-1
votes
0answers
33 views

Show a series converges absolutely [on hold]

Let $\Omega$ be an open subset in $\mathbb{C}$ and $\mathbb{H}=\{f \in L^2(\Omega): f\text{ is holomorphic in $\Omega$ }\}$.We know that $\mathbb{H}$ is a Hilbert space under the inner product ...
0
votes
2answers
34 views

solving $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}|$ graphicly [on hold]

how to solve $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}| $ in the graphic method?
0
votes
1answer
22 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
3
votes
1answer
64 views

asymptotical behavior of integral

I'm interest in the asymptotical of $$\int_{-\pi}^{\pi}\exp\Big((\cos z+i\alpha\sin z-1)t\Big)dz\hspace{3mm}\text{as}\hspace{2mm}t\to\infty$$ for $-1<\alpha<1$. Numberical result suggest that ...
8
votes
1answer
248 views

Evaluation of $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

I need some hints, clues for getting the closed form of $$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
1
vote
1answer
30 views

Complex measures vs. Positive Measures

In his real and complex analysis, Rudin writes that the right hand side of the expression $\mu(E) = \Sigma \mu(E_i)$ must necessarily converge for any countable partition $\{E_i\}$ of a measurable E, ...
1
vote
1answer
21 views

Is the characteristic function of a multivariate normal distribution a real analytic function?

The characteristic function of a multivariate normal distribution with mean $\mu \in \mathbb R^n$ and covariance $\Sigma \in \mathbb R^{n \times n}$ is given by \begin{align*} e^{it^T\mu - ...
0
votes
2answers
33 views

Vanishing moments and integrability

Is this correct? $\int_\mathbb{R}x^m f(x) dx=0 \iff \int_\mathbb{R}x^m \overline{f(x)}\,dx =0$. If yes then please tell the conditions under which this holds.
2
votes
3answers
64 views

value of an integral depending on a parameter in complex plane

For each $z\in\mathbb{C}$, evaluate the integral $$ \int_0^1\int_0^{2\pi}\frac{1}{re^{i\theta}+z}d\theta dr. $$ How to evaluate it? Thanks.
3
votes
3answers
79 views

convergence to exponential with order 1/n

We know that limit $\left(1+\dfrac{x}{n}\right)^n$ converges to $e^x$ but how can we prove that limit $\left(1+\dfrac{x}{n}+o(\frac{1}{n})\right)^n$ converges to $e^x$.
0
votes
3answers
78 views

Complex Roots and calculations

roots of the equation $z^6 =1-\sqrt3 i $ are $$z_1,z_2,z_3,z_4,z_5,z_6 $$ calculate:$$|z_1|^3 +|z_2|^3+|z_3|^3+|z_4|^3+|z_5|^3+|z_6|^3$$ also calculate: $$z_1^6 +z_2^6+z_3^6+z_4^6+z_5^6+z_6^6$$ ...
1
vote
0answers
34 views

Evaluation of Real Integral (with pole) using Complex

For research, I’m attempting to solve this real integral through the use of complex variables. I’m stuck on the approach of the actual integration because the pole lies on the real axis. I started ...
1
vote
0answers
27 views

Exchanging Limits in Series inversion

I have the Lagrange Bürmann formula as follows: $$\sum^{m-1}_{n=1}\frac{1}{n!}(w-b)^n\lim_{z \rightarrow a} \frac{d^{n-1}}{dz^{n-1}} \left(f^{'}(z)\frac{z-a}{g(z)-g(a)}\right)^n$$ Where g(z) is the ...
2
votes
4answers
66 views

$f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set $f(A)$

Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $$f(x,y)=(x^2-y^2,2xy).$$ Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set ...
4
votes
0answers
80 views

Clarification on tetration

So far when I looked at tetration I noticed it had a recursive relation. It's $t_2=2^{(t_1)}.$ For example if we start at point $(0,1)$, we can take the x-value of $0$, and $2^0=1$, then we take $1$ ...
1
vote
1answer
43 views

How to show that a complex-valued function is uniformly continuous?

should a function be uniformly continuous in both arguments if it should be uniformly continuous as a complex-valued function. For example how can I proove that ...
0
votes
0answers
22 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
3
votes
0answers
67 views

Why do fields seem to be a prerequisite for calculus?

