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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12 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
17 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
232 views

Mathematical Way To find how many complex numbers

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
25 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 ...
4
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1answer
35 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
28 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
17 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
24 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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0answers
22 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
30 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 $f(2+i)=3i+1$.

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
27 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 ...
2
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1answer
31 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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0answers
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
34 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
47 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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0answers
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
36 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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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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55 views
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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
37 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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17 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
67 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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0answers
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. ...
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1answer
54 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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23 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 ...
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3answers
66 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 ...
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1answer
12 views

Derivatives of $(z-a)^kf(z)$ at $a$ knowing that $f\in H(D(a,r)\setminus \{a\})$

If $g(z)=(z-a)^nf(z)$ with $f\in H(D(a,r)\setminus \{a\})$. Can we said that $g^{(k)}(a)=0$ for all $k\in \{0,1,...,n-1\}$?
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1answer
24 views

Laurent series: how to join the 2 sums for $f(z)= \frac{1}{(z-1)(z+1)}$ about z = 1 for $0 < |z − 1| < 2$

We are to find the Laurent series for f(z) about $z = 1$ for $0 < |z − 1| < 2$: $f(z)= \frac{1}{(z-1)(z+1)}$ Assumptions: $|\frac{z−2}{1}| < 1 ⇔ |z − 1| < 2$ For $\frac{1}{(z-1)}$ we ...
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44 views

Homework: Awkward formulation of the exercise

The exercise: Let $U$ be open and connected, let $\bar D$ be a closed disk contained in $U$ and let $f : U \to \mathbb{C}$ be analytic. Denote by $γ$ the circle which is the boundary of $D$. Suppose ...
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1answer
84 views

Complex integral $1/(z^2+1)$ along unit circle [on hold]

I want to compute the complex integral $$\int_{|z|=1}\frac{1}{z^2+1}dz$$ both i and -i lie on c , what can I do ? I tried caushy , series , i used def of contour integral along c = z(t)=exp(it) and ...
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1answer
24 views

Laurent series to converge in $0<|z-1|<R$

Question: Determine the largest number $R$ so that the Laurent series of $$f(z)=\frac2{z^2-1} + \frac3{2z-i}$$ about $z=1$ converges for $0<|z-1|<R$. Attempt: I really don't understand this ...
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2answers
40 views

What's wrong with this reasoning? (Cauchy integral theorem)

Asumme that $f$ is analytic and for $z\in \overline{B(x,r)}$: $$|f(z)|\leq d$$ Then for $z\in B(x,r)$: ...
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69 views

Determine $\int \limits_0^{\infty} \frac1{x^4+1}dx$

Let $$f(z)=\frac1{1+z^4}$$ (a) Find the sinularity of $f(z)$ in the first quadrant where $Re(z), Im(z) \ge 0$. (b) Find the residue of the singular point found in the first quadrant. (c) Let ...
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20 views

Help with $\int _{R_0<|z|<R_1}\frac{1}{z} dz$.

Consider the integral in $\mathbb{C}\simeq \mathbb{R}^2$ $$ \int_{R_0<|z|<R_1} \frac{1}{z}\; dx_1 dx_2 $$ where $0<R_0<R_1$ and $z=x_1+i x_2$ and $|z|=(x_1^2+x_2^2)^{\frac{1}{2}}$. So ...
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1answer
33 views

Finding $\lim \limits_{R \rightarrow \infty} \int _{\Gamma_R} \frac1{(z+i)^2 (z-i)^2}dz$

Let $\Gamma_R $ be the half circle centred at $0$ and radius $R>3$ with $Im(z) \geq 0$. Show that $$\lim \limits_{R \rightarrow \infty} \int _{\Gamma_R} \frac1{(z+i)^2 (z-i)^2}dz=0$$ Is this ...
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2answers
32 views

Integral on the real line between 0 and infinity using contour integration

For part (a) I have that the singularity is at $(1+i)/root2$ and it is a simple pole? For part (b) I have that the residue at $f(z)$ at that point is $-(1+i)/4root2$ For part (c) I used the ML ...
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2answers
26 views

Region of Convergence of power series

The power series $\sum_{n=0}^\infty 2^{-n} z^{2n} $ converges if a)$|z|\le 2$ b)$|z|\lt 2$ c)$|z|\le\sqrt2$ d)$|z|\lt\sqrt 2$ I tried this problem,my answer is d).I am not sure whether it is correct ...
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0answers
16 views

Proving max-mod principle by contradiction

This is a homework exercise I have to make which I am kind of stuck on. First let $U$ be open and connected, $\overline{D}$ be the closure of the disk $D$ contained in $U$ and ...
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1answer
13 views

Prove that $\sum_{n=0}^\infty e^{-nz}$ is analytic in the right half plane $\text{Re}(z)>0$

Consider$$\sum_{n=0}^\infty e^{-nz}$$ Using Weierstrass theorem, prove that the series is analytic in $\text{Re}(z)>0$. I know that $f$ is analytic if it satisfies Cauchy–Riemann equations. Could ...
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0answers
8 views

anlyalytic paths through convergent cauchy sequence II

Assume we have a Cauchy sequence $\{\vec{a_i}:i\in\mathbb{N}\}$ converging to $\vec{0}$ in $\mathbb{C}^n$ such that $|\vec{a_i}|<|\vec{a_j}|$ whenever $i>j$. Can we find an analytic path ...