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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5 views

Residues and poles proof

Let the degree of the polynomials $P(z)=a_0+a_1z+a_2z^2+...+a_nz^n$ $a_n\neq0$ and $Q(z)=b_0+b_1z+b_2z^2+...b_mz^m$ $b_m\neq 0$ be such that $m\geq n+2$. Show that if all the zeros of $Q(z)$ ...
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2answers
16 views

Complex analysis, residues and integrate

Let $C$ denote the circle $|z|=1$ oriented counterclockwise. Show that i)$\int_Cz^ne^{\frac{1}{z}}dz=\frac{2\pi i}{(n+1)!}$ for $n=0,1,2$ ii)$\int_C e^{z+\frac{1}{z}}dz=2\pi ...
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3answers
48 views

Prove without Liouville's theorem: $f$ is entire, $\forall z \in \mathbb C: |f(z)| \leq |z|$, then $f=a \cdot z$, $a \in \mathbb C, |a| \leq 1$

Prove without Liouville's theorem: $f$ is entire, $\forall z \in \mathbb C: |f(z)| \leq |z|$, then $f=a \cdot z$, $a \in \mathbb C, |a| \leq 1$ What I tried so far: $f$ is entire, so $f(z)= \Sigma ...
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29 views

Not understanding how to use the following information: $f$ is entire, and $\lim _{|z| \rightarrow \infty} \frac{f(z)}{z^2}=2i$

I do not understand how to use the following information: If $f$ is entire, then $$\lim _{|z| \rightarrow \infty} \frac{f(z)}{z^2}=2i.$$ So if $f$ is entire, it has a power series around $z_0=0$, ...
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1answer
40 views

Can someone please point out the flaw in my proof?

Let $f:X \to Y$ be a proper map.Show that $f$ takes discrete sets to discrete sets. Proof:Let $A$ be discrete in $X$ and let $K$ be compact in $Y$ then $f(A) \cap K=f(A \cap f^{-1}(K))$,is finite ...
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1answer
8 views

Convexity of the complex ellipsoid

Let $p_1,\dotsc,p_n$ be positive integers. Define the complex ellipsoid $$\Omega(p_1,\dotsc,p_n)=\left\{(z_1,\dotsc,z_n)\in\Bbb C^n:\sum\limits_{i=1}^n{\left|z_i\right|^{2p_i}}<1\right\}.$$ I ...
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1answer
42 views

Prove $f(z)=z^9+3z+1$ has 8 distinct zeroes in $A=\{z \in \mathbb{C} \mid 1 < |z| < 2 \}$

I know how to prove $f$ has 8 zeroes in $A$ using Rouche's theorem, but I still want to prove the zeroes are distinct.
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15 views

laplace convolution with Bessel function

$$\int_{0}^{t}e^{a\tau}J_{0}(b(t-\tau))d\tau.$$ This functions appears in the Laplace inverse transform. It is the Laplace convolution of exponetial and Bessel function. Is there anyone know some ...
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14 views

Zero-free parabolic region of the exponential partial sums.

As a corollary of the principal theorem on this paper (see eq. (1.4) and 2. A parabola theorem) it is shown that the parabolic region $$ P=\{ z=x+iy : y^2<4(x+1)\} $$ is a zero-free region for any ...
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0answers
28 views

Non Existence of a proper holomorphic map from the punctured unit disc to an Annulus

Show that there is no proper holomorphic map from the punctured unit disc to an annulus $A_r=\{z \in \mathbb C:1 <|z| < r \}$. Def:A map $f: X \to Y$ is called proper if $f^{-1}(K)$ is compact ...
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1answer
41 views

Computing $\int _C \frac {1}{z^3(z-1)^2}$, $C: |z-2|=5$

How do I compute $\int _C \frac {1}{z^3(z-1)^2}$, $C: |z-2|=5$? I can't seem to use Cauchy's Formula, because both $0$ and $1$ are in the formula. There is this theorem, saying that $\int _C= ...
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1answer
54 views

Help with the proof that the sum of all the roots of a complex number is zero

If a complex number $z \neq 0$ has n roots, then each root can be expressed as: $$z_j=(\sqrt[n]{r}) e^{ {i (\theta +2\pi j) }/{n} } $$ For $j=0,1,2,...,n-1$ Thus, the summation of all the roots ...
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2answers
37 views

Why is the closedness of the set on which $f = g$ immediate, when proving the Identity Theorem?

