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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4
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2answers
132 views

“Close-to-analytic”, but “not quite”, functions on the complex numbers.

I heard of a way by which one can say that a given complex function is "close to analytic", namely if its Wirtinger partial $\frac{\partial f}{\partial \bar{z}}$ is small, meaning it "depends only a ...
6
votes
2answers
155 views

Analytic functions close to $\bar{z}$

Is there an analytic function $f\colon\Bbb{C}\longrightarrow \Bbb{C}$ such that for any $z$ on the unit circle $\lvert f(z) - \overline{z}\rvert < 1 $?
0
votes
1answer
49 views

Mapping question related to complex analysis?

In what line of the plan $C_W$ is the circle $|z|=1$ mapped using the function $W=√(z+1)$? How about the mapping of the circles $|z|=r$ using the function $W= z + (1/z)$ and $W= z- (1/z)$ Now,the ...
2
votes
1answer
73 views

Real solution to a complex equation

I have some trouble solving this equation. Let $a\in\mathbb R$, $a>1$. I want to show that there is a unique solution of $ze^{a-z} = 1$, with $|z|<1$ and that this solution is real and ...
1
vote
1answer
114 views

Bound on the derivative of a holomorphic function at $0$

Suppose $f(z)$ is holomorphic in $|z|< 1$ and $\operatorname{Re}f(z) > 0$ in this region. Furthermore, suppose $f(0) = a > 0$. Then why does $|f'(0)| \leq 2a$? The hint I have is to consider ...
1
vote
1answer
84 views

Questions about Titchmarsh's proof of the Phragmen-Lindelöf Theorem

The discussion of the Phragmen-Lindelöf Theorem on page 176 of Titchmarsh's book Theory of Functions starts off with the following result: Let $C$ be a simple closed contour, and let $f(z)$ be ...
2
votes
2answers
87 views
4
votes
2answers
62 views

Rotations around the origin

The problem is to find the number of rotations around the origin for the function $$f(z)=z^{2013}+2z+1 $$ when $z$ moves through $\left\{|z|=1\right\}$. I tried to solve it with the help of argument ...
1
vote
1answer
53 views

Dealing with partial derivatives in a function space

Please read the following details below. Question: I want to show now that if $r>s>0,f \in F_s (\Omega), $ and $u \in F_r (\Omega)$, then for any $i$, $$f \frac{\partial u}{\partial z_i} ...
0
votes
1answer
74 views

Can anyone help me to solve this example of Blaschke Product, please?

Can anyone help me to solve this example, please? Example of Blaschke Product: Show that we cannot have a bounded analytic function on B(0,1) with its zeros being 1/2, 2/3 , 3/4, 4/5 ,... (hint: ...
1
vote
0answers
57 views

Existence of entire function with simple zeros satisfying that estimate.

Let $u(z) = u(Re z)$ - piecewise linear function, $x_k$ are points of derivative's leaps, is there entire function $f(z)$ with simple zeros, satisfying $|u(z)-\ln|f(z)||=O(\ln(z)), |z| \rightarrow ...
4
votes
5answers
483 views

Rigorous Textbook for Introduction to Complex Numbers/Analysis?

Does anybody know where I can find a rigorous textbook on developing complex numbers/analysis? I'm currently working through Needham's Visual Complex Analysis, which is interesting but non-rigorous. ...
0
votes
3answers
817 views

A Möbius transformation maps circles and lines to circles and lines. What exactly does that mean?

The title pretty much says it all. I am also looking for a concrete example if possible. I have looked at the proof, but I'm not exactly sure what it means because I am kind of confused on what the ...
0
votes
2answers
102 views

Help with complex logarithms

For real $x$ what does $-\ln(1-e^{2\pi i x})$ equal so that it agrees with the series expansion, how would I find the real and imaginary parts. $$-\ln(1-e^{2\pi ix})=\sum_{n=1}^\infty\frac{e^{2\pi i n ...
1
vote
0answers
155 views

Complex logarithm and Cauchy integral formula

I am stuck with a little exercise I wanted to do in order to check I understand the complex logarithm. This is defined as $$ \log z := \ln |z| + i \arg(z) \qquad \text{for } z \in \mathbb{C} ...
-1
votes
2answers
35 views

Find the set of points for which :

So, I have to find the set of points for which $\frac\pi4 <\arg(z+i)<\frac\pi2$ so $z=x+i\times y$ and I'm thinking about finding the tangent of every side $\tan(\frac\pi4) ...
3
votes
1answer
121 views

Integrating around a pole

I brought this up in another thread. Is my observation correct? Let $f(z)$ have a Laurent expansion at $z=z_{0}$ of the form $$ f(z) ...
8
votes
4answers
633 views

How many zeros does $z^{4}+z^{3}+4z^{2}+2z+3$ have in the first quadrant?