I was in my Complex Analysis class, and the professor said that we should look for a field, rather than a group, to do calculus over. Why is this the case? I understand that we gain another operation ...
7
votes
1answer
175 views

Using normal families to bound a complex integral

I am trying to prove that $$\int_{\partial T(Q)} |F'(z)| \,ds(z) \lesssim \int\int_{T(Q)} |F'(z)| |\varphi'(z)|^2 \log \frac{1}{|z|} \,dx\, dy$$ This is an estimate on page $6$ of this paper by ...
24
votes
8answers
2k views

Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
0
votes
2answers
34 views

evaluating an integral with complex exponential (spectral density)

I am having a hard time figuring out how to evaluate this integral from a book that I am reading. Here's the background info but I doubt it's highly relevant to the problem at hand: $X$ is a real ...
1
vote
0answers
38 views

Is there a function $f : D^2 \setminus 0 \to R$ such that $f(x) x \to 0$ but $f(x)$ not bounded near $0$?

I just showed that, for $f$ holomorphic on the punctured disc, $zf(z) \to 0$ implies that $f$ has a removable singularity at $0$. This in turn implies that $|f(z)|$ is bounded near 0. (The other ...
4
votes
1answer
140 views

Double integral containing $e^{(b+ic)/z^2}$

I want to solve the two integrals \begin{aligned} I_3\,& = \int_{0}^{\infty} ze^{a/z^2 - z^2} dz\\ I_4\,& = \int_{0}^{\infty} \frac{1}{z}e^{a/z^2 - z^2} dz. \end{aligned} where ...
0
votes
1answer
21 views

Prove that the functions $g_k(z) = f_k \circ h_k(z)$ form a normal family.

I am having a bit of trouble with the following complex analysis question which originates from a qual. Some help would be awesome. Let $f_k :\mathbb{D} \rightarrow \mathbb{C}$ be a normal family of ...
1
vote
1answer
42 views

Open subgroup and group algebra

Let $G$ be a locally compact group and $H$ be an open subgroup of $G$. Consider the group algebras $L^1(G)$ and $L^1(H)$ with convolution product and consider $L^1(H)$ as a subalgebra of $L^1(G)$ ...
4
votes
3answers
298 views

A difficult integral evaluation problem

How do I compute the integration for $a>0$, $$ \int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx? $$ I want to find a complex function and integrate by the residue theorem.
13
votes
2answers
223 views

The 3 Integral $\int_0^\infty \frac{x\,dx}{\sqrt[\large 3]{(e^{3x-1})^2}}=\frac{\pi}{3\sqrt 3}\big(\log 3-\frac{\pi}{3\sqrt 3} \big)$

Hi I am trying evaluate this integral and obtain the closed form:$$ I:=\int_0^\infty \frac{x\,dx}{\sqrt[\large 3]{(e^{3x}-1)^2}}=\frac{\pi}{3\sqrt 3}\left(\log 3-\frac{\pi}{3\sqrt 3} \right). $$ The ...
13
votes
2answers
420 views

Integral $\int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma x)\,dx$

I am having trouble showing this equality is true$$ \int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma ...
5
votes
2answers
115 views

Integral $\int_0^{\pi/4}\frac{x^2\tan x}{\cos^2 x}dx=\frac{\log 2}{2}-\frac{\pi}{4}+\frac{\pi^2}{16}$

Hi I am trying to evaluate the definite integral which has a closed form given by: $$ \mathcal{I}=\int_0^{\pi/4}\frac{x^2\tan x}{\cos^2 x}dx=\frac{\log 2}{2}-\frac{\pi}{4}+\frac{\pi^2}{16}. $$ We can ...
1
vote
1answer
49 views

Construct a non-constant analytic function $f : \Omega_1 \to \Omega_2$ or show that this is impossible.

