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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, isolated zeros principle...

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1answer
7 views

How do I get $a_{n-1} = -a_{n}(d_{1}z_{1} + d_{2}z_{2} + \cdots + d_{r}z_{r})$?

Show that if the polynomial $p(z) = a_{z}z^n + a_{n-1}z^{n-1}+ \cdots + a_0$ is written in factored form as $p(z) = a_{n}(z-z_1)^{d_1}(z-z_2)^{d_2}\cdots (z-z_r)^{d_r}$, then $a_{n-1} = -a_{n}(d_{1}...
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0answers
51 views

Could this problem be similar to the Riemann Hypothesis?

I've seen the below equivalence. For $a,b,x\in\mathbb{C}$, provided that there are no singularities on the right-hand side: $$\sum_{k=2}^{\infty}\sum_{j=1}^{\infty}\frac{x^k}{(ai j+b)^k}=-\frac{x^2}{...
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0answers
23 views

How to get the integral representation of $f'$ from that of $f$? (Cauchy's Integral Formula)

I know that if $f$ is holomorphic on an open set $\Omega$ containing a circle $C$ and it's interior, then for any $z_0$ in the interior of $C$ we have $$f(z_0) = \dfrac {1}{2 \pi i} \int_{C} \dfrac {...
2
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1answer
40 views

Show $\int \frac{\sin(x^p)}{x} dx = \frac{\operatorname{Si}(x)}{p} $

I was messing around with the Fresnel integral and the Sine integral and found that $\int_{0}^{\infty} \frac{\sin(x^2)}{x}dx=\frac{\pi}{4}$ but I dont see how to extend to irrational powers.
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2answers
34 views

Square root of holomorphic

For what type of sets the square root of a holomorphic function is also holomorphic? What is the restriction?
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3answers
46 views

Find all entire functions that satisfy following inequality

Find all entire functions that satisfy following inequality: $$ |f(z)| \leq |z| e^{\Re(z)} $$ for all $ z \in \Bbb C $
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1answer
39 views

Laurent series problem on $f(z)=\frac{ z }{ z^2-z-2 }$

I have problems with computing Laurent series of the function $f(z)=\frac{ z }{ z^2-z-2 }\quad$ in the ring centered in $0$ containing point $1+i$. I also have to find the radius of convergence of ...
1
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0answers
22 views

Union of thin sets is also thin

Question: Let $D\subset\mathbb{C}^N$ be open and let $A_1, A_2 \subset D$ be thin. Show that $A_1 \cup A_2$ is thin. The definition of thin that I'm using is the following. A set $A\subset D$ is ...
7
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1answer
287 views

A power series with decreasing positive coefficients has no zeroes in the disk

Let $\sum_{n=0}^\infty a_n z^n$ be a formal complex power series, with $a_n$ strictly decreasing to 1 as $n\to \infty$. It is easy to see that the radius of convergence is 1, so that this power series ...
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1answer
20 views

How $\text{Log}_a(Z)$ = $\frac{\text{Log}(Z)}{\text{Log} (a)}$ ? where $a,z \in \mathbb C$.

Can anyone tell me how $\ \text{Log}_a(Z)$ = $\frac{\ \text{Log}(Z)}{\ \text{Log} a}$ where $a,z \in \mathbb C$ ? I was told that the right hand side is not the definition of the left hand side. It ...
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1answer
30 views

Show that $g$ only have removable singularities in $\mathbb{D}$

I've been asked to show that $g(z)=\frac{\varphi_b (f(z))}{h(z)}$ only have removable singularities in $\mathbb{D}$, where $f: \mathbb{D} \rightarrow \mathbb{D}$ holomorphic with $f(0)=0$, $h(z) = \...
1
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3answers
44 views

Proving $e^{-z}=\frac{1}{e^z}$

Prove $$e^{-z}=\frac{1}{e^z}$$. Taking $z=z_1+z_2i$ then: $e^{-z}=e^{-(z_1+z_2i)}=e^{-z_1-z_2i}=e^{-z_1}.e^{-z_2i}$ I know the first part is easy to see $e^{-z_1}=\frac{1}{e^{z_1}}$ since $z_1$ is ...
2
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2answers
62 views

evaluate the integral by using Cauchy's residue theorem

evalulate the integral by using Cauchy's residue theorem $$\int_{|z|=1} \frac{1}{e^z -1-2z}dz$$ MY attempt : $ f(z) =\frac{1}{e^z -1-2z}$, now put $z= 1$ we get $f(z)=-1$ so $$\int_{|z|=1} \frac{...
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2answers
24 views

