Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

Questions tagged [complex-analysis]

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, ...

0
votes
0answers
2 views

Geometric interpretation of $\{z\in \mathbb{C} : \text{Im}(\frac{z-a}{b})>0\}$

I'm trying to find a geometric interpretation for the set mentioned in the question where $a$ and $b$ are given complex numbers that are not zero. What I've done so far: $$\text{Im}\frac{z-a}{b}=\...
0
votes
0answers
3 views

Show that $f$ is locally constant function.

$f: D \to C$ is analytic, Denote $f = u+iv$ and if $\exists$ an $\mathbb{R}$ differentiable function $g: R \to R$ where $u = h \circ v$. Show that $f$ is locally constant function. My idea is to ...
0
votes
0answers
8 views

Prove $|f^{\prime}(a)|< \frac{Im f(a)}{Im a} $ for analytic self mapping on upper half plane

$f:H \to H$ is a analytic mapping, where $H$ is the upper half plane, then $a \in H$, prove the inequality : $|f^{\prime}(a)|< \frac{Im f(a)}{Im a} $. I tried to use Schwarz Lemma to solve it, but ...
0
votes
0answers
12 views

Show that a complex Differentiable function f with $|f'|\leq 1$ is a contraction.

I am working on the proof of the following statement; Suppose $f$ is analytic on a a rectangle $R$ and $|f'(z)|\leq 1$ for all $z \in R$. Then $f$ is a contraction on $R$, that is $$ |f(b)-f(a)|\leq |...
1
vote
0answers
29 views

Determine the Laurent series of $1/z(z-1)$ on $2 < |z+2| < 3$.

Determine Laurent's serie of the function $$f(z)=\dfrac{1}{z(z-1)}$$ on the set $A:=\{z\in\mathbb{C};2<|z+2|<3\}$. My approach: Note that if $u=z+2$, then $2<|u|<3$. So, $$f(z)=\dfrac{...
1
vote
2answers
28 views

Show $f$ is constant given $g=\overline{f}$

I am trying to show that if $f$ and $g=\overline{f}$ are both differentiable in a domain, then $f$ is constant on that domain. My attempt: Let $$f(z)=u(x,y)+iv(x,y)$$ $$g(z)=r(x,y)+is(x,y)$$ Equating ...
0
votes
1answer
31 views

Why can't I just find the residue of the function?

I was solving the contour integral $$\oint \frac{z\sin z}{z^{2}+4} \ dz$$ in the upper half of the complex plane using the residue theorem and I couldn't figure out why I needed to convert it to $$Im\...
1
vote
1answer
17 views

Bounded analytic function on a punctured region 2

Let $G \subset \mathbb{C}$ be a region, $a \in G$ and $f : G \backslash\{a\} \to \mathbb{C}$ be injective and $\Omega = f(G \backslash\{a\})$ be bounded. Then $f$ has a removable singularity at $a$ ...
0
votes
1answer
13 views

Application of Schwarz Lemma Gamelin IX problem 2

I am trying to solve the following problem in Gamelin's Complex Analysis. Suppose that $f(z)$ is analytic and satisfies $|f(z)| \leq 1$ for $|z| <1$ . Show that if $f(z)$ has a zero of order $m$ ...
1
vote
1answer
14 views

Does $\text{Sp}(f(\lambda_n)) = \text{Sp}(f(\lambda_0))$ for all $n$ and $\lambda_n \to \zeta$ $\implies$ $\text{Sp}(f(\zeta))=\text{Sp}f(\lambda_0)$?

Let $f$ be an analytic function from a domain $D$ of $\mathbb{C}$ into a Banach algebra $A$. Suppose that $\text{Sp}(f(\lambda))\subseteq \mathbb{R}$ for every $\lambda \in D.$ Let $\lambda_0 \in D$. ...
0
votes
1answer
33 views

conformal map from croissant to disk

This is a problem from a test that I took few hours ago - and the test was finished just before. Let $U = \{|z-4i|<4\}\cap \{|z-i|>1\}$ be an open subset of $\mathbb{C}$ (which looks like a ...
0
votes
0answers
15 views

How to evaluate analytically the inverse Laplace transform of the following parameterized expression?

I am trying to evaluate analytically the inverse Laplace transform of the following expression: \begin{equation} F(p) = \frac{1}{ p^2 \left[ 1 + \alpha S_p^{-1/2} \tan \left( \frac{\pi}{2} S_p^{-1/2} ...
1
vote
0answers
16 views

Existence of subspace which is totally non-invariant under unitary transformation.