I am having a lot of difficulty with the following past qualifying exam problem. Any help would be awesome. Thanks. Let $\Omega_1 = \mathbb{C}\setminus \left \{\{0\} \cup \{\dfrac{1}{n}:n\in \Bbb ...
5
votes
0answers
76 views

Generating function of the squared Riemann zeta function

It's a well known fact that $$\sum_{k=2}^{\infty} \zeta(k) x^k=-x \psi(1-x)-x\gamma \space (|x|<1) $$ but I didn't meet yet a version for squared Riemann zeta function $$\sum_{k=2}^{\infty} ...
2
votes
0answers
25 views

Show that $P_a(z)=0$ iff $z=N(a)$ for polynomial $P_a$

Let for $a_0=(a_0,a_1,...,a_n)\in\mathbb C^{n+1}$ the polynomial $P_{a_0}=\sum_{k=0}^na_kz^k$ and $z_0\in\mathbb C$ with $P_{a_0}(z_o)=0$ and $DP_{a_o}$ (the differential matrix) invertible. Show ...
6
votes
3answers
152 views

Integral $\iint \limits_{{x,y \ \in \ [0,1]}} \frac{\log(1-x)\log(1-y)}{1-xy}dx\,dy=\frac{17\pi^4}{360}$

Hi I am trying to integrate $$ \mathcal{I}:=\iint \limits_{{x,y \ \in \ [0,1]}} \frac{\log(1-x)\log(1-y)}{1-xy}dx\,dy=\int_0^1\int_0^1 \frac{\log(1-x)\log(1-y)}{1-xy}dx \,dy $$ A closed form does ...
2
votes
1answer
66 views

series of an arbitrary sequence multiplying 1/n

Let $(r_n)_{n=1}^\infty$ be an arbitrary sequence of numbers in $[0,1]$. The series $\sum_{n=1}^\infty\frac{1}{n^2\sqrt{|x-r_n|}}$ converges for almost all $x$ in $[0,1]$. Is it true or not true? I ...
5
votes
4answers
196 views

Integral $\int_0^1\frac{dx}{\sqrt{\log \frac{1}{x}}}=\sqrt \pi$

Hi I am trying to prove this result below $$ \mathcal{J}:=\int_0^1\frac{dx}{\sqrt{\log \frac{1}{x}}}=\sqrt \pi. $$ The result is quite interesting however I realized I am not familiar with working ...
0
votes
0answers
45 views

Complex analysis winding number

We have that $f:\mathbb S^1 \rightarrow \mathbb S^1 $ and $f(z)=f(1)\widehat{\phi}(z)$ with $\widehat{\phi}(\exp{2\pi it}) = \exp(2 \pi i \phi(t)),$ where $\phi:I \rightarrow \mathbb{R} $ is a ...
10
votes
3answers
153 views

Integral $\int_0^\infty \log(1-e^{-a x})\cos (bx)\, dx=\frac{a}{2b^2}-\frac{\pi}{2b}\coth \frac{\pi b}{a}$

$$\mathcal{J}:=\int_0^\infty \log(1-e^{-a x})\cos (bx)\, dx=\frac{a}{2b^2}-\frac{\pi}{2b}\coth \frac{\pi b}{a},\qquad \mathcal{Re}(a)>0, b>0. $$ I tried to write $$ \mathcal{J}=-\int_0^\infty ...
10
votes
1answer
142 views

Integral $\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3$

Hi I am trying to prove this below. $$ I:=\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3 $$ where $$ \zeta_3=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I am ...
0
votes
0answers
67 views

a curious quasi-polynomial

Consider the quasi-polynomial $$ \lambda + 1-\alpha e^{-\lambda\tau_1}+\beta e^{-\lambda\tau_2}=0\;, $$ where $\lambda\in\mathbb{C}$, $\alpha,\beta\in\mathbb{R}$, and $\tau_1,\tau_2\in\mathbb{R}^{+}$. ...
6
votes
3answers
156 views

Integral $\int_0^{\pi/4}\frac{dx}{{\sin 2x}\,(\tan^ax+\cot^ax)}=\frac{\pi}{8a}$

I am trying to prove this interesting integral$$ \mathcal{I}:=\int_0^{\pi/4}\frac{dx}{{\sin 2x}\,(\tan^ax+\cot^ax)}=\frac{\pi}{8a},\qquad \mathcal{Re}(a)\neq 0. $$ This result is breath taking but I ...
2
votes
2answers
126 views

Integral $\int_0^\infty \frac{\sin a x-\sin b x}{\cosh \beta x}\frac{dx}{x}$

Hi I am trying to prove this interesting integral $$ \mathcal{I}:=\int_0^\infty \frac{\sin a x-\sin b x}{\cosh \beta ...
4
votes
1answer
143 views

$\int_0^\infty \frac{x\cos ax}{1+x^2}\coth \frac{\pi x}{4} dx=\frac{\pi}{2}e^{-a}+\cosh a\log \coth \frac{a}{2}+2\sinh a \arctan(e^{-a})-2$

