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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The Lipschitz property of an upper envelope

I'm self studying (for fun) the book "Functional Analysis, Calculus of Variations and Optimal Control" (by Clarke), and I'd appreciate some feedback for my proposed solution for an exercise from the book (I'm not comfortable with it, for reasons I…
Borbei
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recurrence relation with trigonometric function

What is the explicit formula for the sequence? $a(n+1) = \sin ( \frac{\pi a(n)}{2} )$ and $a(1) = 0.5$ What should I read to solute such questions?
David
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Positive and negative Parts

we denote by $u^+=\max(u,0)$ and $u^-=\max(-u,0)$ the positive and the negative parts of $u$ we have that $u=u^+-u^-$ my question is : what is $u'$ using $u^+$ and $u^-$ ? and what is $\int_{\Omega} p(t) u'^2(t) dt$ using $u^+$ and $u^-$ ? Please…
Vrouvrou
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How to proof the limit is convergent for arbitrary initial state?

$\{a_n\},\{b_n\},\{c_n\},\{d_n\}$ is series. And $d_n=c_{n-1},c_n=b_{n-1},b_n=a_{n-1},a_n=b_{n-1}+c_{n-1}$ how to proof for any $a_0,b_0,c_0,d_0$ belong to $Z^+$, $\lim\limits_{n\rightarrow \infty} \dfrac{(a_n-a_{n-1})}{a_{n-1}}$ is existent? In…
Farmer
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Calculate the integral $\int_1^{10}x^xdx$

$$\int_1^{10}x^xdx$$ I figured it with a relative error of 1% and I have a response $$≈0.3*10^{10}$$ But I don´t know how to accurately calculate it ...
Mirzodaler
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A problem in the space $C[a,b]$

Let $E=C[a,b]$ provide with the $\max$ norm. Let $S\neq \emptyset$, let and $D(t,\lambda)$ be a continuous function (for each $\lambda\in S$), from $[a,b]$ to $\mathbb{R}$, such that $\displaystyle D=\sup_{\lambda}\int_{a}^b…
Valent
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Find $x$ for which the series converges

Find the $x$ s for which $$\sum_{n=2}^{\infty} \frac{x^{n^2}}{n \log(n)}$$ converges. How can I do this? My attempt is to write $\sum_{n=2}^{\infty} \frac{x^{n^2}}{n \log(n)}$ in the form $\sum a_n(x-\xi)^n$. How can I do it? Do I have to use the…
user159870
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Pointwise Lipschitz constant and exchanging supremum and limit

I have come across the following definition: Let $(X, d)$ be a metric space . Given a function $f \colon X \to \mathbb{R}$, the pointwise Lipschitz constant of $f$ at a non isolated point $x \in X$ is defined as $$Lipf(x)=\limsup_{y \to x}…
Student
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Shouldn't the values be the same?

$$\text{ Let } f_n(x)=n^px(1-x^2)^n, x \in [0,1], \text{ where } p \in \mathbb{R} \text{ a parameter } \in \mathbb{R}. $$ $$\text{ Prove that } \forall p, (f_n) \text{ converges pointwise to an f in [0,1]. }$$ $$\text{ For which values of p,is the…
evinda
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finite dimention subspaces of infinite dimention hilbert space are closed?

Let H is a hilbert space of infinite dimention, and $ V \subset H $ of finite dimention, can we show that V is closed? I know if H is of finite dimention, then V is closed, but what if H is infinite dimention?
annimal
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At which points is the function $f(x)= \lim\limits _{n \to \infty} \frac{x}{1+(2\sin x)^{2n}}$ discontinuous?

Possible Duplicate: Testing continuity of the function $f(x) = \lim\limits_{n \to \infty} \frac{x}{(2\sin{x})^{2n}+1} \ \text{for} \ x \in \mathbb{R}$ I cannot figure out, at which points the function is discontinuous, the only thing came to my…
Gigili
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Do I need the First Mean Value Theorem For Integals?

Let $f$ be a continuous function on $[0,1].$ Suppose that $\int_0^1 f(x) g(x) dx = 0$ for every integrable function $g(x)$ on $[0,1].$ Prove that $f(x) \equiv 0$ on $[0,1]$ This proof is easy to write out if $g(x) \ge 0.$ If it is integrable on…
Travis
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How to apply an implicit function theorem

I have a function $\phi(x,\omega) \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ and I know that $\nabla_{x} \phi \neq 0$ and $h(x,\omega) > 0$, where $$ h(x,\omega) = \det \left( \frac{\partial^2 \phi(x,\omega)}{\partial x_{i} \partial…
Appliqué
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A question regarding continuous curves.

Say $f:\Bbb{R^2}\to\Bbb{R}$ is a continuous map. Now take the fibre of $a\in\Bbb{R}$, which is $f^{-1}(a)$. Will it always be a continuous curve in $\Bbb{R^2}$? I tried constructing examples. Clearly $(x,y)\to x^2+y^2$ satisfies this condition, as…
freebird
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Is the image of this operator on $\ell^2$ closed?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: \ell^2 \to \ell^2: (x,y) \mapsto \sum_{n=1}^\infty…
Jeroen
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