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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2
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0answers
25 views

Behaviour of $\zeta(s)$ near $s=1$

I would appreciate if somebody could run this over and see if it works out? any suggestions or pointers would be appreciated. I denote the standard eta function $\eta$ by $\zeta^{*}$. I have not used ...
4
votes
0answers
22 views

Uniform limit of injective analytic functions is injective

I'm stuck on the following problem: Let $f_n$ be a sequence of injective analytic functions on the unit disc $D$ such that $f_n$ converges uniformly to $f$ on compact subsets of $D$. Show that ...
2
votes
1answer
35 views

$f: \Omega \rightarrow \Omega$ bounded, $f(z_0) = z_0$, show $|f'(z_0)| \leq 1$

I'm stuck on the following problem: Let $\Omega$ be a bounded domain, and $f: \Omega \rightarrow \Omega$ analytic such that $f(z_0) = z_0$. Show that $|f'(z_0)| \leq 1$. What I did so far was ...
0
votes
1answer
27 views

Why does Liouville's Theorem imply that there are no 1-1 holomorphic maps from $\mathbb{C}$ to $D(0,1)$?

Liouville's Theorem say that every bounded entire function is constant. I can't see why one of the consequences of it is there is no 1-1 holomorphic maps from $\mathbb{C}$ onto $D(0,1)$, where ...
2
votes
1answer
41 views

In which sense does Cauchy-Riemann equations link complex- and real analysis?

On page 12 of Stein, Shakarchi textbook 'Complex analysis', the authors state that the Cauchy-Riemann equations link complex and real analysis. I have completed courses on real and complex analysis, ...
0
votes
0answers
38 views

What is Laurent series expansion of $\frac{1}{e^z-1}$ around $z=0$? [duplicate]

Consider $$f(z)=\frac{1}{e^z-1}$$ I want to expand it over $z_0=0$ in a Laurent series. We know that $$e^z=1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots$$ And I know that ...
0
votes
0answers
28 views

What f(z) will make your 2d self right again? [on hold]

You accidentally got turned inside out while exploring the world of complex functions. What F(z) will make you right again and inverse the "inside out" ? z=x+iy and if I give more info, I will answer ...
0
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2answers
38 views

$f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|\leq 2\}$ such that $\iint_D=|f(z)|^2\,dx\,dy\leq 3\pi$. Maximize $|f''(0)|$

Determine the largest possible value of $|f''(0)|$ when $f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|<2\}$ with the property that $\iint_{D}|f(z)|^2\,dx\,dy\leq 3\pi$. I ...
1
vote
1answer
28 views

Schwarz Reflection Principle on a unit disk

Suppose $f$ is a analytic function defined on $\bar{D}(0;1)$ and has real value on the boundary. I'm trying to show $f$ can be extended to entire plane by $$g(z) = \begin{cases}f(z) &, \lvert ...
6
votes
5answers
65 views

If $|f| \le |g|$, does analytic continuation of $g$ imply analytic continuation of $f$?

Let $f,g$ be two holomorphic functions on a domain $D$ such that $|f(z)| \le |g(z)|$ for all $z \in D$. Further suppose that there is an analytic continuation of $g$ to a bigger domain $D'$. Does that ...
6
votes
2answers
35 views

Show that $f$ is continuous at $0$ and it satisfies the Cauchy Riemman conditions but it is not differentiable.

Let $f:\Bbb{C}\to \Bbb{C}$ be defined as $$f(x+iy)= \frac{x^{3}-y^{3}+i(x^{3}+y^{3})}{x^2+y^2} \text{ if} x+iy \neq 0$$ and $f(x+iy)=0$ if $x+iy=0$ Show that $f$ is continuous at ...
0
votes
2answers
36 views

Finding a homography in $\hat {\Bbb {C}} $

Let $z_2,z_3,z_4$ be distinct points of $\hat{\Bbb{C}}$. Show that there exists a unique homography $T:\hat{\Bbb{C}} \to \hat{\Bbb{C}}$ of the form $T(z)=\frac{az+b}{cz+d}$ with $ad-bc \neq 0$ such ...
1
vote
1answer
33 views

What is the difference between the following $2$ sets?