It's a short proof given on Wikipedia, and I understand the argument for why the set on which $f = g$ must be open, but I'm not sure why closedness of the set is obvious. Apparently, this comes from ...
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1answer
20 views

Complex integrate and residues

Evaluate the integral of that $f(z)=\frac{z+1}{z^2-2z}$ around the circle $|z|=3$ oriented counterclockwise First I found that singularity points are $z=0,z=2$ ...
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1answer
17 views

Complex analysis, residues

Find the residue at $z=0$ of $f(z)=\dfrac{\sinh z}{z^4(1-z^2)}$. I did \begin{align} \frac{\sinh z}{z^4(1-z^2)} & =\frac{1}{z^4}\left[\left(\sum_{n=0}^\infty ...
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0answers
21 views

Uniform convergence of harmonic functions to $0$ on compact subsets

Let $D \subset \mathbb{C}$ be an open, connected set and let $\{ u_n \}$ be a sequence of harmonic functions with $u_n: D \longrightarrow (0, \infty)$. Show that if $u_n(z_0) \rightarrow 0$ for some ...
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2answers
309 views

Mathematical Way To find how many complex numbers [on hold]

Suppose that $a$ and $b$ are integers and that $ |a + bi| \leq 5 $, then how many complex numbers $a + bi$ are there? Is there a mathematical way to do this?
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1answer
29 views

Are there different ways to find (the) residue(s) for a function with one simple pole vs. a function with several simple poles?

Regarding evaluation of residuals for functions with simple poles. Let's say $m$ represents the order of the pole, then in order to find the residual at each pole/the pole (if only one pole) we have ...
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1answer
40 views

Complex analysis integration with logs

$$\int_C \operatorname{Log}\left(1-\frac 1 z \right)\,dz$$ where $C$ is the circle $|z|=2$ I don't even know how you would begin doing this. I understand $\operatorname{Log}(z)=\ln|z|+i\arg(z)$, ...
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2answers
31 views

Compute the maximum of $|f(z)|$ when $|z| \leq 1$ and $f(z)=\sin (z)$

Compute the maximum of $|f(z)|$ when $|z| \leq 1$ and $f(z)=\sin (z)$ So since $f$ is holomorphic on $|z| \leq 1$, we know we'll find the max of $|f(z)|$ on $|z|=1$. So: ...
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1answer
18 views

Specific requirements for Runge's theorem to hold

This question is exercise 8.2 in Conway's Functions of One Complex Variable I. It states: Let $\mathbb{D}\subset\mathbb{C}$ be the open unit disk, and let $K=\{z\in\mathbb{D}: \frac{1}{4}\leq ...
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1answer
25 views

Laplace transform of $H(-t)$

How to compute the Laplace transform of $H(-t)$, where $H$ is the Heaviside step function? Does it exist? Basically, I want to compute the Laplace transform of $e^{2t}H(-t)$. I know how to compute ...
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1answer
30 views

Given the holomorphic maximum modulus principle, prove Hopf's lemma

To smooth out my lecture notes, I'm looking for a derivation of Hopf's lemma for harmonic functions $u \colon D \subset \mathbb{R}^2 \to \mathbb{R}$ from the maximum modulus principle (and mean value ...
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2answers
33 views

Prove or disprove: If $f$ is entire and $\lim_{|z| \rightarrow \infty}\frac {f(z)}{z^2}=2i , f(1)=-2, f(i)=3$ then $f(2+i)=3i+1$. [on hold]

