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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Notation regarding sigma algebra's definition

I saw the following definition of sigma algebra in Amann/Escher's Analysis iii. A subset $\mathcal{A}$ of $\mathfrak{P}(X)$ is called a $\sigma$-algebra over X if it satisfies the properties i) $X \in \mathcal{A}$; ii) $A \in \mathcal{A} \Rightarrow…
Mirabelle
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I(A) is the set of all isolated points of A

I(A) is the set of all isolated points of A (isolated point means point $x$ such that $x \in A$ but $x$ is not a limit point of A) ${S_k}$ is a sequnce such that For all $k\in N$, $I(S_k)=\varnothing$ $S=\cap S_k$ Then $I(S)=\varnothing$? I couldn't…
user1014131
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How do I prove I found every element in the closure of $A$?

Suppose we have this interval $A=]1, 5].$ We know that $A$ is in the closure of $A$. If we add an arbitrary number $b > 0$ to $1$ we get: $1 + b > 1$, so the intersection of the sphere around $1$ with radius $b$ with $A$ is non-empty. So $1$ is in…
D.G. van de Schepop
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Prove $|f''(\xi)|\geq\frac{4}{(b-a)^2}|f(b)-f(a)|$

Suppose that f is twice differentiable on $[a, b]$ and $f'(a)=f'(b)=0$ show $\exists\xi\in(a,b)$ such that $|f''(\xi)|\geq\frac{4}{(b-a)^2}|f(b)-f(a)|$ There's an answer available on this site but it uses integration which I cannot use here I tried…
DanielKel
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Decouple Three Partial Differential Equations

How to decouple this system so I can get three decoupled equations of the following forms: $q_1$= function of $q_1$, $q_2$= function of $q_2$ and $p$= function of…
Xiazhi Hao
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Proving integral identity via Substitution

my task is to prove that $\int_{0}^{\infty} \frac{t^a}{sinh(t)} \frac{x}{x^2+t^2} dt= x^a \int_{0}^{\infty} \frac{t^a}{sinh(xt)} \frac{1}{1+t^2} dt$ so starting with the integral on the right hand side and by substituting u=xt one gets $x^a…
hello
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Let $X$ be a normed space and $x_0, x_1 \in X$. Let $l \in (0,1)$ be fixed. Prove, that the sequence $x_{n+1}=l x_{n}+(1-l) x_{n-1}$ converges.

I'm trying to prove the following: Let $(X, ||.||)$ be a normed space and $x_0, x_1 \in X$. Let $l \in (0,1)$ be a fixed constant. Prove, that the sequence $x_{n+1}=l x_{n}+(1-l) x_{n-1}$ converges. I started with noting, that because $X$ is a…
Hasarewa
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a simple question about an inverse application

Bonjour to everybody. I have to explain some notations before asking a simple question quoted from my favorite exercise book. Sorry about that. First of all $\mathbb R$ is the set of real numbers. Use $z$ to denote a complex number, with real part…
user43414
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How do I find the average distance to a point within a polygon?

Specifically, the polygons are Voronoi/Thiessen polygons created from the points, and I want to find the average distance from within the polygon to the point within. A more general solution is welcomed. If this is confusing, you may think about it…
David Giacomin
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prove the limit inferior of $(x_n)$ where $n \in\mathbb{N}$

The problem states let $(x_n)$ be a bounded sequence for each $n \in\mathbb{N}$. Let $t_n=inf\{x_k: k\geq n\}$. Prove that $(t_n)$ is monotone and convergent. After a little research because I was confused what limit inferior really was this was the…
user60887
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If $x=(x_n), s_n=\sum_{k=1}^n x_k$ and $\sup_n |s_n| =1$ then $\sup_n|x_n|=2$

In this case, $x\subseteq\mathbb{C}$. I can visualize the problem graphically, if I have a complex number $z$ inside the ball with radius 1, centered at the origin, I can add a lot of other complex numbers without leaving the ball, all of them with…
Byag
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Compute the sum $ \sum_1^{\infty} \ln{ \frac{ \left( n + 1 \right) \left( 3n + 1 \right) }{n \left( 3n + 4 \right)} }$

I need to prove that the sum $ \sum_{1}^{\infty} \ln \left( \frac{ \left( n + 1 \right) \left( 3n + 1 \right) }{n \left( 3n + 4 \right)} \right) $ converges and equals $ \ln(\frac{4}{3})$. I tried expanding the fraction into four terms, which…
sleepydrmike
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If $\ 0<\varepsilon<\vert x\vert,\ $ prove that only finitely many numbers of the form $\ \frac{(-1)^{k}}{2^{k}}\ $ are in $\ N(x;\varepsilon).$

If $\ 0<\varepsilon<\vert x\vert,\ $ prove that only finitely many numbers of the form $\ \frac{(-1)^{k}}{2^{k}},\ k\in\mathbb{N},\ $ are in $\ N(x;\varepsilon).$ My proof is : Let $x>0.$ Let $$S = \left\{ \left(-\frac{1}{2}\right)^k:\ k\in…
user1014131
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