Given a complex Hilbert space $\mathcal{H}$ of dimension $d$ interpreted as vectorspace over $\mathbb{R}$ with dimension $2d$ and the space $L(\mathcal{H},\mathbb{C}^4)$ of all linear operators from $\...
0
votes
1answer
22 views

Let $G$ be a connected open set and let $f: G \to \mathbb{C}$ be an analytic function. Then the following are equivalent statements:

Let $G$ be a connected open set and let $f: G \to \mathbb{C}$ be an analytic function. Then the following are equivalent statements: a) $f \equiv 0$ b) There is a point $a \in G$ such that ...
-2
votes
0answers
24 views

Help me get a book for understanding [on hold]

Which would be a good book for complex analysis(INDIAN AUTHOR)
0
votes
1answer
32 views

Is this analytic function linked to Barnes multiple gamma function ? Is it entire?

I come across this series of following analytic functions $$ z\mapsto\frac{1}{\Big(\Pi_{k=0}^{n-1}\Gamma(1+ e^{2i\frac{k}{n}\pi}z)\Big)^{\frac{1}{n}}} $$ One can get easily a local (around zero) ...
-1
votes
0answers
30 views

Complex analysis books required [duplicate]

Which would be a good book for complex analysis for understanding concepts and problems as well? CONWAY OR AHLFORS (Ps- Books with geometrical approach can be suggested)
1
vote
1answer
22 views

path integral using complex logarithm

Compute the following integral $$\int_{-1}^{0} \! \frac{i}{1+it} \, \mathrm{d}t$$ using a determination of the complex logarithm. Rewriting I get $$\int_{-1}^{0} \! \frac{i}{1+it} \, \mathrm{d}t = i\...
1
vote
2answers
55 views

Is $f(z)$ entire?

I am trying to determine if the the following is entire $$f(z)= \begin{cases} e^{-z^{-4}} & z\neq0 \\ 0 & z=0\\ \end{cases} $$ My attempt: Consider $z\ne 0$. $f(z)=e^{-z^{-4}...
0
votes
0answers
17 views

Variation of Energy Momentum Tensor in CFT

I wish to compute the variation of the energy momentum tensor under an infinitesimal conformal transformation, $w(z)=z+\epsilon (z)$. I am following di Fransesco, who starts from the conformal ward ...
0
votes
1answer
28 views

Conditions under which $az+b\overline{z}+c=0$ in one complex unknown has one solution

In a solution that I found to the problem in my title, I'm unsure how they rewrote $a(x+iy)+b(x-iy)+c=0$ as equations $(1.6a)$ and $(1.6b)$. Why do we get $2$ equations? And I'm just starting to ...
3
votes
2answers
110 views

How to find all the $z$ that satisfy $(1+i)z^4=(1-i)|z|^2$?

Would you please help me solve this? I need all the $z$ that satisfy the equality $$(1+i)z^4=(1-i)|z|^2.$$ I tried doing this: $$ \begin{aligned} (1+i)z^4&= (1-i)z\overline z\\ (1+i)z^4 -(1-i)z\...
0
votes
1answer
42 views

Integrating the geometric series

We Know ($ z \in \textbf{C} $): $\dfrac{1}{1-z} = \sum_{n=0}^{k} z^{n} + \dfrac{z^{k+1}}{1-z}$ Integrating this along the straight line $L$ from $0$ to $z$: $ -ln(1-z) = \sum_{n=0}^{k}\dfrac{z^{n+1}...
2
votes
1answer
28 views

Are two real, two variable polynomials, satisfying the Cauchy-Riemann equations, a complex polynomial?

