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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Showing this integral from complex analysis is an integer without residues

I've already proven that the winding number is an integer, now I want to show that, given the following assumptions: The function $f$ is holomorphic on the domain $D$ $\gamma$ is a ...
2
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2answers
368 views

Möbius transform which completely preserves circles (how to map a circle?)

(remmert theory of complex function) I am trying to solve this exercise, however it seems impossible because I don't know how to map a circle, and I will be very thankful if somebody points out to ...
3
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1answer
132 views

Handling Cross ratios ( Fractional linear transformations )

According to remmert there is a relationship between the crossratios: $$C(z,u,v,w) = \frac{(z-v)(u-w)}{(z-w)(u-v)} \text{ and } C(z,v,u,w)= \frac{(z-u)(v-w)}{(z-w)(v-u)}$$ where $z,u,v,w \in ...
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2answers
157 views

Computing $\int_{|z|=1}|z-1|\cdot|dz|$.

I'm trying to teach myself some complex integration. I'm doing some exercises, and want to compute $$ \int_{|z|=1}|z-1|\cdot |dz|. $$ I parametrize with $z(t)=e^{it}$ on the interval $(0,2\pi)$, so ...
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1answer
350 views

Suggestions for a Global Analysis book

can somebody tell me some good books or lecture notes in "global analysis" ? I am a newcomer in this subject. thanks in advance. greetings trito
2
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1answer
150 views

Question on analytic functions

I can prove parts $a$ and $b$ of this question using the Cauchy Riemann equations. However, I can't see how to prove part $c$. Does anyone know how to do it?
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2answers
106 views

Complex polynomial root

I have done the first part of this question and proved the statement is true by induction. But I am not sure about the second part. Use induction on $n$ to erify that $$1 + z + \cdots + z^n = ...
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1answer
528 views

$\sum \frac{z^{2n}}{1-z^{n}}$ normally convergent in $\mathbb{E}$

I tried to solve this exercise (Remmert Theory of Complex Functions, p. 107, exercise 1 ), but I didn't get very far: Proposition: $$\sum \frac{z^{2n}}{1-z^{n}}$$ is normally convergent in ...
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1answer
265 views

$f_{n}=\frac{1}{1+z^{n}}$ uniform convergence

$\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\C}{\mathbb{C}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\s}{\sigma}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\F}{\mathbb{F}}$ I am trying to show ...
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2answers
568 views

Why are $\log$ and $\ln$ being used interchangeably?

A definition for complex logarithm that I am looking at in a book is as follows - $\log z = \ln r + i(\theta + 2n\pi)$ Why is it $\log z = \ldots$ and not $\ln z = \ldots$? Surely the base of the ...
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1answer
298 views

Power series, entire functions

Some questions about power series: i) Why the series $\sum_{n\geq 1} \frac{(-1)^{n}}{z + n} $ converges uniformly on compacts in $\mathbb{C}\setminus { \{ -1, -2, -3, \ldots \} } \ $ ? ii) Let f be ...
2
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1answer
106 views

Determine a holomorphic function by means of its values on $ \mathbb{N} $

This is exercise 5, page 236 from Remmert, Theory of complex functions For each of the following properties produce a function which is holomorphic in a neighborhood of $ 0 $ or prove that no such ...
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1answer
472 views

Cauchy Riemann equations - polar form question

Doing some exercises from a book and it says the following - I dont get this. The bottom line differentiates both sides with respect to r and $ru_{r}$ becomes $ru_{rr} + ru_r$. Where is the ...
2
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3answers
260 views

Harmonic conjugate with tough integral

I am trying to find the harmonic conjugate of $u(x,y) = \dfrac{y}{(x^2 + y^2)}$ I have got $Ux = Vy = \dfrac{-2xy}{(x^2 + y^2)^2}$ And now I need to integrate Vy with respect to y to find V. ...
2
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1answer
2k views

Every harmonic function is the real part of a holomorphic function

Is there a way to show that every harmonic function is the real part of a holomorphic function without using integration equations if later theorems are allowed also?
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255 views

Characterization of rotations of the Riemann sphere?

Out of curiosity, is there a nice characterization of the linear fractional transformations which give rotations of the Riemann sphere? My thinking was a rotation of the Riemann sphere rotates about ...
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0answers
59 views

Is a conformal mapping always (partially) differentiable over its domain?

Consider a conformal mapping (whose domain and codomain may be either subsets of $\mathbb{R}^n$ or of $\mathbb{C}^n$). I wonder if it is always differentiable over its domain? Is it always partially ...
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1answer
50 views

Is a map that satisfies $|z||w|\langle T(z),T(w)\rangle = |T(z)||T(w)|\langle z,w\rangle $ and isn't the 0 map an injection?

