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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Fixed point of Lipschitz mapping

Let X be a complete metric space containing the point $p_0$ and let r be a positive real number. Define $K=\{p\ in\ X\ \big|\ d(p,p_0) \le r \}$. Suppose that $T:K \to X$ is Lipschitz with Lipschitz constant c. Suppose also that $cr +…
Nik
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Is the following limit true or not? (function convergence)

Let $x\in(0,1)$, let $p>1.$ I was wondering if $\lim_{n\to\infty} n^{\frac{2}{p}}x^{\frac{n}{p}}=0$ is true or not? I plotted some values and it seems to be true but I am struggling to prove this. Intuitively, one thing I can think of is that…
UnsinkableSam
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prove squeeze theorem

Can anyone prove Squeeze theorem for me, which says: (1):if $f(x)< g(x) < h(x)$ for all $x$ near $a$, and $\lim\limits_{x\to a}f(x) = \lim\limits_{x \to a}h(x) = L$ then $\lim\limits_{x \to a}g(x)=L$. (2): If $f$ dominates $g$, that is, ($f(x) \geq…
LoveMath
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Sign of a function

Suppose $\gamma$ is a real number with $|\gamma|\ll1$. The function $$ \theta(s)=s-\frac{\sin \left( \sqrt{1+\gamma} \, k \, \pi \, s \right)}{\sin \left( \sqrt{1+\gamma}\, k \, \pi \right)}, \qquad k = 1,2,3\dots $$ is negative on $[0,1]$ when…
simon
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Prove the existence of a limit

Suppose $f_0$ is a Riemann integrable function on $[0,1]$, $f_n(x)=\sqrt{\int_0^xf_{n-1}(t)dt}$, find $\lim_\limits{n \to \infty}f_n(x)$. It is easy to see if the limit does exist and is differentiable on $[0,1]$, $\lim_\limits{n \to…
Y.Wayne
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On the limits, prove that and find its limit

Can any one tell me how to prove that: The sequence $x_{1}=\sqrt{2}$ , $x_{2}=\sqrt{2+\sqrt{2}}$, $\cdots$, $x_{n}=\underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n \text{ times}}$ converges and compute its limit. Thanks
LoveMath
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Proof with a family of function

I have to prove that $$f_a(x)=\ln x^2 +\frac{a}{x}$$ has no zeros if for $a>0$ it holds that $a\cdot e>2$. I tried to find the zeros but I came to no result.
calculatormathematical
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Give an example of a measurable space $(X, S)$ and a function $f :X \to \mathbb{R}$ such that $|f|$ is $S$ -measurable but $f$ is not $S$ -measurable.

Give an example of a measurable space $(X, S)$ and a function $f:X \to \mathbb{R}$ such that $|f|$ is $S$-measurable but $f$ is not $S$-measurable. I have considered a function on $(\mathbb{R},\mathcal B)$ where $\mathcal B$ is the collection of…
DJCham
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Rudin Chapter 1 Exercise 10 Hang-Ups?

Suppose $z = a + bi$, $w = u + iv$, and $$a = \left(\frac{|w| + u}{2}\right)^{1/2}, b = \left(\frac{|w| - u}{2}\right)^{1/2}$$ Prove that $z^2 =w$ if $v \geq 0$ and that $(\bar{z})^2 = w$ if $v \leq 0$. Conclude that every complex number (with…
sqrtoiler
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analysis sequence induction series

Let $k$ be a positive real number with $k < 1$, and let $f$ be a $k$-contraction. Let $c\in \mathbb{R}$ be given, and define a sequence $(x_n)n≥1$ by $x_1 = c$ and $x_{n+1} = f(x_n)$. Set $C = |x_1-x_2 |$. By induction on $n$ prove that…
miguel barnes
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distorted fourier transform is isometric

I need a proof for this theorem that says : Assume that $V \in L_{1}^{1},$ and that $V$ has no bound states, then $\tilde{\mathcal{F}}$ is an isometry on $L^{2}$, $$ \|\widetilde{\mathcal{F}} f\|_{L^{2}}=\|f\|_{L^{2}}, \quad \forall f \in…
Mohamad Janbein
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A question about the integral of convex function

Let $F:\mathbb{R}\to\mathbb{R}$ be a strictly convex function. Let $u:[0,1]\to\mathbb{R} $ be a continuous function, with $$\int_{0}^{1}u(x)\,dx=0$$ Show that $$\int_{0}^{1}F(u(x))\,dx\leqslant\frac{F(\| u\|_\infty)+F(-\| u\|_\infty)}{2}$$ …
xuhong liu
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For differentiable $f$ and $g$ with $f(x)g'(x)-f'(x)g(x)\not=0$, there is always a root of $g$ between two roots of $f(x)$

If $f(x)$ and $g(x)$ are differentiable in an interval $I$ and: $$f(x)g'(x) - f'(x)g(x)\not=0 \qquad\forall x \in I$$ then there is always a root of $g$ between two roots of $f$. How do I prove this?
analysisgb
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Asymptotic analysis for sum

Let $f,g,h,j$ be functions of $x$. Suppose $f\sim g$ as, say $x\to\infty$ and $h\sim j$ as $x\to\infty$ which, by definition, means that $$ f-g=\mathcal{o}(g)\text{ and }h-j=\mathcal{o}(j)\textrm{ as }x\to\infty. $$ Does this imply that $f +h\sim…
Rhjg
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Showing a function is not differentiable at $(0,0)$

Let $\displaystyle f(x,y)=\begin{cases} \frac{x^3+y^4}{x^2+y^2} \text{ if } (x,y) \neq (0,0)\\ 0 \text{ if } (x,y)=(0,0). \end{cases}$ Show this is not differentiable at $(0,0)$. My strategy is to compare $D_uf(p)$ and $\nabla f(p)$ and…
emka
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