Let $u, v \in \mathbb{R}[x,y]$ satisfying $u_{x} = v_{y}$ and $u_{y} = -v_{x}$ everywhere in $\mathbb{C}$. Is the function $f(x + iy) = u(x,y) + iv(x,y)$ a polynomial in the variable $z = x + iy$? I ...
1
vote
1answer
33 views

Proof Understanding : Complex Holomorphic function

I am reading Complex analysis By Stein and Shakarchi. In that I came across one proposition which I do not understand I thought for some point but Do not know I correct or wrong . If f is ...
0
votes
0answers
23 views

Proving that the following convergence is uniform over compact sets of $\mathbb{D}$

I would like some help with the following: Suppose we have a function $f:\mathbb{D}\to\mathbb{D}$, holomorphic, and let $f^{(n)}$ denote the $n$-th iterate of $f$, that is $f^{(n)}=f\circ\dots\circ f$,...
1
vote
2answers
23 views

Finding a conformal map from this domain into the unit disc

Problem: Find a conformal map $f$ from $ A = \left\{ z \in \mathbb{C} \mid \text{Im}(z) > 0, |z| > 1 \right\}$ into the unit disk. Attempt: I started off with a map $F_1: z \mapsto z + \frac{1}{...
0
votes
1answer
43 views

Evaluating the integral $\int_0^\infty \mathrm{d}k e^{-i( t - x)k}k^{-i\omega/a}$

I am attempting to compute the vacuum expectation value for the energy density of a particular system (Quantum Field Theory). I come across the following integral $$\int_0^\infty \mathrm{d}k \: e^{-...
2
votes
1answer
42 views

In which sense are the Cauchy-Riemann equations elliptic

I often read that the rigidity and smoothness properties of holomorphic functions can be explained by the fact that the Cauchy-Riemann equations are elliptic. In which sense is that true? Obviously ...
0
votes
0answers
38 views

Complex Integration by parts

Suppose that f is analitic in an opened region $G$, which has $0$. Suppose $C$ is a closed curve in $G$, which encloses $0$, also it is positive and it begins and ends in $z_0$. Show that $\oint_C (...
0
votes
1answer
17 views

Two variable differentiable function

Let $f,g:\mathbb{R}^2 \to \mathbb{R}$ be two differentiable functions such that $f(x+1,y)=f(x,y+1)=f(x,y)$ and $g(x+1,y)=g(x,y+1)=g(x,y)$ for all $(x,y)\in \mathbb{R}^2$ Choose the correct statements ...
2
votes
3answers
43 views

Write as the sum of a series

The question asks to write $\dfrac{1}{1-i-z}$ as the sum of a series such that $\left|-z-i\right|<1$ but I genuinely have no idea how to do it, or even where to start.
2
votes
1answer
29 views

Question on square roots of complex functions

LeI have a function $f(x)$ where $x\in R$ and $|x|<1$. Now, I can write $f(x)=\sqrt{1-x^2}g(x)$ for some non-divergent $g(x)$ in the domain of definition of $x$. Can I write the following: $$\sqrt{...
0
votes
3answers
33 views

Is the set $S = \{z \in \mathbb{C} : |z − 3 − 2i| \lt 4 , \operatorname{Re}(z) \gt 0 \}$ open?

I have made a sketch of the set $S = \{z \in \mathbb{C} : |z − 3 − 2i| \lt 4 , \operatorname{Re}(z) \gt 0 \}$ on an Argand diagram and the region described by $S$ has no boundary points, since the ...
0
votes
2answers
29 views

How is the set $S = \{ z \in \mathbb{C} \: |z − 2| \lt 3|z|\}$ simply connected?

I have sketched the set $ S = \{ z \in \mathbb{C} \: |z − 2| \lt 3|z|\} $ and to me it appears to be the complement set to the closed ball $B(\frac14, \frac34)$. As a result I can think can come up ...
1
vote
1answer
128 views

What's remarkable about this transfer of structure from $\Bbb Z[\frac12]$ to $\Bbb C$?

What, if anything, is remarkable about this transfer of structure from $\Bbb Z[\frac12]$ to $\Bbb C$? $Y=\Bbb Z[\frac12]\setminus0$ $X=\Bbb Z[\frac12]\cap(\frac12,1]$ $N=\{\ldots2,1,\frac12,\frac14,...
0
votes
1answer
20 views

Finding a conformal map from this region into the unit disk [duplicate]

Problem: Find a conformal map from $\left\{ z \in \mathbb{C} \mid -1 < \text{Re} (z) < 1 \right\}$ to the unit disk. Attempt: What I did, was to first shift everything to the right with one ...
0
votes
0answers
31 views

How to show that $\exists N \in \mathbb{N}$ s.t $|a_n|^{1/n} < \frac{1}{R} <\frac{1}{r}$, where $\limsup |a_n|^{1/n} = \frac{1}{R} $

In the book of Functions of One Complex Variables by Conway, at page 31, it is claimed that However, I cannot understand the existence of such an $N$ that makes $|a_n|^{1/n} < 1/r$. I mean as far ...
-1
votes
2answers
45 views

How can we prove that $z^{n+1} \to 0_{\mathbb{C}}$ as $n\to \infty$?

