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 .

42884 questions
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Understanding the equation $|x+1|=x^2 -1$

I want to understand the equation $$|x+1|=x^2 -1$$ $$\Leftrightarrow x^2 - |x+1| - 1 = 0$$ Case $1$: $$x+1 \geq 0 \Rightarrow x^2 - x-2 = 0$$ $$x_{1,2} = \frac{1}{2} \big( 1\pm \sqrt{1-4\cdot(-2)} \big) = \frac{1}{2}(1\pm3) \Rightarrow x_1 = 2, x_2…
stepbysteptomathpro
  • 685
2
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1 answer

How to show $f\equiv 0$?

$f(x)$ is differentiable, and for any $x\in \mathbb R$, $|f'(x)|\le \lambda |x|$, then how to show $f\equiv 0$ ? This is a question my student ask me, but I don't know how to deal it. So ask help here. Thanks for any hint or answer.
Farmer
  • 1,535
2
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2 answers

Suppose $f (x)$ is bounded and has a primitive on $[a,\,b]$, and $g\in C \big([a,\,b]\big) $. Must $f (x)\cdot g (x)$ have a primitive?

Suppose $f (x)$ is bounded and has a primitive on $[a,\,b]$, and $g\in C \big([a,\,b]\big) $. Must $f (x)\cdot g (x)$ have a primitive? Suppose it has a primitive, then it's necessary to be Riemann integrable in $[a,\,b]$.
Knt
  • 1,649
2
votes
1 answer

Open or Closed set in sup norm

$ B=\big\{x= (x_n)_{n\in\mathbb{N}}:|x_n|<\epsilon\text{ for all }n\in \mathbb{N}\big\}$ , where $\epsilon>0$ is given in $(\ell^{\infty},||\cdot||_{\infty})$ How do I go about showing if this set is open or closed?
Mathsstudent147
  • 1,495
2
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1 answer

Is it possible that $n^{\alpha}$ is rational for all $n \in \mathbb{N}$ with $0 < \alpha <1$?

I think that is impossible, i.e., for each $\alpha \in (0,1)$, there exists $n_0 \in \mathbb{N}$ such that $n_0^{\alpha}$ is irrational. Or equivalently, there is no $\alpha \in (0,1)$ such that $n^{\alpha}$ is rational for each $n \in…
Hsing
  • 305
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1 answer

Evaluate $\lim_{x\to0+}\sum_{n=0}^{\infty}2^{-n}\frac{\sin (2^nx)}{x}$.

Suppose $\displaystyle f(x)=\sum_{n=0}^{\infty}2^{-n}\sin(2^nx)$, evaluate $$ \lim_{x\to0+}\frac{f(x)-f(0)}{x}. $$ $f(x)$ converges uniformly on $\mathbb R$. But I don't know how to evaluate $$ \lim_{x\to0+}\sum_{n=0}^{\infty}2^{-n}\frac{\sin…
Knt
  • 1,649
2
votes
1 answer

A homeomorphism of the unit disk that cannot be extended to the boundary of its domain

I have a problem with the following exercise: Let $D^2$ be the unit disc and S^1 be the unit circle. Show that the function $ h: {D^2\setminus{S^1}} \to {D^2\setminus{S^1}} \\ h(re^{it})= \begin{cases} 0 &\text{if}\, r=0 \\ r \cdot…
Polymorph
  • 1,225
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3 answers

Evaluate $ \lim_{n\to+\infty}\sum_{k=1}^{n}(e^{\frac{k^2}{n^3}}-1)$.

Evaluate $$ \lim_{n\to+\infty}\sum_{k=1}^{n}(e^{\frac{k^2}{n^3}}-1). $$ Since $\sum_{k=1}^{n}(e^{\frac{k^2}{n^3}}-1)\leq n(e^{\frac{1}{n}})-n$, which implies that the limit is no more than $1$. But I met some problems in the method of enlarging…
Knt
  • 1,649
2
votes
2 answers

Find the explicit form of $ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1} $.

Find the explicit form of $$ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}. $$ Let $S(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}$. It has radius of convergence $1$. Let $S_1(x)=xS(x)$. Then…
Knt
  • 1,649
2
votes
2 answers

Prove $\frac{2}{\pi}\int_0^{\infty}\frac{\sin^2u}{u^2}\cos(2ux)\,du=\begin{cases} 1-x,&\mbox{$x\in[0,\,1]$}\\ 0,&\mbox{$x>1$}. \end{cases}$

Prove $$ \frac{2}{\pi}\int_0^{\infty}\frac{\sin^2u}{u^2}\cos(2ux)\,du=\begin{cases} 1-x,&\mbox{$x\in[0,\,1]$}\\ 0,&\mbox{$x>1$}. \end{cases} $$ Let $$ I(x)=\frac{2}{\pi}\int_0^{\infty}\frac{\sin^2u}{u^2}\cos(2ux)\,du. $$ It's easy to check that…
Knt
  • 1,649
2
votes
3 answers

Some estimation a series by integral

Let $a \in (0,1)$. Does there exist a constant $C>0$ or function $C(a)>0$, which may be a function of $a$ but not $k$, such that $$ \sum_{n=1}^\infty a^n n^k \leq C(a) \int_0^\infty a^x x^k dx $$ for all $k \in \mathbb N$?
Alex
  • 2,191
2
votes
1 answer

Does in invertibility of Hessian matrix $H_{F(X)}$ implies $v^tH_{f(X)}v\neq 0$?

Let $U$ be an open convex subset of $\mathbb{R}^n$, and $f: U\rightarrow\mathbb{R}$ be a function having continuous first and second partial derivatives on $U$. Let $H_{f(X)}$ denote the Hessian of $f$ at $X\in U$. If $\det(H_{f(X)})\neq 0$ for all…
Mohana Krishna
  • 21
2
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0 answers

Evaluate $\sum_{n=1}^{\infty}T_n\frac{x^n}{n!}$ where $T_n$ is the number of distinct partitions of $\{1,\cdots,n\}$

A partition of set $\Omega_n=\{1,\cdots,n\}$ is a family of nonempty sets $\{B_i\}$ where $\bigcup_iB_i=\Omega_n$ and $B_i\cap B_j=\emptyset\,(j\ne i)$. $\{\Omega_n\}$ is also a partition of $\Omega_n$. Evaluate…
Knt
  • 1,649
2
votes
1 answer

Show that a function over a mesurable set is the center of mass function

First of all excuse me because my english is not good enough. I need some help with this excercise, I tried to solve it for 1 hour but nothing ocurred. I know you are very altruist. Thank you.
Strauca
  • 189
2
votes
4 answers

Open interval, prove that you can derive new numbers

Prove If $I$ is an open interval, and if $x\in I$ , then there is some $d > 0$ such that $[x-d; x+d ] \in I$ I, for the life of me can't figure this one out. Despite being preceded by an easy exercise, and being seemingly intuitive, I just can't…
Roberto Singer
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