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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23 views

Is the following complex function Conformal ? In which points?

So is the complex function $f: C -> C$ defined by $f(z) = e^z$ conformal? In which points? Correct me if I'm wrong but from what I know a complex function is conformal if it has a nonzero ...
2
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1answer
51 views

Showing $|f(z)| \leq |z|$ inside the unit ball, given some conditions?

Let $f$ be a holomorphic function inside the unit disc, so that $f(0) = 0$ and $|f(z)| \leq 1$ for $|z| = 1$. Show that $|f(z)| \leq |z|$ inside the unit ball. Does this follow immediately from the ...
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1answer
16 views

Determining whether functions could be the real part of a holomorphic function?

Which one of the following functions $u$ could be the real part of a holomorphic function $f(z)$ with $z = x + iy$? For those that could, find $f$. $u(x, y) = \frac{x}{y}$ $u(x, y) = xy$ My ...
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1answer
38 views

Proving that f is a constant on a nonempty open disc

If you start with $f: \Bbb C \to \Bbb C$, and we let it be entire. And we let $\Bbb D$ be a nonempty open disc on $\Bbb C$, such that $f(z)$ does not attain any values in the disc. Is it possible to ...
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64 views

Evaluating $\int_0^\infty e^{-x}\cos(x)dx$

By integrating over the contour around an appropriate sector, how does one solve $$\int_0^\infty e^{-x}\cos(x)dx$$
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1answer
20 views

Proving the range of $Log(z+i)$

I have the function $Log(z+i)$ and I am told the range is $$ R=\{x+iy : 0<y<\pi,\;\; x<\log(2\sin y)\} $$ How do I go about showing that this is true?
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1answer
11 views

Complex integral, absolute value of integrand

I want to integrate $f(z)=\frac{1-\mathrm{e}^{\mathrm{i}z}}{z^2}$ over the indented semicircle in the upper half-plane positioned on the $x$-axis as pictured below. The book (Complex Analysis by ...
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1answer
86 views

Determine the following integral (very difficult)

So we let $\sqrt{z}$ be the principal value square root of $z$ (i.e. with $\sqrt{1} = 1$ and branch cut along the negative real axis), also let $a \in \mathbb{R}^+$. Determine the following integral: ...
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2answers
47 views

Does $\frac{\overline{z}^2}{z}$ go to zero as z goes to 0?

I am trying to prove something and I need $\frac{\overline{z}^2}{z}$ go to zero as z tends to 0 to get the result. Is this true? z is a complex number Thanks
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1answer
24 views

Singularity and residue in z = 0 - Complex Analysis

Let's say we have the following two functions: f(z) = cos(1/z) * (z+1)^2 g(z) = 1/(1+1/z) For each function classify the singularity in z = 0 and determine the respective residues in z= 0. Can ...
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1answer
20 views

Mobius Tranformation

Show that two non intersecting circles can always be mapped by a suitable Mobius transformation to two concentric circles. I wanted to map the centre of the first circle to the centre of the second ...
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1answer
28 views

Complex integration with real integral

If $\gamma$ is unit circles $A(0,1)$ parameterization of one positive rotation and $a\in\mathbb{R}$, $0<a<1$. Show that $$ \int\limits_0^{2\pi} \frac{dt}{1+a^2-2a \cos t}=\oint\limits_{\gamma} ...
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2answers
23 views

Proof of Casorati-Weierstrass?

In Stein's Complex Analysis, he presents the following statement and proof of Casorati-Weierstrass: Suppose $f$ is holomorphic in the punctured disc $D_r(z_0) - \{z_0\}$ and has an essential ...
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0answers
15 views

Why are these triangles formed by the product of two complex numbers similar?

I was trying to understand Eulers formula from this link and I came across this image on the second slide: I'm trying to understand why the specified triangles are similar. One intutive ...
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1answer
19 views

Let $T$ be a mobius transformation such that $T(3i)=5$ and $T$ maps circle $\{|z-i|=4\}$ onto circle $\{|z-2|=2\}.$ Determine all values of $T(9i) $

Let $T$ be a mobius transformation such that $T(3i)=5$ and $T$ maps circle $\{|z-i|=4\}$ onto circle $\{|z-2|=2\}.$ Could anyone advise me how to find all possible values of $T(9i) \ ?$ A mobius ...
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0answers
62 views

Find the principal part of $\tan z + \tan(z-a)$

How can I find the principal part of $\tan z + \tan(z-a)$ ? Do I have to find the Laurent Expansion of each function, then do the sum and after that find the principal part? Finding the Laurent ...
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2answers
65 views

Deriving $\frac{d}{dz} = \frac{1}{2}(\frac{d}{dx}-i\frac{d}{dy})$ Intuitively

Without the usual math (i.e. the usual algebra you use to define these things), why should we want to define $$\frac{d}{dz} = \frac{1}{2}(\frac{d}{dx}-i\frac{d}{dy})$$ $$\frac{d}{d\bar{z}} = ...
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0answers
33 views

How to prove that this function is delta function when t $\to$ 0?