In the book of Function of One Complex Variable by Conway, at page 31, it is given that However, normally, if $z$ was a real number, we could argue that $z^{n+1}$ goes to zero as $n \to \infty$ if $|...
1
vote
1answer
19 views

Is there a Dirichlet series that can be analytically continued to every point on the abscissa line?

Let $$f(z)=\sum_{n=1}^\infty \frac{a_n}{n^z}$$ be a Dirichlet series with convergence half plane ${\rm Re}z > \sigma_c$ where $\sigma_c\in\mathbb{R}$. My question is: is it possible that for each ...
1
vote
2answers
7 views

How do I get $\frac{\partial u}{\partial x}(x^*, y_0 + \Delta y)=\frac{\partial u}{\partial x}(x_0, y_0) + \epsilon$ using continuity?

Because the partial derivatives exist in $G$, the mean-value theorem says that there is a number $x^*$ between $x_0$ and $x_0 + \Delta x$ such that $u(x_0 + \Delta x, y_0 + \Delta y)-u(x_0,y_0 + ...
0
votes
1answer
27 views

Prove that the function can be continued into a larger domain

Prove that the function $f(z)=\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{z^n}{n}$ can be continued into a larger domain by means of the series $$\ln2-\frac{1-z}{2}-\frac{(1-z)^2}{2\cdot 2^2}-\...
-3
votes
0answers
53 views

How to evaluate the following complex integral? [on hold]

How can I evaluate the following integral? my working part 1 my working part 2 I have attached my working, and did arrive at an answer but unsure if my approach is correct.
0
votes
1answer
23 views

Show that there exists a holomorphic function $\Gamma$ that agrees with a curve $\gamma$

This problem was on an old qualifying exam and I was looking for ideas on how to get started: Let $\epsilon>0$, $I= (-\epsilon, \epsilon) \subset \mathbb{R}$ and $\gamma: I \rightarrow \mathbb{C}...
0
votes
0answers
35 views

Maximum and minimum of complex functions [on hold]

Let $f(z) = z^7 + z^6 + z - 2018, $ where $z = a + bi.$ The function $g(z) \colon \mathbb{C} \to \mathbb{R}$ defined by $g(z) = |f(z)|$ admits a maximum $x$ and minimum $y$ in the compact $A_n = \{ z ...
0
votes
1answer
44 views

Can I write $(ix+\sqrt{1-x^2})=\sqrt{ix+\sqrt{1-x^2}}*\sqrt{ix+\sqrt{1-x^2}}$ when $x \in (-1,1)$?

Can I write $(ix+\sqrt{1-x^2})=\sqrt{ix+\sqrt{1-x^2}}*\sqrt{ix+\sqrt{1-x^2}}$ when $x \in (-1,1)$? If not, then what is the rule for breaking up complex functions in square roots? Sorry if this is ...
1
vote
1answer
24 views

Family of analytic function in the unit disk problem

We denote by $D$ the open unit disk and consider the family $$\mathcal{F}=\{f: D\to D, f \text{ is analytic and } f'(0)=\tfrac{1}{2}\}.$$ Prove that there is a function $g\in \mathcal{F}$ such ...
1
vote
0answers
20 views

Composition of a periodic holomorphic function and logarithm in complex analysis

I am trying to solve a problem: Let $f(z)$ be a smooth real-valued harmonic function on the punctured unit disc $\mathbb{D}^* = \mathbb{D} \setminus \{0\}$. Show that \begin{align} f(z) = \mathrm{Re}(...
1
vote
2answers
31 views

Find the Laurent expansion $f(z)=\frac{1}{z(1-z)^2}$.

Find the Laurent Series expansion of $$f(z)=\frac{1}{z(1-z)^2}$$ at $z=1$. How do I do this when I am not given any region?
1
vote
1answer
72 views

Why the conditions $w(0)=0$ and $w(2)=\infty$ map the region $|z-1|<1$ onto the region $\Re w>0$?

I want to find a linear fractional transformation which maps the region $D$ of the $z$-plane onto the region $G$ of the $w$-plane, where $D=\{z;|z-1|<1\},~G=\{w;\Re w>0\}$ This is an exercise ...