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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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} ...
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16 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 ...
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0answers
20 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 ...
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24 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 ...
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15 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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1answer
196 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 ...
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2answers
68 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 ...
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24 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 ...
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3answers
24 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 ...
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81 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 : ...
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35 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?
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
75 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 ...
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1answer
51 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 ...
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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 + ...
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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 $\gamma_1 \times \gamma_2$ ...
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28 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 ...
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32 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 ...
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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
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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 ...
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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 ...
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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) = ...
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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 ...
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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? ...
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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 ...
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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 ...
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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) = ...
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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)| ...
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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
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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 ...
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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 ...
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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 ...
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2answers
110 views
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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 ...
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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 ...
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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 ...
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1answer
29 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} : ...
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60 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 ...
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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 ...
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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 : ...
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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 ...
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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 ...
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3answers
82 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?
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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
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1answer
84 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 ...