# Tagged Questions

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### “Angle-preserving” equivalent to conformal?

I'd like to investigate the common turn of phrase that conflates "angle-preserving map" with "conformal map". Let $f:\Bbb R^2\to\Bbb R^2$ be a continuous function. I'll define $f$ to be ...
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### Conplex/real Integration and poles of function

So I am working on the following problem: Let $\Delta$ be the unit disk centered at origin, and $f$ is holomorphic on $\Delta-\{0\}$. If $$\int_\Delta|f|dxdy<\infty$$ show that $f$ has at most a ...
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### (absolute) Convergence of a series

I want to prove that the following series is convergent for $x>0$: $$\sum_{n=1}^\infty \left( \prod_{p\mid n} \frac{1}{p-1}\right) n^{-x}$$ I tried to estimate the product but I didn't get so ...
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### asymptotics from Laplace transform

Suppose I know that a non-negative random variable with density $f$ has the following Laplace transform: $$\hat{f}(s)=\int_0^{\infty}e^{-st}f(t)dt=\frac{1}{\cosh(\sqrt{2s}x)}$$ where $s>0$ and ...
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### What does monotonically convergent mean in this example.

Suppose that $$f(x)=\sum_{n=1}^{\infty}f_n(x)\,\,\,\,\,\,\,\,\,\,(x\in X)$$Where $f_n:X\rightarrow [0,\infty]$ for $n=1,2,3...$ Let $g_N=f_1+...+f_N$. Then the sequence $\{g_N\}$ converges ...
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### $f_n$ converges uniformly on $\overline\Omega$

Suppose $\Omega$ be a bounded region and $\{f_n\}_{n\in\mathbb N}$ a sequence of continuous functions on $\overline\Omega$ which are holomorphic in $\Omega$ and $f_n$ converges uniformly on the ...
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### Fourier transform and inverse transorm

I have to prove that for $f\in L^1(\mathbb{R})$ $$\check{\hat{f}}=\hat{\check{f}},$$ where $$\hat{f}(\xi):=\int\limits_{\mathbb{R}}e^{-i\xi x}f(x)\mathrm{d}x$$ and ...
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### separate vs joint real analyticity

Let $$f(x,y) := xy\exp\left(-\frac{1}{x^2+y^2}\right),$$ if $(x,y)\neq (0,0)$ and $f(0,0):=0$. I read the claim that $f$ is (a) separately real analytic on $\mathbb{R}\times\mathbb{R}$ (i.e. for ...
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### If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
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### For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
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### how to determine the existence of double limit?

Let $f(x,y)$ be a function of two variables. Are there any criterions to determine the existence of double limit $$\lim_{(x,y)\to(x_0,y_0)} f(x,y)?$$ If for all $y\in(y_0-\delta,y_0+\delta)$, ...
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### show that a function belongs to $L^1(m)$

Let m be normalized lebesgue measure on $d{\Bbb D}$ and for $|z|<1$ and $|w|=1$, let $p_z(w)=(1-|z|^2)/|1-\bar{z}w|^2$. I need to show that a- $p_z \in L^1(m)$ b- for every $x^*\in (L^1(m))^*$, ...
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### To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
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### The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools, ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
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### How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$?

I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of ...
### solving $|(x-3)(x-1)| $$\le |\frac{1-x}{x-3}| graphicly [closed] how to solve |(x-3)(x-1)|$$\le$ $|\frac{1-x}{x-3}|$ in the graphic method?
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, ...