Complex function to Taylor and Laurent series

I am trying to express a function with Taylor and Laurent series. I've been reading my textbook and also various online resources, but I still can't follow any of the example problems. Here's what I ...
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0answers
17 views

On finding holomorphic anti-derivative of a function from the vanishing of path integral over any closed, piece-wise linear path

Let $f=u+iv: \mathbb C \to \mathbb C$ is a function such that $u,v$ are continuously differentiable functions. Suppose that for every piece-wise linear curve $\gamma : [a,b]\to \mathbb C$ with $\gamma(...
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0answers
20 views

For which subsets of $\mathbb{C}$ can a non-constant analytic function have an accumulation point of zeros?

Define $f$ to be complex analytic on some subset $S$ of $\mathbb{C}$ if $f$ is given locally by a power series (I'm not requiring $S$ itself to be open.) If $S$ is connected and open, then if there ...
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2answers
32 views

Integrate $\int_{|z|=1}z^{3}e^{1/z}dz$ - verification

I integrate over a circular path centered at 0 with radius 1 $\int_{|z|=1}z^{3}e^{1/z}dz=\int_{|z|=1}z^{3}\sum\limits_{n=0}^{\infty}\frac{1}{n!z^{n}}dz=\int_{|z|=1}\sum\limits_{n=0}^{\infty}\frac{1}{...
1
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1answer
34 views

winding number of paths

Let $c:[0,1]\to\mathbb{R}^2\backslash\{\mathbf{0}\}$ be a closed path with winding number $k$. Let $\tilde{c}=\rho(t)c(t)$, where $\rho:[0,1]\to(0,\infty)$ is function satisfying $\rho(0)=\rho(1)$. ...
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0answers
15 views

Understanding the proof of inversion formula for density using characteristic function

The formula is: $f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i\lambda x}\hat{f}(\lambda)d\lambda$ where $\hat{f}$ is the characteristic function, $f$ is continuous bounded on $R$ and both $f,...
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1answer
25 views

Geometric Interpretation of "$A,B\subseteq \mathbb{C}$, there exists a point $a∈A$ such that $∀x∈A,y∈B $ there exists $b∈B$ such that $|a−b|≤|x−y|.$

Suppose $A,B$ are subsets of the complex plane $C$ with $A$ compact. Then there exists a point $a∈A$ such that $∀x∈A,\ y∈B $ there exists $b∈B$ such that $|a−b|≤|x−y|$. This is not a duplicate of ...
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3answers
29 views

Integrate $\int_{|z|=1/2}\frac{e^{1-z}}{z^{3}(1-z)}dz$ verification

I integrate over the edge of a circle $K$ with radius 1/2 $\int_{|z|=1/2}\frac{e^{1-z}}{z^{3}(1-z)}dz=\int_{|z|=1/2}-\frac{e^{1-z}}{z^{3}}\frac{1}{(z-1)}dz$ By the Cauchy Integral form $f(w)=\frac{...
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1answer
21 views

convergence of bounded, holomorphic functions on the disk

I want to show that a sequence of holomorphic, zero-free functions on the disk converges uniformly to zero on compact subsets of the disk if $|f_n| < 1$ and $\lim_{n \rightarrow \infty} f_n(0) = 0$....
2
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1answer
30 views

Convergence of Euler's definition of the Gamma function

I was reading the wikipedia article on the Gamma function and found out that the original definition of it was... quite clever actualy. Here's the article. Anyway the definition is $$ \Gamma(z) = \...
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0answers
8 views

how functional reduce to Y(x,y,y') = y(x) + e x n(x)

How this functional can be expressed as above linear function . what math concepts and topics clear this fully and meaningful mathematical explanation.
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2answers
63 views

Computations with complex numbers and trigonometric functions

I have the following expression: $$\frac{(\cos x + i \sin x)-(\cos nx + i \sin nx)(\cos x + i \sin x)}{1-(\cos x + i \sin x)}$$ $$=\frac{\sin (\frac{nx}{2})}{\sin (\frac{x}{2})}\cos\biggl(\frac{(...
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5answers
57 views

$\sum_{\omega\in \mathbb{Z}(i)^*} |\omega|^{-2}$ does not converge.