Let $f(z) = z^{4}+z^{3}+4z^{2}+2z+3$. I know that $f$ has no real roots and no purely imaginary roots. The number of zeros of $f(z)$ in the first quadrant is $\frac{1}{2\pi ...
2
votes
2answers
64 views

Can I apply Cauchy's integral formula here?

Could any one tell me how and what is the value of $\int_{|z|=2}ze^{3\over z}$ where the contour is oriented in anti clockwise direction. Can I apply Cauchy integral formula here?
2
votes
1answer
90 views

Integration of $z^{1/2}$ along a contour winding the origin twice, without introducing Riemann surface

If we introduce a Riemann surface, it is easy to show that the integral of $z^{1/2}$ along a contour winding the origin twice is zero. An anti-derivative exists everywhere, so the integral depends ...
0
votes
1answer
87 views

Complex analysis mapping question?

So I am giving this important exam on complex analysis on September the 12th and I'm preparing for it.I found this exercise in a book: In what lines of the plan $C_w$ are the mapped: a)The ray ...
1
vote
2answers
74 views

To argue that a point is not an accumulation point of a given set

I want to show that $\mathbb Z^2$ has no accumulation points in $\mathbb R^2\backslash\mathbb Z^2$. Is this argument correct? In particular, have I correctly invoked the density property of $\mathbb ...
0
votes
1answer
56 views

If a set in a general metric space consistes entirely of isolated points, can it still have any accumulation points in its complement?

It seems not in $\mathbb R ^n$ (correct?), but how about in a general metric space? On the other hand, I'm not so sure about my claim above regarding $\mathbb R^n$: surely you can have points outside ...
5
votes
2answers
136 views

Expand $(4+i)(5+3i)$ and show $\pi/4=\arctan{1/4}+\arctan{3/5}$

I can't remember ever having done this before so if someone could help me out that would be great. The question is expand $(4+i)(5+3i)$ and hence show that $\pi/4=\arctan{1/4}+\arctan{3/5}$. ...
3
votes
1answer
521 views

If $u$ is harmonic and bounded in $0 < |z| < \rho$, show that the origin is a removable singularity

This is a reworking of a previous question here which was marked as a duplicate. Some nice folks have referred me to solutions to similar problems. I still have a couple of questions, since one of the ...
1
vote
1answer
60 views

Computation of the Wirtinger derivative of a product (continuation)

Let us have a real function $f=(A/2)\phi\bar{\phi}$, where $\phi, \bar\phi$ are complex fields. When looking for the stationary state of this function, we can either treat $\phi, \bar\phi$ as ...
2
votes
2answers
75 views

Please help me check my metric definition of isolated point

I translated the word definitions into the more symbolic form below, but as they aren't mere negations of each other, it was a little tricky. Is there any mistake below (especially for 'isolated ...
1
vote
0answers
63 views

Help with an integral involving zeta and digamma

Consider the integral: $$f(s)=\frac{1}{2\pi i}\int_{\tau-i\infty}^{\tau+i\infty}\frac{\zeta(z)}{z}\left[\psi\left(\frac{z}{s} \right ) +\frac{s}{2z}-\log\left(\frac{z}{s} \right )\right ...
4
votes
2answers
167 views

Why is $\sum_{k=-\infty}^\infty \frac{1}{(z-n)^2}$ uniformly convergent in $|y| \geq 1$

In the Complex Analysis text by Ahlfors', he says that it's easy to see that the series $$\sum_{k=-\infty}^\infty \frac{1}{(z-k)^2}$$ converges uniformly in the set $\{x+iy:|y| \geq 1\}.$ I can't see ...
1
vote
1answer
46 views

Complex analysis question help?

I have to describe the set of points of $\operatorname{Im}(z)=\frac15$. So since $\operatorname{Im}(z)=y$. I have $1=5y$ and so $y =\frac15$. So what does $y=\frac15$ represent?
2
votes
1answer
56 views

Growth rate of Taylor convergents near pole

For any fixed $z_0\in\mathbb{C}\setminus \{0\}$ and $\beta\in\mathbb{R}^{+}$, prove that $$\left.T_n\left(\log^{\beta}z;z_0\right)\right|_{z=0}\sim\log^{\beta} n$$ Note: I observed that this holds ...
3
votes
1answer
163 views

Is this region simply connected?

Let $ \gamma (t) = t + it^2$ be a curve in the complex plane, $0 \leq t \leq \infty$. My question is: Is $ D = \mathbb{C} \setminus \gamma $ simply connected? The curve separates the complex plane ...
3
votes
1answer
360 views

contour integration of logarithm function

I'm new to contour integral involving branch point and stuck on this particular integration. Here is the problem: $$\int_{\mathcal{C}}\log z\,\mathrm{d}z,$$ where $\mathcal{C}$ is a closed square ...
3
votes
1answer
60 views

a property of Analytic functions

Let $D$ be the open unit disk. How can I find all analytic functions $f: D\longrightarrow D$ such that $f(\frac 14) = \frac 14$ and $f'(\frac 14) = \frac 7{15}$ ?
4
votes
3answers
2k views