Remmert page 15 chapter 0 it says that angle preserving mapping is R-linear and injective. We want to prove: Given $$T:\mathbb{C}\rightarrow \mathbb{C}$$ a $\mathbb{R}$ linear map which satisfies ...
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0answers
104 views

Showing inequalities in complex variables

Remmert chapter 1 page 13 and 14 Set $z:= x+iy, w:= u+iv$: $$\langle z,w\rangle^2+\langle iz,w\rangle^2 = Re(w\bar{z})^2+Re(iw\bar{z})^2=(ux+vy)^2+(uy-vx)^2= (x^2+y^2)(u^2+v^2)= |z|^2|w|^2$$ so ...
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0answers
105 views

C-linearity, conformity, bijectivity of a function

$\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\C}{\mathbb{C}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\s}{\sigma}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\F}{\mathbb{F}}$ Remmert chapter 0 ...
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2answers
237 views

When do three points determine the same orientation?

I'm trying to understand the following claim: If $z_1,z_2,z_3,z_4$ are points (as complex numbers) on a circle, then $z_1,z_3,z_4$ and $z_2,z_3,z_4$ determine the same orientation iff ...
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1answer
477 views

Examples of function that are differentiable in R but not in C

What are some examples of functions that are differentiable (everywhere) in $\mathbb{R^2}$, but that are not differentiable in the complex plane? We got an example for homework, $f(z)=2xy$, and I was ...
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1answer
187 views

Determining transformation sending $|z|=2$ to $|z+1|=1$.

I'm trying to teach myself complex analysis, and am reading about linear fractional transformations. I want to find the transformation carrying the circle $|z|=2$ into $|z+1|=1$, $-2$ into the ...
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4answers
720 views

Holomorphic function on bounded domain.

Let $\Omega$ be a bounded domain of $\mathbb{C}^n$ and $f$ be a holomorphic function defined on $\Omega$. Is it possible that $L^2$-norm of $f$ is bounded but $f$ itself is unbounded?
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94 views

Definition of stereoprojection and Möbius maps

@WillieWong has kindly pointed out that there are 2 definitions of stereographic projection. One with the unit sphere placed on top of the plane, the other where the plane is at the equator of the ...
3
votes
3answers
735 views

Radius of convergence of complex power series.

http://www.math.harvard.edu/~ctm/papers/home/text/class/harvard/213a/course/course.pdf Why the following is true? Corollary 1.4: An analytic function has at least one singularity on its circle of ...
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1answer
111 views

Equality on absolute values in the complex plane.

Something I've been wondering. Suppose $z_1,z_2,z_3,z_4$ are consecutive vertices of a quadrilateral that lie in a circle in the complex plane. Why does $$ ...
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2answers
114 views

What can be said about a complex polynomial $f$ with $\dfrac{\partial}{\partial \bar z} f^2 = 0$?

The hypothesis is similar to that of a previous question of mine, namely that we have a complex polynomial $f: \mathbb{C} \to \mathbb{C}$ with $\dfrac{\partial}{\partial \bar z} f^2 = 0$ . (In the ...
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1answer
57 views

How to construct a polynomial with minimum deviation from zero on the complex region?

I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
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1answer
293 views

Absolute value of $\sin z$ on square

Show that $|\sin z|\geq 1$ at all points on the square with vertices $\pm (N+1/2)\pi\pm(N+1/2)\pi i$, for any positive integer $N$.
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140 views

A proof (?) of a Cauchy Integral Theorem type claim

I want to show the following: Suppose $f\in C(|z|\leq 1)\cap O(|z|< 1)$, where $O(|z|< 1)$ means that $f$ is holomorphic in the open unit disk $D$. Then $$\int_{|z|=1} fdz=0$$ (Note: I ...
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0answers
121 views

Stereographic projections and cross-ratios

Would anybody shed some light on question 2.11 in Wilson's Curved Spaces? The numbers $p,q\in \hat{\mathbb{C}}$ are stereographic projections of points $P,Q$ on the unit sphere. The spherical ...
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0answers
554 views

Complex analysis - using defn of limit to prove statment

Let $a,b,c$ denote complex constants. Use the definition of a limit to show that $$\lim_{z \to z_0} (az + b) = az_0 + b$$ Here is what I have done - $$|az + b - (az_0 + b)| =$$ $$|az - az_0 ...
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1answer
89 views