Prove or disprove: If $f$ is entire and $\lim_{|z| \rightarrow \infty}\frac {f(z)}{z^2}=2i , f(1)=-2, f(i)=3$ then $f(2+i)=3i+1$. How do I approach to this type of question? Thank you for any ...
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1answer
32 views

Non Existence of a proper holomorphic map from the unit disc onto the complex plane

It is well known that there is no proper holomorphic map from complex plane onto disc by Liouville's theorem.Does there exist a proper holomorphic map $f$ from the unit disc onto the complex plane?I ...
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1answer
33 views

Reference for amazing generalized version of Morera's Theorem

I recently came to know about following amazing generalized version of Morera's Theorem: Theorem:Let $f$ be a continuous function on the complex plane and suppose that there exist numbers $ r_1,r_2 ...
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14 views

Bounded function on $\Bbb C$. [on hold]

Consider functions $ f,g:\Bbb C\rightarrow \Bbb C$ defined by $f(z)=e^z,g(z)=e^{iz}.$ Let $ S=\{{z\in\Bbb C:Re(z)\in[-\pi,\pi]}\} $.Then a)f is a onto entire function. b)g is a bonded function ...
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1answer
38 views

Why can't branch cut pass through poles?

In the wiki article, Example (IV) – branch cuts. Why can't we can't we choose the contour so that the branch cut is on the negative x axis. If we choose this, the two residual is out of the contour, ...
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2answers
50 views

Taylor series about $[Log(1-z)]^2 $

I'm having some trouble proving that the Taylor series about the origin of the function $[Log(1-z)]^2$ to be $$\sum_{n=1}^\infty \frac{2H_n}{n+1}z^{n+1}$$ where $$H_n = \sum_{j=1}^n \frac{1}{j}$$ So ...
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0answers
22 views

Analytic continuation of ln(z) counterclockwise about the unit circle,

We write ln(z) as ln(1+z-1) = ln(1+(z-1)) to utilize the familiar expansion that is: (z-1) - (z-1)^2 / 2 + ... which converges for |z-1| < 1, i.e., we get convergence of ln(z) in an open Taylor ...
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26 views

Poisson Kernel - Complex Analysis

This is a problem from Ahlfors, Complex Analysis, pag. 171 #5. "Show that the mean value formula $u(z_0)=\frac{1}{2 \pi} \int_0^{2 \pi} u(z_0 +re^{i \theta} ) d \theta$ remains valid for $u=log ...
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0answers
38 views

Representing a function as a Poisson Integral.

This is a question I came across in Ahlfors' book Complex Analysis. It is found on page 171 of the 3rd Edition, Exercise 2. "Prove that a function $T(z)$ which is harmonic and bounded in the upper ...
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0answers
31 views

Complex Analysis - Poisson Integral [on hold]

This problem is from the book "Complex Analysis", Ahlfors, pag. 171 #4 as an application of Schwarz's Theorem : " If $C_1$ and $C_2$ are complementary arcs on the unit circle, set $u=1$ on ...
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2answers
22 views

In complex variables, why is |z-1| < 5 an open disk centered at +1, where the boundary is a circle of radius 5?

How can I justify this basic concept? Use the definition of the modulus? Write z = $e^{i\theta}$? ...and why is |z+1| < 5 ...centered at -1 and not +1? Thanks, Edit: it is always the basic ...
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1answer
55 views
2
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1answer
49 views

showing an isolated singularity is removable

Suppose $f$ is analytic for $0 < |z| < 1$. Further, suppose that there exists $C>$0 and $m \geq$ 1 with $|f^m(z)| \leq C/|z|^m$ for $0 < |z| < 1$. Show that $f$ has a removable ...
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2answers
41 views

ML-inequality: Why does this hold $|e^{-3y+3ix}| = e^{-3y}$ during numerator estimation of $f(z) = \frac{e^{3iz}}{z^2 + 1}$ [duplicate]

Given the following related to an ML-inequality for $R > 1$: Estimation of the numerator from the function $f(z)$ is supposed to develop as follows: I'm wondering why and how exactly the ...
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31 views

Fundamental theorem of Algebra using ideas of complex singularities

Below is an excerpt from Arnold's Theory of Catastrophes (I haven't got an American edition, so translating from Russian). Where I can read about it in more detail? Not only regarding polynomials. ...
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16 views

how to find singularities of complex functions.

how to find singularities of this function; i) $f(z)= 1+\frac{1}{\sqrt{z}}$ and ii) $f(z)=e^{1+\frac{1}{\sqrt{z}}}$ As the z is in the denominator the function (i) has pole at z=0 but what's ...
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1answer
32 views

complex integral of 1/z independent of choice of ellipse?