What is the difference between the following two sets? $\{s\in\mathbb C:\Re(s)\ge1+\delta\},\quad\delta>0$ $\{s\in\mathbb C:\Re(s)>1\}$ I read that $\displaystyle\sum\limits_{n\in\mathbb ...
1
vote
0answers
20 views

Integral of ratio of complex polynomials

Let $p(z),q(z) \in \mathbb{C}[z]$ two polynomials with coefficients in $\mathbb{C}$ s.t. $deg(p) = m$, $deg(q) = n$ and $n \ge m +2$. I need to show that $$ \lim_{R \to \infty} \int_{|z| = R} ...
1
vote
0answers
23 views

Help with the integral $\int_{0}^{\infty}\frac{x^{y}}{\Gamma(y)}\cos(y)dy$

We have the integral : $$\int_{0}^{\infty}\frac{x^{y}}{\Gamma(y)}\cos(y)dy$$ We have: $$\frac{1}{\Gamma(y)}=\frac{i}{2\pi}\int_{C}(-t)^{-y}e^{-t}dt$$ Where the path $C$ encircling 0 in the positive ...
1
vote
0answers
36 views

Possible Connections between Harmonic Analysis, Potential Theory and Analytic Capacity for a Fourier Analyst

So, Folks, here's the deal: After looking at this question, posted a little earlier on this site, and getting quite inspired by the beauty of this kind of result, I have got quite interested on this ...
2
votes
1answer
35 views

Question About Filled Julia and Julia Sets

Question: Let $Q_{c}(z) = z^{2} +c $ which $ c \in \mathbb{C}$ and suppose that $z_{0} \in K _{c}$ for the filled Julia Set, $K_{c}$ of $Q_{c}$. Suppose further that $z_{1} = Q_{c}(z_{0})$ and it ...
1
vote
1answer
25 views

Equivalence of branched covers of the Riemann sphere

Consider the functions $f(z)=z^4$ and $g(z)=z^4+1$, branched covers of $S^2$. These functions have the same branch data, so they should be equivalent in some way. In what way are they equivalent?
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0answers
8 views

Continuity of the Loewner flow (SLE theory).

In the SLE paper "Basic Properties of the SLE" from Rohde and Schramm, it is mentioned on page 898 that the map $$(y,t)\mapsto g_t^{-1}(iy+\xi(t))$$ is clearly continuous on $y>0,t>0$, where ...
3
votes
1answer
204 views

Prove that the Mandelbrot Set Is A Closed Set

The Problem: Suppose we define the Mandelbrot Set as the following For $c \in \mathbb{C}$ , $\mathbb{M}$ = $({c:|c| \leq 2}) \cap ({c: |c^2 + c| \leq 2}) \cap ({c: |(c^2+c)^2 +2| \leq 2}) \cap ...
2
votes
2answers
79 views

$\zeta(2n)$ proof [duplicate]

Can anybody pass me on a good source to see the steps in proving, \begin{equation} \zeta(2n) = \frac{(-1)^{k-1}B_2k (2 \pi)^{2k}}{2(2k)!} \end{equation} I know how we start by looking at the product ...
1
vote
0answers
26 views

simplifying complex expression

Hi I am trying to simplify the following expression:$$ \left|\frac{1}{a+ib}\left(\frac{J_1(c x)}{J_1(c b)}-x\right)\right|^2,\quad a,b,x\in \mathbb{R}, \ c\in \mathbb{C} $$ Is there a simple way of ...
0
votes
3answers
26 views

Limit of a complex valued function.

Let $f(z) = (\frac{z}{\bar{z}})^{2}$ , be a complex valued function , we need to prove that $\lim_{z \to 0} f(z)$ does not exists. So , to prove that its limit doesn't exists , we approach (0,0) from ...
1
vote
2answers
84 views

Complex Analysis ( Limits at a point ).

