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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How to prove a continuous bijection on a compact set is a homeomorphism?

Let $E\in R^n$ be a compact set. $f$ is a continuous bijection on $E$. Try to prove that $f$ is a homeomorphism from $E$ to $f(E)$.
weak solution
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The asymptotic of a integral.

Let $\alpha>0$. Then please prove that: $$\lim_{x\rightarrow+\infty}\left(\int_0^{+\infty}t^{-\alpha t+x}dt\right)\left[\sqrt{\frac{2\pi}{e^\alpha}} x^{1/2\alpha} \exp\left(\frac{\alpha}{e}x^{\frac{1}{\alpha}}\right)\right]^{-1}=1,$$ where…
DATO
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Tao, Analysis I, Definition of "$\epsilon$-closeness": Why weak inequality ($\leq$) instead of strict inequality ($<$)?

From Tao, Analysis I, p. 87, bottom: Why does he put $d(y,x)\leq \epsilon$ instead of $d(y,x)< \epsilon$ (which I think is more usual in these contexts)? What does this move achieve?
user986614
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Tao, Analysis I: Proof of trichotomy of integers--is the second paragraph necessary?

From Tao, Analysis I, pp. 77-78: I don't quite understand why the second paragraph is necessary. In the first paragraph, we use an earlier result---trichotomy of natural numbers: if $a$ and $b$ are natural numbers, then exactly one of (i) $a>b$;…
user986614
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Calculate the sum of the series $\sum_{n=1}^{\infty} \frac{n+12}{n^3+5n^2+6n}$

They tell me to find the sum of the series $$\sum a_n :=\sum_{n=1}^{\infty}\frac{n+12}{n^3+5n^2+6n}$$ Since $\sum a_n$ is absolutely convergent, hence we can manipulate it the same way we would do with finite sums. I've tried splitting the general…
HornyPigeon54
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Prove $\lim_{n\to \infty}{nb^n}=0$

Prove $\displaystyle\lim_{n\to \infty}{nb^n}=0, 0
user60887
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prove that The function h(x) = g(f(x)) is convex. given g(x) and f(x) are convex

SO, I have been reading on convex functions and came across the property that the function h(x) = g(f(x)) is convex. given g(x) and f(x) are convex, could someone give me the proof? given g is non-decresing
Riya
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Proofing Convergence of Cauchy Sequence in Metric Space with Cluster Point

Prove that if a Cauchy sequence in a metric space (not assumed to be complete) has a limit point, then it has a limit. My idea is using Cauchy sequence definition and limit point definition to proof convergence but having difficulties on writing it.
actsci stud tries2learn math
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$f\in C([a,b])$ Each closed subinterval of $[a,b]$ contains an open interval on which $f$ is constant. Is $f$ then constant on $[a,b]$?

Suppose $f: [a,b]\longrightarrow \mathbb{R}$ is a continous function. If $\forall [c,d]\subset[a,b]$ there exists an open interval $U\subset[a,b]$ s.t. $f$ is constant on $U$, then can we infer that $f$ is constant on $[a,b]$? Motivation: Some days…
Asigan
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$f$ real continuous map in $[a,b]$. $∀ x_0∈[a,b], ∃ (c, d) : x∈ (c, d)⊂ [a,b]$ s.t. $x$ is minimum for $f$ in $(c,d)⇒f$ is constant map

If $f(x)$ is continuous at the closed interval $[a,b]$, and $\forall x_0\in [a,b]$, there is an open interval $U$ containing $x_0$ s.t. $f(x_0)$ is none bigger than $f(\eta)$ for any $\eta\in U$, then prove that $f$ is constant on $[a,b]$. I think…
Asigan
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How do you explain this step in the following integral due to Stieltjes?

I am working through John Todd's Introduction to the Constructive Theory of Functions (1963). On page 47, he computes the following integral (due to Stieltjes): \begin{align*} \mu_n&=\int_0^\infty x^ne^{-x^{1/4}}\sin(x^{1/4})\;dx\\ …
InterestingFool
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If f and g are differentiable in a show that $$ is also differentiable in a

Suppose that $f,g: U\to \mathbb{K}$ where $U$ is an open set around $a$ are differentiable in $a$. Show that $: U \to \mathbb{K}$ where $(x) = $ is also differentiable i $a$. I know $f$ is differentiable in $a$ iff there is a $T…
Mina
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Proof of inequality with sum and product

$$ \forall n \in \mathbb{N}\backslash{0} \quad \forall x_1...x_n \in [0,1] $$ $$ 1-\sum_{i=1}^n{x_i} \leq \prod_{i = 1}^{n} {(1-a_{i})} $$ I'm trying to solve this one for a long time, I've tried to use induction but then I realized it can't be…
MeepMeep
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Showing that $\hat f(\xi) = \int_{\mathbb R} f(x) \exp (-2 \pi i \xi x) dx$ is continuous

Assume that $f(x) \leq \frac A {1+x^2}$ on $\mathbb R$. I now want to show that $$ \hat f(\xi) = \int_{\mathbb R} f(x) \exp (-2 \pi i \xi x) dx $$ is continuous. By some simple calculations I get $$ |\hat f(\xi) - \hat f(\xi+h)| \leq A…
user42761
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Prove this inequality concerning integral average.

Let $f\in L^1([a,b])$, and extend $f$ to be $0$ outside $[a,b]$. Let $$ \phi(x)=\frac{1}{2h}\int_{x-h}^{x+h}f $$ How to prove $$ \int_a^b\left | \phi\right | \leqslant\int_a^b\left|f\right| $$ To @martini: I asked this question because I find…
hxhxhx88
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