think I need to work with Liouville's theorem and ...
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1answer
42 views

Derivative of complex root

Suppose we fix the branch definided in $\mathbb{C}-(\infty,0]$ and that satisfies $f(1)=-i $ for $f(z)=z^{1/4}$. We are asked to compute $f'(e^{i 2\pi/3})$. The derivative es $f'(z)=\frac{1}{4} ...
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2answers
26 views

Integrate a sum (geometric series) round |z| = 1

This is a question from a text book (Saff and Snider, Complex analysis for mathmatics science and engeneering, page 203). Let $$ f(z) = \sum_{k=0}^\infty (k^3/3^k)z^k $$ Evaluate $$ ...
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1answer
23 views

Evaluate the integral using CIF

Evaluate $$ I=\int_C \frac{dz}{z(z-1)(z-2)}$$ where $C = \{z \in \mathbb C: |z| = r\}, r \gt 0$. I have splitted the integrals using partial fractions and applied Cauchys integral formula and found ...
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0answers
23 views

Obtain the Laurent Series expansion of these two complex functions.

I have obtained the Laurent Series expansion for the first one. I am stuck on the answer for the second one as it doesn't seem as straight forward method wise. Now, I know that $e^z$ can be ...
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2answers
34 views

Singularities, essential singularities, poles, simple poles

Could someone possible explain the differences between each of these; Singularities, essential singularities, poles, simple poles. I understand the concept and how to use them in order to work out ...
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2answers
31 views

Radius of convergence of a complex power series question

I know the general idea for ROC (radius of convergence) is to use the ratio test, or lim sup, however how would i go about solving this?
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2answers
107 views

Integration of $\dfrac{x}{\sinh x}~dx$ from $-\infty$ to $\infty$

Problem from set on recidues: Evaluate $$\int_0^\infty \frac{\log x}{(x-1) \sqrt{x}}~dx.$$ After the substitution $x = e^u$ and easy computations. The integral becomes $$\int_{-\infty}^\infty ...
4
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1answer
45 views

Proof of Cauchy Schwartz inquality from Terry Tao's notes doubt

Hi I was reading Tao's proof Cauchy Schwartz by exploiting certain inherent symmetries and making some transformations. He says that first we use the fact that the norm is positive ie. ...
2
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1answer
48 views

How i can find the fourier transform of $\frac{\sinh(ax)}{\sinh(\pi x)}$ where,$ |a| < \pi$

Using a rectangular contour in the complex plane, bypassing the poles at $z=0$ and $z=i$, i got $$\int_{-\infty}^{+\infty}\frac{\sinh(ax)e^{ikx}}{\sinh(\pi x)}dx - ...
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1answer
28 views

Essentially self adjoint operator

Given a linear operator a self adjoint operator A
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1answer
64 views

Are functions with singularities, but no poles, manipulable by Residue Calclus?

Consider these two functions $$\int_0^{\infty} {{1} \over {(x^2+a)^{3/ 2}}} \ dx$$ And $$\int_0^{\infty} {{1} \over {x^2+\cos(x)}} \ dx$$ They have singularities, however, wolfram alpha says that ...
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0answers
53 views

3D surface intersections

I tried to look at 3D Hypersurface intersections of 4D this way based on four Mathematica (circular) trigonometric parametrization combination selections. No hyperbolic functions are directly ...
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1answer
29 views

Proving that $\inf_{\nu}\varphi_{\nu}$ is subharmonic

Let $\Omega\subseteq\Bbb C$ open, $\varphi:\Omega\to[-\infty,+\infty[$ is subharmonic on $\Omega$ if $\varphi$ is uppersemicontinous on $\Omega$ For all $K\Subset\Omega$ compact, and for all ...
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3answers
57 views

Let $f:U \to V$ be a bijective holomorphic function. Show that inverse of $f$ is also holomorphic.

Suppose $U$ and $V$ be domains(i.e., open and connected) in $ \mathbb C$.Let $f\colon U \to V$ be a bijective holomorphic function. Show that the inverse of $f$ is also holomorphic. By Open Mapping ...
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2answers
45 views

How to integrate this integral using Cauchy? [closed]

How can i find Solution use Cauchy Integrate? \begin{align*} \int_{|z-1|=1}\frac{1}{z^3-1}dz \end{align*}
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votes
1answer
48 views

Function never equal to 0

Let $p,q,x,y:(a,b)\to\mathbb{R}$ be $C^1 ((a,b))$ functions. Knowing that for $(u,v), (\tilde{u},\tilde{v})\in (a,b)\times (0,1)$ we have that: $$\begin{cases} ...
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3answers
121 views

antiderivative of $\frac{1}{z(z-1)}$, complex logarithm

I have the domain $\mathbb{C} \backslash [0,1]$ and want to show that $$\int_\gamma \frac{1}{z(z-1)}dz = 0$$ for all closed curves $\gamma$. I want to accomplish this by explicitly finding an ...
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2answers
34 views

how to find integral of z conjugate

please >>how to find the integral of z conjugate where the path is determined as given here>> I saw that we can write z=x+iy >>but how to continue?
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1answer
38 views

Need a hint for convergence of alternating series

So I have this following sum $1-\frac{\pi^2}{2!} + \frac{\pi^4}{4!} - \frac{\pi^6}{6!} + \space ... \space $ Now obviously this is equal to the following infinite sum $$\sum_{n=0}^\infty (-1)^n ...
0
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1answer
22 views

$n(\gamma_1,0)=n(\gamma_2,0)$ iff $\int_{\gamma_1}f(z)dz=\int_{\gamma_2}f(z)dz$

Use Laurent decomposition to prove the next equivalence, assuming that $\gamma_1,\gamma_2$ are closed curves such that they live in the annulus $r<|z|<R$: ...
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2answers
17 views

An upper semicontinous function which is not subharmonic.