We need to prove that $ \lim_{z \to z_{0}}(z^{2}+c)$ = $z_{0}^{2}+c$ , where c is a complex constant , using $\epsilon - \delta$ definition , where $z , z_{0}$ are complex variables. What I tried : ...
0
votes
0answers
36 views

More elegant way for solving $y(x) = y_{1}(x) + y_{2}(x)$ in $y'' - 10y' + 28y = 29xe^{-x}$

Is there a more elegant way for solving $y(x) = y_{1}(x) + y_{2}(x)$ in $y'' - 10y' + 28y = 29xe^{-x}$ than to use Euler's identity and get the general solution through brute computation?
2
votes
0answers
36 views

isomorphism between function space and complex matrices

How would you show that $\mathcal{L}(X) \cong \mathbb{C}^{n \times n}$, where $X= \mathbb{C}^{n}$. Note that $\mathcal{L}(X)$ denotes the space of linear bounded functions on $X$. Is this a specific ...
-1
votes
0answers
22 views

Prove the result on connected sets in complex analysis. [on hold]

If $B = S \cup \{$some or all of its limit points$\}$, then $B$ is connected.
0
votes
0answers
31 views

Prove that a bijective entire function is uniformly continuous

Let $f$ be a bijective entire function. Prove that $f$ is uniformly continuous. I want a direct proof of this without using the fact that $Aut(\Bbb C)$ is the collection of linear polynomials ...
-1
votes
0answers
16 views

$\int_0^1 \log|x-\zeta|dx\ge (\log|\zeta|+\log|1-\zeta|)/2-1$ [on hold]

I recently came across this inequality: Prove that for any $\zeta\in\mathbb{C}$, $\zeta\ne 0,1$, we have that $$\int_0^1 \log|x-\zeta|dx\ge \frac{\log|\zeta|+\log|1-\zeta|}{2}-1.$$ How do you prove ...
0
votes
0answers
21 views

curl-free, conservative vector fields in complex analysis

I just verified that for the conjugate of an analytic function $\bar{f}$=u-iv, this conjugate function is curl-free - the Cauchy-Riemann equations forces this to be the case. Then we can consider ...
1
vote
0answers
38 views

How to understand the Identity Theorem in complex analysis, from the point of view of power series expansions

The theorem states that if $f$ and $g$ are analytic functions and their values agree on an open set that is contained in a larger, connected domain, then $f$ must equal $g$ on the entire domain. (The ...
3
votes
2answers
78 views

Geometry of images of maps $f: \mathbb R \to \mathbb C$?

I am having trouble seeing what a continuous map $f: \mathbb R \to \mathbb C$ might look like. If it was linear it would look like a line but it's not clear to me what happens if it's any map. I ...
2
votes
1answer
54 views

Is a bijective entire function uniformly continuous?

Let $f$ be an entire function such that $f$ is bijective. Is then $f$ uniformly continuous? I am thinking on this when trying to compute the analytic automorphisms $Aut(\Bbb C)$. I know that ...
2
votes
1answer
54 views

Difference between line integrals in complex analysis and real analysis,

The formula in complex analysis is $$\int f(\gamma(t))\cdot(\gamma'(t)dt$$ and the formula in the real variable setting, for a gradient field, is: $$\int F\cdot dr$$ $$=\int f_x\,dx + f_y\,dy + ...
1
vote
0answers
31 views

Prove that $F_1$ and $F_2$ are continuous and that $\int_{\gamma_1}F_1(z) dz = \int_{\gamma_2}F_2(z) dw$

Let $\Omega_1, \Omega_2 \subseteq \mathbb{C}$ and let $\gamma_1: [a,b] \to \Omega_1$, $\gamma_2: [c,d] \to \Omega_2$ be paths. Let $f$ be a continuous function defined on $\gamma_1 \times \gamma_2$ ...
0
votes
0answers
30 views

Singularity type for ratio of functions

Let $f,g:\mathbb{C}\to\mathbb{C}$ be two functions with a pole of order $n$ in $z_0$. I need to classify the singularity type of $\frac{f(z)}{g(z)}$ in $z_0$. I would say (intuitively) that $f \over ...
4
votes
0answers
33 views

Generalized Trigonometric Functions in terms of exponentials and roots of unity

I am trying to come up with generalized trigonometric functions using the exponential definition that we use today for the trig functions sine and cosine $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}; \cos x ...
0
votes
0answers
13 views