Complex Integral Help

Hello i am confused with this complex integral! It isnt for math homework. $\large\int^{T/2}_{-T/2}p(t)e^{-j2πnt/T}dt $ $p(t)$ is $A$ for $-T/4 < t < T/4$ $-A$ otherwise Its a kind of ...
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4answers
314 views

Finding the n-th root of a complex number

I am trying to solve $z^6 = 1$ where $z\in\mathbb{C}$. So What I have so far is : $$z^6 = 1 \rightarrow r^6\operatorname{cis}(6\theta) = 1\operatorname{cis}(0 + \pi k)$$ $$r = 1,\ \theta = \frac{\pi ...
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1answer
32 views

Trying to figure out a complex equality

An answer to a comlex equation I was working on was $$z = \frac{1}{2} + \frac{i}{2}$$ My teacher further developed it to be $$e^{\frac{i\pi}{4}-\frac{1}{2}\ln{2}}$$ And here's what I tried: $$z = ...
9
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1answer
288 views

interpolating the primorial $p_{n}\#$

The primorial $p_{n}\#$ is given by the product $p_n\# = \prod_{k=1}^n p_k$ (where $p_{k}$ is the $k$th prime) -- is there a natural (a la the gamma function $\Gamma(z)$) way of interpolating it for ...
2
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1answer
118 views

Limit of a complex function defined by an integral

Let $f\in L^1(\mathbb R) \cap C(\mathbb R)$, ie. $f$ is integrable and continuous. For $z \in \mathbb C$ with $Im(z) \not= 0$, define $g(z) = \int_{-\infty}^\infty \frac{f(t)}{t-z} dt .$ I am ...
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2answers
1k views

Any linear fractional transformation transforming the real axis to itself can be written in terms of reals?

I'm trying to teach myself complex analysis, and was reading about linear transformations. I would like to understand why any linear fractional transformation which transforms the real axis into ...
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2answers
519 views

Proving that the “real part” function is continuous

I want to use the definition of a limit, $|f(z) - w_0| < \varepsilon$ whenever $0 < |z - z_0| < \delta$ to prove $$\lim_{z \to z_0} \mathop{\rm Re}(z) = \mathop{\rm Re}(z_0)$$ By intuition ...
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4answers
1k views

Remember trig identities like $\cos(\pi/3) = 1/2$

I have started doing complex analysis and I keep having to switch between rectangular co-ordinates and polar form and I keep running into stuff like - $\cos(\pi/3) = 1/2$. I keep having to look these ...
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2answers
615 views

For an analytic function $f(z)$, $|f(z)^2-1|<1$ implies $\Re f(z)>0$ or $\Re f(z)<0$?

Doing a bit of self study, and I'm unsure about a problem. It says, Suppose $f(z)$ (a complex valued function) is analytic and satisfies the condition $|f(z)^2-1|<1$ in a region $\Omega$. Show ...
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2answers
393 views

Finding Poles and Zeros in the Z-Domain

I have the transfer function$$ H(z) = \frac{z-.75}{.1 z+.15} $$ how do I find the Poles and Zeros?
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3answers
507 views

Weierstrass $\wp$ function doubly periodic

I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough ...
3
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2answers
1k views

Mysterious entities by the name of branch points

Could someone please explain the concept of branch points to me? I have tried searching online and had a read of the textbook Visual Complex Analysis by T. Needham, but I am still not very clear how ...
4
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1answer
445 views

Using the complex logarithm to find the sum of angles in a triangle.

Suppose you have a triangle with vertices $a$, $b$, and $c$. I asked earlier how you can define the angles in a triangle based on the $\log$ function. I received the answer that, for instance, the ...
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1answer
875 views

Question regarding infinite Blaschke product

According to Gamelin's $\textit{Complex Analysis}$, a finite Blaschke product is a rational function of the form $B(z)= e^{i \varphi} (\frac{z-a_1}{1-\bar{a_1} z} \cdots \frac{z-a_n}{1-\bar{a_n} z})$ ...
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2answers
308 views

Algebraic functions are polynomials?

Does any one how to prove that every entire algebraic function is a polynomial? I'm under the impression that this can be achieved by showing that an algebraic function grows no faster than a ...
3
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1answer
153 views

How can one use the logarithm function to define angles?

In dealing with the complex logarithm function, I read that the imaginary part of $\log w$, is also called the argument of $w$, $\operatorname{arg }w$, and it is interpreted geometrically as the angle ...
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3answers
427 views

Expressing the area of the image of a holomorphic function by the coefficients of its expansion

I have the following problem. Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set ...