Can Someone please help me with the following. complex integral of 1/z over an ellipse is independent of choice of ellipse centered at zero. Why is this the case. Is it due to homotopy ...
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1answer
23 views

Maximum modulus of a holomorphic function on a disc within a certain sector

Given the polynomial $$f(z) = az^n + b \qquad (n \geq 2)$$ and a modulus $0 < \rho < 1$, can one find a modulus $0 < r < \rho$ such that there is a point $$w \in \{ |z| \leq r \} \cap \{ ...
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18 views

Extension of biholomorphisms between planar domain to Möbius transformations

Let $\Omega, \Omega' \subset \mathbb{C}$ be two planar domains, i.e. open connected. For simplicity we can assume they are also bounded and with smooth boundary (but I don't know if this is actually ...
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1answer
42 views

The Origins of $Re \lbrace x \rbrace$

I am just starting to take a differentential equations course. We do not use the notation $Re \lbrace x \rbrace$ to describe the real part of a complex number, but I have come accross it in other ...
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1answer
68 views

$e^{2\pi i x} = (e^{2\pi i})^x$: What happens if x is rational? [duplicate]

I'm a bit embarrassed that I've had difficulty on getting around this one: $$e^{2\pi i x}$$ Solving it by itself, we can reduce it down to $(e^{2\pi i})^x = 1^x$ such that $e^{2\pi i x} = 1$ for all ...
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1answer
31 views

Exponential type of $\sin z$

An entire function $f$ is of exponential type if $\,\lvert\, f(z)\rvert\le C\mathrm{e}^{\tau\lvert z\rvert},\,$ for all sufficiently large values of $\lvert z\rvert$. The exponential type of $f$ is ...
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17 views

Distance from $f(0)$ to the boundary of $D$ if $f$ maps open unit disk to $D$ conformally

Let $f(z)$ be a conformal map from the open unit disk onto $D$, which is a domain. I would like to show that the distance from $f(0)$ to the boundary of $D$, denoted $\partial D$, is given by ...
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1answer
41 views

Show that $f(z)$ is constant

If $f(z)=u(x,y) + iv(x,y)$ is an entire function such that $u\cdot v$ is constant then $f(z)$ is constant. I know that I need to use the Cauchy-Riemann equations, but I don't know how to start. ...
3
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1answer
55 views

Let $f$ be a non-constant entire function. Prove that $f(z)=cz^n$ for some constant $c$ and positive integer $n$

Let $f$ be a non-constant entire function satisfying the following conditions: $$f(0)=0$$ for each $M \gt 0$ the set $$\{z \mid \lvert f(z)\rvert \lt M\}$$ is connected. Prove that $f(z)=cz^n$ for ...
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0answers
24 views

Finding the harmonic function at a given point [on hold]

Draw concentric circles of radii $r_1 = |b|$ and $r_2 = |c|$, each centered at $z_0 = a + id$. Suppose $\theta(x,y)$ is a harmonic function inside the washer defined by these circles. The circle with ...
2
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3answers
70 views

Real Methods to Evaluate $2 \int_{-1}^{1}x^2 \sqrt{1-x^2}dx$

I was recently contacted by a friend to find the values of the two following integrals by any means. $$ I=2\int_{-1}^{1}x^2 \sqrt{1-x^2}dx$$ $$ J=\int_{-1}^{1}(1-x^2) \sqrt{1-x^2}dx$$ The first ...