Poles and zero of a function inside an annulus

Let $f(z)=\tan z-\frac{z}{z^2+1}$. How many distinct poles and zeros does $f$ has inside the annulus $(N-1/4)\pi\leq |z|\leq (N+1/4)\pi$ for $N$ arbitrarily large. How large must $N$ be? For poles it ...
3
votes
2answers
41 views

Complex line integrals in increasing directions

The problem I am stuck on is :Evaluate $\displaystyle\int\frac{dz}{z^2+4}$ along the line $x+y=1$ in the direction of increasing x ..... Nothing I have learned in my independent study of this subject ...
1
vote
2answers
35 views

Max Mod Principle

I'm stuck with the following exercise: Let $f$ be holomorphic on an open set containing $\bar{D}$, the closed unit disk. Prove that there exists a $z_0 \in \partial D$ such that ...
1
vote
1answer
42 views

Logarithm Propr

I'm having a bit of trouble proving the following property: Theorem If $Re(z)>0 $ and $ Re(w)≥0$, then $\log(zw)=\log(z)+\log(w)$, where log is the principal branch. I know that $\log (zw) = ...
1
vote
1answer
18 views

Upper Bound on Complex Line Integral

I'm working through the second edition of Complex Variables by Stephen Fisher, and reached a proof involving the upper bound of line integrals, namely $$ \left| \int_\gamma u(z)\;dz \right| \leq ...
1
vote
1answer
49 views

How to Write a Vector Multiplication as a Trace of Matrix?

Let $\mathbf{w}_j\in\mathbb{C}^{M\times 1}$ and $\mathbf{h}_k\in\mathbb{C}^{M\times 1}$ be two complex vectors. How to prove this? ...
1
vote
1answer
35 views

Complex Conjugation problem using the identity $|x|^2=xx^*$

Show that $$|c|^2= \frac{4k^2}{k^2 +\gamma^2}$$ given (1)$$a+b=c$$ and (2)$$ik(a-b)=-\gamma c$$ This was given in a lecture without proof, so there's probably a very simple way of proving the ...
3
votes
1answer
42 views

Singularity type of $\frac{1}{z} e^{-\frac{1}{z^2}} $

I've been asked to compute the singularity type of $f(z) := \frac{1}{z}e^{-\frac{1}{z^2}} $. Here's my reasoning: $$ \frac{e^{-\frac{1}{z^2}}}{z} = z^{-1} \sum_{n=0}^\infty \big( -z^{-2} \big)^n ...
0
votes
1answer
25 views

Combining Moebius transformations

Moebius transformation in this case $\frac{az+b}{cz+d}$ for complex $z$. I have several transformations I want to apply to an initial $z$. For example first transform $f(a,b,z) = z + (a + bi) = ...
1
vote
2answers
39 views

Complex analysis proof about $|f(z)|$

I have to prove the following and have absolutely no idea where to start: If $f$ is holomorphic in $|z|>R$ and its limit at $\infty$ is $0$, then $\exists \; m \in \mathbb{N}$ such that $|f(z)| ...
0
votes
1answer
33 views

Explain about proof

Let $0 \leq R_1 \leq R_2 \leq \infty$ and let $f$ be holomorphic in the annulus $R_1 < |z - z_0| < R_2 $. Then, for any $r_1, r_2, z $ such that $R_1 < r_1 <|z-z_0| < r_2 < R_2$, we ...
3
votes
1answer
58 views

Find all entire function $f$ such that $\lim_{z\to \infty}\left|\frac{f(z)}{z}\right|=0$

If $f$ is an entire function such that $\lim_{z\to \infty}\left|\frac{f(z)}{z}\right|=0$ then find the function $f$. Replacing $z$ by $\frac{1}{z}$, we get $$\lim_{z\to 0}|zf(1/z)|=0$$This shows ...
6
votes
1answer
100 views

Is there a deep reason why replacing $\cos(x)$ with $e^{ix}$ and taking the real part often makes a contour integral work out?

I'm grading a complex analysis course right now and it turns out to involve a lot of contour integration. For instance, students are asked to find the integral $$\int_0^\infty \frac{\cos (ax)}{(x^2 ...