Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

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Solving $\int_{-\infty}^{\infty}\frac{x^2e^x}{(1+e^x)^2}dx$

I am attempting to use residues to solve $\int_{-\infty}^{\infty}\frac{x^2e^x}{(1+e^x)^2}dx$; the answer is $\frac{\pi^2}{3}$. I have tried to split $\frac{x^2e^x}{(1+e^x)^2}$ into two…
Koskarium
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How to find a conformal mapping of the first quadrant.

Find a conformal mapping of the first quadrant onto the unit disc mapping the points $1+i$ and $0$ onto the points $0$ and $i$ respectively. I think that i need to use "the change of variables $w=z^k$" but how? And why do we apply this? Please…
Nrsnr
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Applications of the "soft maximum"

There is a little triviality that has been referred to as the "soft maximum" over on John Cook's Blog that I find to be fun, at the very least. The idea is this: given a list of values, say $x_1,x_2,\ldots,x_n$ , the function…
Tom Stephens
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Hahn-Banach to extend to the Lebesgue Measure

I remember reading an example in a textbook that went something like this: if we take a function $\ell(f) = \int_{0}^{1}f(t)\, dt$, (with this being the Riemann integral) defined only on the set of continuous functions on $[0,1]$, then we may extend…
user2959
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Carathéodory's method gives a complete measure

I would really like to show that the following is true. "Suppose that $X$ is a set and $\theta$ is an outer measure on $X$, and let $\mu$ be the measure on $X$ defined by Carathéodory's method. Then if $\theta E = 0$, then $\mu$ measures $E$." I'm…
Harry Williams
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Optimal assumptions for the differentiation of integrals

Let us consider the following integral: $$I(x)=\int_\Omega f(x, \omega)\, d\omega, $$ where $\Omega$ is a measure space and $f\colon \mathbb{R}\times \Omega \to \mathbb{R}$ is such that $f(x, \cdot)\in L^1(\Omega)$ for all $x$. When can we…
Giuseppe Negro
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Set of discontinuous points

Suppose $f$ is function from $\mathbb{R}$ to $\mathbb{R}$. Let be the set $\mathbf{A}$ that contains all the discontinuous points of $f$.Is $\mathbf{A}$ Borel Measureable?
user95525
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Proving energy conservation for wave equation

Hi guys I have a midterm tommorow and I was doing this practice problem that I need help on. So any hints or solutions would be appreciated. Thank you for your time Problem The head of timpani is constituted by some kind of elastic membrane which is…
david ricardo
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Proof that absolute continuity implies differentiability a.e.

Can somebody recommend a book/resource that provides a proof that absolute continuity of a function implies its almost-everywhere differentiability?
Stromael
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Questions about coercive functions and its implications

Given this definition: A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is $coercive$ if $$\lim_{||x||\rightarrow\infty}f(x) = \infty.$$ Explicitly, this means that for any $M>0$ there is an $R>0$ such that $||x||\geq R$ implies $f(x)\geq M$. I…
RDizzl3
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If a sequence of boundaries converges, do the spectrums of the enclosed regions also converge?

A planar region will have associated to it a spectrum consisting of Dirichlet eigenvalues, or parameters $\lambda$ for which it is possible to solve the Dirichlet problem for the Laplacian operator, $$ \begin{cases} \Delta u + \lambda u = 0 \\…
anon
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Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum.

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum. I want to prove this. This is my proof: Since $X$ is compact then it follows that $f(X)$ is compact…
RDizzl3
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A problem about Cantor set and found when learning dynamical systems.

Consider the family of functions F(x)=$x^3 -\alpha$x, for $\alpha \gt 0$ Prove that if $\alpha$ is sufficiently large, then the set of points |$F^n(x)$| which do not tend to infinity is a Cantor set. Note:$F^n(x)$ means the iteration of the…
ranky
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The local lipschitz condition implies differentiability?

I know differentiability implies the local lipschitz condition. however, I am not sure the converse. Actually, I think it might be. The definition of the local lipschitz condition is that for $$ f: A \subset \mathbb{R}^m \rightarrow…
Block Jeong
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A problem about limit

Problem: Suppose a sequence $\{x_n\}_{n\in\mathbb{N}}$ satisfies that $$\lim_{n\to\infty}\bigg(x_n\cdot\sum_{k=0}^nx_k^2\bigg)=1.$$ Prove that $\lim_{n\to\infty}\sqrt[3]{3n}x_n=1$. I do not have much idea about how to construct precisely this…
OnoL
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