Let's have function $$ \tag 1 f(x, t) = \frac{1}{\sqrt{i\pi t}}e^{i\frac{x}{t}}. $$ How to prove that $(1)$ is delta-function when $t \to 0$? I have tried to use "usual" properties of delta-function, ...
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2answers
34 views

Harmonic functions and Holder's inequality

In the book Real and Complex Analysis by Rudin, it is given that by applying Holder's inequality to the $u(re^{i\theta})=\frac{1}{2\pi}\int_{-\pi}^\pi P_r(\theta-t)f(t)dt$ we get ...
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1answer
28 views

Conformal mapping of nonsimply connected domains

The question asks: Map the complement of the arc $|z|=1$, $y\geq 0$ on the outside of the unit circle so that the points at $\infty$ correspond to each other. How would you construct such conformal ...
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2answers
15 views

Why would the Möbius transform of a segment on a circle be perpendicular to the real axis?

Dear potential answerer, I am slightly confused with the following problem: Given the möbius transformation $w=\frac{z-a}{z-b}$ with $a,b \in \mathscr{C}$ determine the image of the circle that ...
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0answers
20 views

Inverse of the Stereographic projection $\mathbb{CP}^1 \to S^2$

I've some problems with this exercise: Consider the stereographic projection $$ \varphi \colon S^2 \setminus \{ (0,0,1)\} \to \mathbb{CP}^1\setminus \{[1:0]\}$$ $$ (x,y,z) \mapsto [ x+iy:1-z]$$ and ...
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40 views

Complex integration of exponential function

I am asked to find the integral of $z e^{z^2}$. I have applied the formula of multiplication but the factor of exp cannot be eliminated ofcourse. So how can i solve it. Sorry for such a basic question ...
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26 views

How can one derive Stokes lines of the Stokes phenomenon of asymptotics from a differential equation?

Is there a standard technique to calculate Stokes lines and anti-Stokes lines of the Stokes phenomenon of asymptotics for a function defined as the general solution to a differential equation without ...
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0answers
42 views

Analytic region of $f(z)^{1/n}$

Provided that f(z) is analytic, determine the region in which the complex function $f(z)^{1/n}$ is NOT analytic. A: consider the principal branch B: consider all branches I worked out this question ...
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2answers
43 views

Taylor Expansion of complex function $ \frac{1}{1 - z - z^2} $

The problem is to prove that complex function's Taylor expansion: $$ \frac{1}{1 - z - z^2}=\sum^{+\infty}_{n=0} c_nz^n$$ satisfying: $$c_{n+2}=c_{n+1}+c_n$$ for $n \geq 0$ I have tried to transform ...
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24 views

Domain of reciprocal of log in complex plane

Ok so what i already know is that the function f(z) = log(z) is undefined when z = negative or zero which is quite a basic concept. Can someone explain the mystery ...
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1answer
40 views

Find a Mobius transformation $f$ that maps $\mathbb{H}=\{z \in \mathbb{C}:Im(z) >0\}$ bijectively to ball $B(0,2)$ such that $f(i)=1, f(1)=-2 \ ?$

Could anyone advise me on this problem: Find a Mobius transformation $f$ that maps $\mathbb{H}=\{z \in \mathbb{C}:\text{Im}(z) >0\}$ bijectively to ball $B(0,2)$ such that $f(i)=1, f(1)=-2 \ ?$ ...
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0answers
25 views

A branch of logarithm $L$ such that $f(z)=L(z+i-2)$ is analytic at $-i$

How do I find a branch of logarithm such that $f(z)=L(z+i-2)$ is analytic at $-i$ and $f(-i)=log2+2\pi i$? If we take $g(z)=z+i-2$ this is analytic at $-i$. Also, $f(-i)=L(-2)=log2+2\pi i$. I do not ...
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0answers
21 views

Region and Domain

Is every Region a Domain? The definition of Domain is "An open and connected set". $(1,6) \cup (12,23) \cup {9,10}$ is this a domain? If yes then how? I think there is a problem at points 9 and 10 ...
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0answers
38 views

Complex Number 'inversion'

I was playing around with alternative ways of proving that if $\dfrac{iz-1}{z-i}$ is real, then $z$ lies on the unit circle, excluding $i$. However, I noticed something that I'm not sure is of any ...
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2answers
25 views

Simplification of a series so that it converges to a given function

I am trying to rearrange the series $ \frac{1}{1-z} - \frac{(1-a)z}{(1-z)^2} + \frac{(1-a)^2z^2}{(1-z)^3} - \cdots$ In such a way that I can show it converges to $\frac{1}{1-az} $ What I ...
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2answers
80 views

Abstracted Metric and Measure Spaces

As I am just beginning to study general topology and metric spaces in more and more detail, it seems to me that the metric space topology is entirely determined by the properties of $\Bbb R$, since ...
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0answers
51 views

Zeros of a holomorphic function.