I'm trying to prove that the series $$ \sum_{\omega\in \mathbb{Z}(i)^*} |\omega|^{-2} $$ The problem can be viewed as the sum above the fundamental region $\Omega^* = \{m\omega_1 + n\omega_2 : m,n\...
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0answers
20 views

Find the hyperbolic distance between $1$ and $2$ in $S=\{z: |Im(z)|<\pi/2\}$

1) Find the hyperbolic distance between $2$ and $5+i$ in the upper half plane $H=\{ z: Im(z)>0\}$. 2) Find the hyperbolic distance between $1$ and $2$ in $S=\{z: |Im(z)|<\pi/2\}$ Ans: For (1), ...
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1answer
39 views

An analytic doubt from Riemann zeta function-Titchmarsh

Doubt form the Book Theory of Riemann Zeta Function by Titchmarsh I've problem on Theorem $5.8$ in the equation $5.8.2$. When $k=l$ the from Lemma $5.7$ we get, $$ \sum_{n=N+1}^bn^{-it}=O\left(N^{1-1/...
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1answer
46 views

Why is this integral equality true? $\int\limits_{-\pi}^{\pi}|a+be^{it}|dt=\int\limits_{-\pi}^{\pi}\left||a|+|b|e^{it}\right|dt$

I was reading an article, and while proving a proposition, they state that if $a,b\in\mathbb{C}$, then due to periodicity we have $$\int\limits_{-\pi}^{\pi}|a+be^{it}|dt=\int\limits_{-\pi}^{\pi}\left||...
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0answers
36 views

Is $\ln{f}$ well-defined?

Let $f$ be holomorphic and nonzero on some open set $\Omega\subseteq \mathbb{C}$. Then is $\ln {f}$ well-defined? Usually, in order for $\ln {z}$, we need to choose a branch, e.g., $(-\infty,0]^c$, so ...
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3answers
78 views

Solve for $z:$ $(1+z)^n + (1-z)^n = 0$

I'm confused about how to approach this problem. I know that I can rewrite this as $$\left(\frac{1-z}{1+z}\right)^n = -1.$$ However, I do not know where to go from here. Any help would be greatly ...
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0answers
53 views

I think I am gonna get answer here faster sorry for bothering but can you answer me this pls? [on hold]

$$\zeta (\frac{1}{2})=\sum_{x=1}^{\infty }\frac{1}{\sqrt{x}}=?$$ Asking this question btw because I am considering the infinite series that looks like this: $$S = \sum_{x=1}^{\infty} \frac{1}{{x^{bi} \...
0
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1answer
32 views

Prove $|f(z)|\leq C$ whenever $|z|\leq 1/2$

Let $D=\{z\in\mathbb{C}:|z|<1\}$ and $f:D\rightarrow\mathbb{C}$ be a analytic function. Suppose $f(0)=1$ and $f(z)\notin(-\infty,0]$ for all $z\in D$. Prove that there exists a constant $C$ ...
1
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1answer
79 views
+50

Mellin inverse of $\sum_{n=0}^{\infty}\frac{\Gamma(n+s)\zeta(n+s)\zeta(n+1+s)}{\zeta(s)n!}\left(-\omega\right)^{n}$

I am trying to compute the inverse Mellin transform of : $$\sum_{n=0}^{\infty}\frac{\Gamma(n+s)\zeta(n+s)\zeta(n+1+s)}{\zeta(s)n!}\left(-\omega\right)^{n}$$ w.r.t. the complex number $s$. $\omega$ ...
1
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5answers
39 views

Separating real and imaginary parts $\frac{(\cos x + i \sin x) (\cos nx + i \sin nx)(\cos x + i \sin x)}{1-(\cos x + i \sin x)}$

I have the following expression: $$\frac{(\cos x + i \sin x) (\cos nx + i \sin nx)(\cos x + i \sin x)}{1-(\cos x + i \sin x)}$$ I need to separate the imaginary and real parts however I am not ...
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2answers
34 views

$f(z)=2 \sin(\sqrt z)$, how to find $f'(\frac i2\pi^2)$?

$f(z)=2 \sin(\sqrt z)$ and I want to find $f'(\frac i2\pi^2)$. We can take $\sqrt z$ be principal value of function $z^\frac1 2$. Should I start by considering $\sqrt z = e^{\frac12 \text{Log} (z)}$ ...
0
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0answers
24 views

If v is harmonic conjugate of u, then the harmonic conjugate of $3u^2 − v^3$ is harmonic conjugate of

Attempt: I tried to use cr-equations,but integration becomes clumsy. Also, I tried to form another holomorphic function which have real part equal to $3u^2 − v^3$ using any other holomorphic functions,...
0
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0answers
36 views

Meromorphic function satisfying $f(z) = f(z+1) =f(z + \sqrt{2}$) is constant.