Hi I am trying to prove this $$ \int_0^\infty \frac{x\cos ax}{1+x^2}\coth \frac{\pi x}{4} dx=\frac{\pi}{2}e^{-a}+\cosh a\log \coth \frac{a}{2}+2\sinh a \arctan(e^{-a})-2,\qquad a>0. $$ What a ...
3
votes
2answers
110 views

$\int_0^\infty \frac{\cos a x-\cos b x}{\sinh \beta x}\frac{dx}{x}=\log\big( \frac{\cosh \frac{b\pi}{2 \beta}}{\cosh \frac{a\pi}{2\beta}}\big)$

Hi I am trying to prove this interesting integral $$ \mathcal{I}:=\int_0^\infty \frac{\cos a x-\cos b x}{\sinh \beta x}\frac{dx}{x}=\log\left( \frac{\cosh \frac{b\pi}{2 \beta}}{\cosh ...
2
votes
3answers
156 views

Integral $\int_0^1\log(1+x)\frac{1+x^2}{(1+x)^4}dx=-\frac{\log 2}{3}+\frac{23}{72}$

EDIT: Small Typo Fixed now, Thanks to Sir Chen Wang! Hi I am trying to prove this result without using a series approach $$ \int_0^1\log(1+x)\frac{1+x^2}{(1+x)^4}dx=-\frac{\log 2}{3}+\frac{23}{72}. ...
0
votes
2answers
74 views

Homotopy invariance of line integral on manifolds

Consider a 1-form: $\omega\in\Gamma(\mathrm{T}^*M)$ and two differentiable curves: $\gamma,\tilde{\gamma}:[a,b]\to M:\gamma(a)=\tilde{\gamma}(a),\gamma(b)=\tilde{\gamma}(b)$ together with a ...
10
votes
6answers
250 views

Integral $I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}dx=\frac{10\pi^3}{81 \sqrt 3}$

Hi how can we prove this integral below? $$ I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}dx=\frac{10\pi^3}{81 \sqrt 3} $$ I tried to use $$ I=\int_0^1 \frac{\log^2x}{1-x(1-x)}dx $$ and now tried changing ...
3
votes
1answer
89 views

Integral $\int_0^\infty \log \frac{1+x^3}{x^3} \frac{x \,dx}{1+x^3}=\frac{\pi}{\sqrt 3}\log 3-\frac{\pi^2}{9}$

I am trying to prove this interesting integral $$ I:=\int_0^\infty \log \frac{1+x^3}{x^3} \frac{x \,dx}{1+x^3}=\frac{\pi}{\sqrt 3}\log 3-\frac{\pi^2}{9}. $$ I tried using $y=1+x^3$ but that didn't ...
6
votes
1answer
113 views

$\int_0^\infty \log(1+x^2) \frac{\cosh \frac{\pi x}{4}}{\sinh^2 \frac{\pi x}{4}}dx=4\sqrt 2-\frac{16}{\pi}+\frac{8\sqrt 2}{\pi}\log(\sqrt 2+1)$

I am trying to prove this interesting integral $$ I:=\int_0^\infty \log(1+x^2) \frac{\cosh \frac{\pi x}{4}}{\sinh^2 \frac{\pi x}{4}}dx=4\sqrt 2-\frac{16}{\pi}+\frac{8\sqrt 2}{\pi}\log(\sqrt 2+1). $$ I ...
1
vote
3answers
108 views

Integral $\int_0^\infty \frac{\sin^2 ax}{x(1-e^x)}dx=\frac{1}{4}\log\left( \frac{2a\pi}{\sinh 2a\pi}\right)$

How can we prove this ${\it{interesting}}$ integral $$ I:=\int_0^\infty \frac{\sin^2 ax}{x(1-e^x)}dx=\frac{1}{4}\log\left( \frac{2a\pi}{\sinh 2a\pi}\right) $$ I to write $$ I=\frac{1}{2}\int_0^\infty ...
1
vote
1answer
42 views

Find the Fourier series for $f(x) := |\sin(x)|$ and the sum of $\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {4n^2-1}$.

Find the Fourier series for $f(x) := |\sin(x)|$ and the sum of $\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {4n^2-1}$. I have computed $$c_n = \frac 1 {2\pi} \int^{\pi}_{-\pi} |\sin(x)|e^{-inx} dx = ...