Let $X$ be a connected complex manifold with dimension $n \geq 1$ and $f:X \rightarrow \mathbb{C}$ be a holomorphic function. Suppose that on a coordinate open set $U$, the function takes infinitely ...
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1answer
39 views

Sums of complex numbers - proof in Rudin's book

I have one question about a proof in Rudin's book: If $z_1 ..., z_N$ are complex numbers then there is a subset $S$ of $ \{1,..., N \}$ for which $|\sum_{k \in S} z_k| \ge \frac{1}{\pi} \sum_{1}^N ...
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1answer
14 views

Coefficient symmetry of the Laurent expansion of the composition of a function with the Joukowski map

In Sheehan Olver's exposition of how Chebyshev series arise, he lets $f\in C^\infty[-1,1]$ and defines $$g(z)=f(J(z))$$ where $$J(z)=\frac{1}{2}\left(z+\frac{1}{z}\right)$$ is the Joukowsky map. He ...
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0answers
19 views

verifying CRE => f(z) differentiable?

Say we had $f(z) = e^{-y}(x\cos x - y \sin x) + i e^{-y}(y \cos x + x \sin x)$, and I verified that the Cauchy Riemann Equations hold, then does that mean I can use the theorem $f'(z) = \dfrac{ ...
2
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2answers
32 views

CR Equations using Polar Form

I have a question to check whether following function is analytic or not using CR Equations. The question is f(z) = 1/(z-z^5) I just don't know how to start and ...
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0answers
44 views

Rotation Number of Polynomial

I conjectured that Maximum Rotation Number of $n$-th degree polynomial image of unit circle (in the complex plane) is $n$. (for example, if $f(z)=z^n$, then rotation number is $n$) Is it right?
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1answer
30 views

Surjectivity of a map $D^{2n} \to \mathbb{CP}^n$

I'm solving an exercise about the complex projective space, and during a step of the solution I'm asked to find a surjective map $D^{2n} \to \mathbb{CP}^n$. I defined the map in this way $$ ...
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27 views

Using Contour Integration to Integrate $\frac {x^{-z}}{1+x}$

I would like to evaluate the following integral using complex integration: $$\int_0^{\infty}\frac{x^{-z}}{(1+x)}dx$$ where $z \notin \mathbb Z$. I'm given that the answer is $\frac{\pi}{\sin(\pi ...
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0answers
21 views

Finding the singular part of a particular rational function

Find the singular part of $$f(z)= \frac{1}{(1+z^3)^2}$$ at $z=-1$. I tried to compute the Laurent series expansion at $z=-1$, as the term with the negative exponent of $(1+z)$ would be the singular ...
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0answers
50 views

suggestion on how to learn maths effectively

this question is not about a problem.My problem is I was made to read topics such as real analysis,complex analysis,metric spaces,topology,functional analysis,abstract algebra comprising of group ...
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1answer
43 views

Integration by Residue

How do I evaluate $$\int_{C(0,e)} \frac{1-\cos z}{(e^z-1)\sin z}dz.$$ This looks simple but I have a hard time to find the residue at $z=0$. At $z=0$ the function say $f(z)$ is even undefined. I ...
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235 views

Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. ...
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1answer
62 views

Gauss curvature of the graphs of the real and imaginary parts of an analytic function. [closed]

General question about analytic function f of complex variable z: $ f(x + i y) = f(z) $. At any mapped point, are Gauss curvatures in separate 3D plots of real and imaginary parts of $ f(x + i y)$ ...
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1answer
58 views

complex differentiation

It is my first text here. So I have started to look at complex numbers in death. I do Uni know, so adding $3+4i$ and $4+7i$ is now nothing. What I am stuck on is the idea of taking a derivative of a ...
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1answer
36 views

Questions regarding a complex-analytic function

So the question is formulated as follows. Given the analytic function $z \mapsto f(z) = \dfrac{1}{\sin z} - \dfrac{\cos z}{z}$, Is $z = 0$ a pole, an essential singularity, a removable singularity, ...
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2answers
87 views

How To Prove:$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4} = -\frac{7}{{720}}{\pi ^4}$

When I tried to solve this integral: $$\int_0^\infty {\frac{{{x^3}}}{{1 + {e^x}}}} \;{\rm{d}}x$$ I had trouble computing the sieries: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4}$$ Thanks.
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1answer
50 views

How we can drow a Blaschke $3$ ellipse?

Today I read the article Ellipses and Finite Blaschke Products (www.jstor.org/stable/3072367 Blaschke ellipses) by Ulrich Daepp, Pamela Gorkin, and Raymond Mortini. In there they have proved very nice ...