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
3
votes
0 answers

Dirac Delta distribution of discrete variables

There exists the following relation for the Besselfunctions $j_{l}(\alpha\,x)$, $j_{l}(\beta\,x)$ with $\alpha$, $\beta$ $\in \mathbb{R}$ and $x\in \mathbb{R}$ $$\int_{0}^{\infty}dx \, x^{2}\, j_{l}(\alpha\,x)\, j_{l}(\beta\,x) =…
navocane
  • 31
3
votes
2 answers

How to construct a bump function ends at different value?

May I ask how to construct a ''bump'' function ends at different value? For example: $\Psi\colon [0,1] \to [0,1]:$ $$ \Psi (x) = \begin{cases} 0 & \quad \text{for $0 \leq x < 1/3$}\\ ??? & \quad \text{for $1/3 \leq x < 1/2$}\\ 1 &…
1LiterTears
  • 4,572
3
votes
1 answer

The discrete Bessel kernel

Theorem 2 in a paper by Borodin, Okounkov, and Olshanski states that the discrete Bessel kernel $J(x,y,\theta)$ is given by \begin{equation*} \sqrt{\theta} \frac{J_x J_{y+1} - J_{x+1} J_y}{x-y} \end{equation*} where $J_x = J_x(2\sqrt{\theta})$ is…
Zach Conn
  • 5,043
3
votes
2 answers

How prove $h(x)=f(x)-g(x)$

show that: every function $h(x):R\to R$ can be written as $$h(x)=f(x)-g(x)$$ where $f(x),g(x)$ are satisfying the intermediate value property： http://en.wikipedia.org/wiki/Intermediate_value_theorem My try: since $f(x),g(x)$ are all such…
math110
  • 93,304
3
votes
1 answer

How to prove $f'(\xi)=a$ ,if $ f(0)=0,f(a)=a,f'(x_{0})=0$?

let $a\in (0,1)$,and $f(x)$ is continuous on $[0,a]$, and is differentiable on $(0,a)$, and $x_{0}\in (0,a)$,such $f'(x_{0})=0$,and such $$f(0)=0, f(a)=a$$ show that there exsit $\xi\in(0,a)$,such $$f'(\xi)=a$$ My try: I want to…
math110
  • 93,304
3
votes
2 answers

How prove this $g(x)=\sup{\{f(x,y)|0\le y\le 1\}}$ is continuous on $[0,1]$

let $f(x,y):[0,1]\times[0,1]\to R$ is continuous real function. show that $$g(x)=\sup{\{f(x,y)|0\le y\le 1\}}$$ is continuous on $[0,1]$ My try: since $f(x,y)$ is continuous on $D=[0,1]\times [0,1]$, so $f(x,y)$ is Uniformly continuous on…
math110
  • 93,304
3
votes
3 answers

Find $f$ if $f(f(x))=\sqrt{1-x^2}$

Find $f$ if $f(f(x))=\sqrt{1-x^2} \land [-1; 1] \subseteq Dom(f)$ $$$$Please give both real and complex functions. Can it be continuous or not (if f is real)
Szabolcs Sipos
  • 83
3
votes
1 answer

proving a function is differentiable

$g: \mathbb{R} \to \mathbb{R} $ $$g(x) = \begin{cases} x^2\sin(1/x)& \text{if $x\ne 0$}, \\ 0 &\text{if $x = 0$}.\end{cases}$$ Prove that g is differentiable everywhere, and that its derivative $g':\mathbb{R} \to \mathbb{R}$ is continuous on…
user113494
  • 225
3
votes
1 answer

How prove this $f(x)=\sum_{n=1}^{\infty}\left(\frac{\sin \frac{1}{x-r_n} }{2}\right)^n$ such follow condition?

Question: let $r_{1},r_{2},\cdots.$ is the interval $[0, 1]$ rational sequence in a line, and define: $$f(x)=\begin{cases} \displaystyle\sum_{n=1}^{\infty}\left(\dfrac{\sin{\left(\dfrac{1}{x-r_{n}}\right)}}{2}\right)^n &x\in…
math110
  • 93,304
3
votes
3 answers

Identify when $f(x) = 0$.

I have the following problem to solve. My attempt a. $\int_0^{\pi} x^n f(x) dx =0$ $\forall$ $n \ge 0$ gives $ x^n f(x) dx =0$ almost everywhere in $[0,\pi]$ $\forall$ $n \ge 0$. Putting $n = 0$ we shall get $f(x) = 0$ almost everywhere. As $f(x)…
Supriyo
  • 6,119
  • 7
  • 32
  • 60
3
votes
2 answers

Is indicator function integrable?

f is discontinuous on [0,1] so it's not integrable? but I think the answer is yes, it is integrable. But i dont know how to prove that. any hint would be great. thanks
Dafty
  • 509
3
votes
3 answers

How to determine the sum of the series $\,\sum_{n=1}^{\infty}\frac{n+1}{2^n}$

I am stuck on the following problem: I have to determine the sum of the series $$\sum_{n=1}^{\infty}\frac{n+1}{2^n}$$ My Attempt:…
learner
  • 6,726
3
votes
3 answers

Prove that inf(A+B) = infA + infB

A+B={a+b} I proved that the set A+B is bounded below. Now I'm stuck on how to prove that inf(A+B) = infA + infB
user104906
  • 39
3
votes
1 answer

sup of A Union B

Assumgin A,B, two set are upper-bounded. I need to prove that A Union B is also upper bounded and the supremum is max(supA, supB). This question can be explained intuitivly, but how do you prove it in a formal/mathemtical way? Thanks!
captain dragon
  • 759
3
votes
4 answers

Let $a_n$ and $b_n$ be sequences of real numbers. If $b_n$ is bounded and $\lim_{n \to \infty} a_n = 0$, then $\lim_{n\to\infty} a_{n}b_{n} = 0$

Prove the below statement: Let $a_n$ and $b_n$ be sequences of real numbers. If $b_n$ is bounded and $\lim_{n \to \infty} a_n = 0$, then $\lim_{n \to \infty} a_n b_n=0$ When I read this question, I read that $b_n$ may or may not converge, so taking…
david stocker
  • 95
1 2 3
…
99 100