Type of singularity of $\log z$ at $z=0$

What type of singularity is $z=0$ for $\log z$ (any branch)? What is the Laurent series for $\log z$ centered at 0, if exist? If the Laurent series has the form $\sum_{k=-\infty}^{\infty} a_kx^k$, ...
2
votes
0answers
77 views

$u(x,y)$ harmonic and bounded in punctured disc; show $0$ is a removable singularity [duplicate]

I'm working on a problem from p. 166 of Lars Ahlfors' Complex Analysis: If $u$ is harmonic and bounded in $0 < |z| < \rho$, show that the origin is a removable singularity in the sense that ...
1
vote
1answer
52 views

Expressing a function as exponential of another function in a given domain

Can the functions $h_1(z) = 1+z^2$ and $h_2(z) = 1+\log(4+z)$ be expressed as the exponential of a function $f(z)$, where $f$ is holomorphic in the set $D = \{z : |z| < 2 \}$ ? More generally, ...
-1
votes
1answer
622 views

Describe the set of points on the complex plane…

Describe the set of points on the complex plane for which $|z-2| + |z+2|=4$... So, I know you can solve this instantly, just by using definition, but I want to do it the long way.. So, $$|x- i*y ...
0
votes
1answer
84 views

If $f$ is an entire function with $|f(z)|\le 100\log|z|$ and $f(i)=2i$, what is $f(1)$?

Let $f$ be an entire function with $|f(z)|\le 100\log|z|,\forall |z|\ge 2,f(i)=2i, \text{ Then} f(1)=?$ I have no idea how to solve this one! $g(z)={f(z)\over \log|z|}$ Then Can I say $g$ is ...
11
votes
3answers
528 views

Integrating $\int_0^\infty \frac{\log x}{(1+x)^3}\,\operatorname d\!x$ using residues

I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,\operatorname d\!x.$$My first attempt involved trying to take a circular contour with the branch cut being the positive ...
1
vote
1answer
103 views

Showing that a function is Lipschitz.

Let $w$ be complex $n$-dimensional and suppose $f(w)$ is analytic and bounded on the open "rectangle" $$R: |w-w_0|<b, \quad b>0.$$ Is $f$ Lipschitz on $R$? What I know is that since $f$ is ...
0
votes
1answer
58 views

Complex analysis question?

Describe the points of the plane where the following zone is mapped: the zone defined by the cardioid $ρ=2 (1+ \cosθ)$ from the analytic branch of the function $w= \sqrt{z}$ ,which takes positive ...
4
votes
1answer
123 views

An inequality involving arctan of complex argument

I have the following conjecture: \begin{equation} \text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0. \end{equation} Which seems to be ...
0
votes
2answers
104 views

Co-efficient of $(z-\pi)^2$ Taylor Series Expansion of $f$ around $\pi$

Co-efficient of $(z-\pi)^2$ Taylor Series Expansion of $f$ around $\pi$ is $1.1/2$ $2. -1/2$ $3.1/6$ $4.-1/6$ where $f(z)={\sin z\over z-\pi};z\ne \pi$ and $-1$ at $z=\pi$ Could any one tell me ...
2
votes
0answers
74 views

quadrature formula for a lune

I have been reading about quadrature formulas in the complex plane. On the set $\mathbb{D} = \{|z| < 1\}$ we have $\int_{\mathbb{D}} f(z) dA = \pi f(0)$. Standard result in Harmonic functions. ...
4
votes
2answers
167 views

Find all complex solutions to an exponential equation

Find all complex numbers z such that $e^{-2iz}/4 + e^{-iz}/2 + 1 + 2e^{iz} + 4e^{2iz} = 0 $ Rewriting the left-hand side using Eulers formula doesn't seem to get me anywhere. Need some help with this ...
3
votes
1answer
108 views

Is my proof correct? (a generalization of the Laurent expansion in an annulus)

I want to see if my solution to the following problem in Ahlfors' Complex Analysis text is correct. The problem reads: Let $\Omega$ be a doubly connected region whose complement consists of the ...
2
votes
0answers
60 views

Roots of a polynomial plus a logistic equation

I would like to know if there are any methods to find the roots (analytically) of complex valued equations of the following form: $$ f(z)=P(z)+\frac{e^{-z}}{(1+e^{-z})^2} $$ where $P(z)$ is a ...
1
vote
1answer
58 views

Using Möbius transformation to change $B\left(a;R\right)$ to halfplane

In Conway's Functions of One Complex Variable, there is a proposition which says: Let $f$ be analytic in the disk $B\left(a;R\right)$ and suppose that $\gamma$ is a closed rectifiable curve in ...
0
votes
1answer
44 views

What is the value of the complex integral?

What is the value of $$\displaystyle\int_\gamma\frac{z^3e^{\sin\pi z}}{(z-1)^3}dz,~\gamma(t)=2e^{it},0\le t\le 2\pi$$