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, ...

learn more… | top users | synonyms (2)

2
votes
1answer
80 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 ...
0
votes
2answers
39 views

$\zeta(2n)$ proof

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
18 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
23 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
70 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
33 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
34 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
27 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
19 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
72 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
48 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
52 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
26 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
26 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
31 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
39 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
34 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
48 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
34 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
23 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
98 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 ...
2
votes
0answers
36 views

Integral using Cauchy's integral formula and residue theorem

So, I'm having trouble getting the correct value for the integral $\int_0^{2\pi} \frac{\cos^2(3\theta)}{5-4\cos(2\theta)}\mathrm{d}\theta$. I substitute the exponential form of cosine into the ...
3
votes
2answers
108 views
+200

On every simply connected domain, there exists a holomorphic function with no analytic continuation.

I am working on a question that requires me to prove that on every simply connected open subset of $\mathbb{C}$, there exists a holomorphic function that cannot be extended to a holomorphic function ...
5
votes
1answer
54 views

Is there a holomorphic function $f$ on the unit disc such that $|f(z)|\rightarrow\infty$ as $|z|\rightarrow 1$?

When I learnt that there exists a holomorphic function on the unit disc $D$ that cannot be continuously extended to a domain that is strictly larger $D$, I was taught the example ...
10
votes
1answer
124 views

why is $\zeta(1+it) \neq 0$ equivalent to the prime number theorem?

Reading through Titchmarh's book on the Riemann Zeta Function, chapter 3 discusses the Prime Number Theorem. One way to prove this result is to check the zeta function has no zeros on the line $z = 1 ...
5
votes
1answer
28 views

Prove that the given condition implies analytic continuation

Here is an old qual problem I'm working on, I have some idea, but I'm not sure if I'm correct or not. I would be happy if anyone could possibly confirm or correct me: Let $U=\{z\in \mathbb{C} : ...
1
vote
0answers
59 views

The Hessian quadratic form of a real function can't be complex

Let $\Omega\subseteq\Bbb C^n$ open, $z_0\in\Omega$, $r:\Omega\to\Bbb R$ twice real differentiable. We know $\Bbb C^n\simeq\Bbb R^{2n}$ is an isomorphism of vect.sp. So we think $\Bbb C^{n}$ as an ...
0
votes
0answers
24 views

How to show $\Im\{z\cot(z)\}$ is not $0$ in the first quadrant?

I know that $\Im\{z\cos(z)/\sin(z)\}$ is non-zero in the open first quadrant of the complex plane, $\Im z > 0$, $\Re z > 0$, but somehow I cannot seem to show it directly. I think I must be ...
2
votes
1answer
45 views

Contour integral of $\frac{1}{\sqrt z}$ with branch cut

I am a physicist who usually doesn't need to care about the fact that square root is not single-valued on the complex plane. But I would like to give a meaning to and compute the contour integrals : ...
2
votes
1answer
76 views

Complex integration on NON-simple closed curve

Compute the following integral with the help of Cauchy's residue theorem. $$\int_C\cot z\,dz$$where , $C:z=4e^{4i\theta}$ , $-\pi\le \theta\le\pi$ Here , singularities of are given by $\sin ...
6
votes
1answer
40 views

Proving Plancherel's theorem using Cauchy integral formula

Plancherel's theorem says that $f(x) = \frac{1}{2\pi} \int^\infty_{-\infty} F(k) e^{ikx} dk$ where $F(k) = \int^\infty_{-\infty} f(x)e^{-ikx}dx$. I'm wondering if we can prove this using Cauchy's ...
2
votes
3answers
81 views

$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$

Prove that for all $\xi \in \mathbb{C}$, $$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$$ I don't really know how to compute this integral. Can you please help me?
1
vote
1answer
60 views

What is the solution to this integral?

In some calculation, I encounter an integral of the form \begin{equation} \int_{-\infty}^\infty \text dz\ \frac{1}{z-i\varepsilon}e^{- a z^2+i b z}, \end{equation} where $a>0$ and $b$ are some ...
3
votes
3answers
55 views

Complex integration by Cauchy's residue theorem

Evaluate the following integral by Cauchy's Residue Theorem $$\int_C\frac{2z^2-z+1}{(2z-1)(z+1)^2}\,dz$$where , $C:r=2\cos \theta$ , $0\le \theta \le \pi.$ I have problem about the contour ...
14
votes
1answer
83 views

Is the ring of holomorphic functions on $S^1$ Noetherian?

Let $S^1={\{ z \in \Bbb{C} : |z|=1 \}}$ be the unit circle. Let $R= \mathcal{H}(S^1)$ be the ring of holomorphic functions on $S^1$, i.e. the ring of functions $f: S^1 \longrightarrow \Bbb{C}$ which ...
2
votes
1answer
44 views

Quartic equation or Sextic equation? And how to solve it?

In this arxiv paper (p. 11, eq. (3.2)) the authors claim that equation (3.2) is ... a quartic equation [...] which can be solved explicitly. The equation in question is \begin{equation} ...
2
votes
0answers
48 views

(Theoretical) Complex Analysis Textbooks

Most books I've seen on complex analysis do not develop it theoretically, which can be somewhat infuriating for the budding pure mathematician. What I am looking for are some comprehensive, rigorous ...
5
votes
2answers
66 views

geometrical interpretation of a line integral issue

I was wondering : if the geometrical interpretation of a line integral is that the line integral gives the area under the function along a path, then why the line integral is equal to zero when the ...
5
votes
0answers
22 views

Radial limits of composition of functions

Is it true that if $f\in H(U)$ is a holomorphic function whose nontangential limits exist a.e and $g\in H^\infty(U)$ is a nonconstant function whose range is in $U$ and whose radial limits exist ...
2
votes
3answers
90 views

Compute the integral

Compute the integral: $$\frac{1}{2\pi i}\int_{|z| = 1}\frac{(z-b)^m}{(z-a)^n}dz$$ where $|a| < 1 < |b|$; $m, n \in \mathbb{Z}$ My approach is using Cauchy integral formula, we have ...