Let $\Delta\subset\Bbb C$ be the open unitary disk. Let $\varphi:\Delta\to\Bbb R$ defined as follows: $\varphi(z)=1$ if $\Re z\ge0$, $\varphi(z)=0$ otherwise. So $\varphi$ is upper semicontinous. In ...
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1answer
22 views

Using Picards theorem to find unique interval

Consider the initial value problem: $\frac{dy}{dx}$= $xy - x^2 + 1$ with $y(0) = 0$ In order the find the unique interval we first find that $f(x,y)$ and $f_y$ are continuous in the rectangle: ...
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1answer
17 views

Show that if a Mobius transformation has 3 fixed points then it is the identity map.

I have that any non trivial Mobius transformation has at most 2 fixed points since f(z)-z=0 has at most 2 roots. But I cannot deduce why it must then be the identity.
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2answers
49 views

Residue theorem classical example - doubts

I'm having trouble following one of the steps in the classical example of the residue theorem presented in Wikipedia: http://en.wikipedia.org/wiki/Residue_theorem $$\int_{-\infty}^{+\infty} ...
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0answers
22 views

Plotting the images of complex functions?

I am new to complex analysis and its pretty hard for me to visualize complex functions. In particular I was trying to visualize the branches of the Lambert W function from complex $z$ plane to $w$ ...
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0answers
19 views

Does $\gamma$ closed in $\mathbb C \setminus \{0 \}$ imply that $\gamma$ is piece-wise differentiable?

Let $\gamma$ be a closed curve in $\mathbb C \setminus \{0 \}$. I want to calculate $$\omega(\gamma,0):=\frac 1 {2 \pi i} \int_{\gamma} \frac 1 {z-z_0} dz$$ As far as I know this number is only ...
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1answer
19 views

How do I find the set $U$ on which this series defines a holomorphic function?

I have just come across a question that asks me to find the set $U$ on which this series defines a holomorphic function. I have trawled through my notes but I can't find anything, any help on how I ...
3
votes
1answer
51 views

What is a branch point?

I am really struggling with the concept of a "branch point". I understand that, for example, if we take the $\log$ function, by going around $2\pi$ we arrive at a different value, so therefore it is a ...
2
votes
1answer
42 views

Finding holomorphic map on Riemann surface from a map between two Riemann surfaces

I have a non-constant degree two map between Riemann surfaces $R$ and $S$, $f: R \to S$. I'm trying to find a holomorphic homeomorphism $\tau: R \to R$ such that $f(\tau) = f$ and $\tau^2$ is the ...
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1answer
23 views

Please verify correctness of $H_2(e^{j\theta}) = \sum_{n=-\infty}^{\infty} h_2[n] \cdot e^{-j\theta}$

This frequency spectrum of a signal $h_2[n]$ is bugging me. I am not sure if what I've done here is correct. It's the sum \begin{align*} \sum_{n=-\infty}^{\infty}(-1)^{n-1} \end{align*} in ...
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1answer
39 views

Phase-spectrum: $arg(\cdot)$ function

I came to this frequency spectrum for a signal $h_1[n]$: \begin{align*} H_1(e^{j\theta}) &= \sum_{n=-\infty}^{\infty} h_1[n] \cdot e^{-j\theta} \\ &= \sum_{n=-\infty}^{\infty} (0.3 \cdot ...
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2answers
67 views

Holomorphic function definition. Am I missing something very obvious?

I'm reading a book of complex analysis in which the definition of holomorphic function is given as follows: Definition: If $V$ is an open set of complex numbers, a function $f:V \to \mathbb C$ is ...
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0answers
21 views

Sequence of functions converging uniformly to $\infty$

In an effort to demonstrate a particular family of functions is normal, I'm trying to show that if I have a sequence of functions $\{f_n\}$ that converges uniformly to the meromorphic function ...
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8 views

Showing that a function is bounded on a strip based on its value on the boundary.

Assume that $f$ is an entire function of finite order. Prove that if $| f ( z ) |≤ 1$ for all $z$ on the boundary of the horizontal half-strip $S = \{ z ∈ C : Re( z ) ≥ 0 , | Im( z ) |≤ 1 \}$ , then ...
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2answers
39 views

holomorphic function and simply connected region

I knew that if $f$ is holomorphic on region $A$ which is simply connected and $f$ has no zero in $A$, then there exists $g$ s.t. $g$ is holomorphic on $A$ and $f$ is the $n$th power of $g$ for some ...
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0answers
25 views

How to prove $f$ is outer, when $Re f$ >0?

This is an exercise problem, which can be found in most of the book which deals with Hardy Space. Any kind of suggestion or hint is also welcome. The question is following: Let $f$ be a function in ...
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0answers
21 views

Extrapolation: Richardson , Taylor

Does a first order approximation only use $f(x)$ and $f’(x) $ with $f’’(\theta)$ in the error? While second order would include$ f’’(x)$?