I was looking this question and answer Take $f$ and entire function such that $f(z) = f(z+1)=f(z+\sqrt{2})$, then it is constant and I wonder what would happen if $f$ is not entire but meromorphic. ...
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3answers
32 views

Find the map from $\{ z: - \pi/2 < Im(z)<\pi/2\}$ to the vertical strip $\{ z: 0 < Re(z)<\log 2\}$.

Find the map from $\{ z: - \pi/2 < Im(z)<\pi/2\}$ to the vertical strip $\{ z: 0 < Re(z)<\log 2\}$. Using the map $f(z)=i (2/\pi)(\log 2) z$ we get the image of $f$ as $\{ z: \log 1/2 &...
4
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0answers
38 views

Solving a variant of the Poisson Boltzmann equation

The following is an equation I've derived in my personal research: $$ \frac{d^2V}{dx^2}=e^{\alpha x} \sinh(V) $$ I'm looking to solve it explicitly for V(x). It's a variant of the Poisson-...
2
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1answer
27 views

Prove that $\int_{\mathbb{R}}|x+iy|^{-\alpha}dx=c_\alpha y^{-\alpha}$.

Prove that $\int_{\mathbb{R}}|x+iy|^{-\alpha}dx=c_\alpha y^{-\alpha}$. where $c_\alpha$ is a constant to be determined. I tried to prove the problem using mathematical induction and it is okay for ...
1
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0answers
40 views

$\int_0^{+\infty} f(x) dx = - \lim_{a\rightarrow 0^+} \frac{1}{a} \sum Res_{\Bbb{C} \setminus \Bbb{R}^+} (f(z)z^a)$?

I find integrals on the real axes computed by complex number techniques and looking for a generalization working just on the positive semiaxes. I tried to put together this general result, but I am ...
1
vote
1answer
34 views

Proof of theorem of connectedness of an open set

I was watching an online lecture about complex analysis and in one if the first videos: The following theorem is stated: Let $G$ be an open set in $\mathbb{C}$. Then $G$ is connected if and only if ...
0
votes
1answer
23 views

root of complex number - which quadrant / find theta (or phi) in [-pi,pi]

how do I calculate the phi (/theta/angle/degree/no idea how it translates correctly) when calculating the roots of a complex number? I have seen the "formula" (principal argument for complex numbers) ...
1
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2answers
36 views

Simplifying 3rd root of (24 * sqrt(3))

I have problems following a solution towards simplifying a given polynomial. $$Polynomial: p(x)=x^5+{\sqrt 3}x^4+24{\sqrt 3}x^2+72x$$ the zeros of this function (Polynomial roots? English isn't my ...
1
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0answers
46 views

Proof verification using Schwarz Lemma

Let $f : \mathbb{D} \rightarrow \mathbb{D}$ be an holomorphic function in $\mathbb{D}$ with $f(0)=0$. Let $a_1, a_2,\ldots,a_n$ be $n$ different points in $\mathbb{D}$ with $f(a_j)=b \; \forall \; j =...
0
votes
1answer
49 views

Good textbook for questions/answers?

Does anyone know of any good textbooks which contain a lot of exercises and solutions? A lot have exercises, and that's useful but I really would prefer having solutions if I'm going to be doing 100+...
1
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1answer
38 views

Riemann Sphere Mapping

this is my first post so sorry if my question is too vague. I didn't see a related question posted, hence why I'm asking. I can't find any resources on it, but there's supposed to be a bijective ...
0
votes
1answer
28 views

How do you prove the sum of the roots of a complex polynomial is a ratio of two coefficients?

I am trying to follow Proof 1 from here: https://proofwiki.org/wiki/Sum_of_Roots_of_Polynomial. It proves that a complex polynomial $p(z)=a_nz^n+a_{n-1}z^{n-1}+...+a_0 $ with complex roots $z_1,....
0
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0answers
27 views

How to deal with the case that $f(z_0)=0$ for some $z_0\in C_R$?

In the content labeled by red box, if $f(z_0)=0$ for some $z_0\in C_R$, where $C_R$ denotes the circle of radius $R$ centered at origin, how to deal with it?