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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Lagrange-mean like inequality

I had this problem on an assignment a while ago, but I don't quite understand the formulation of the problem nor the purpose: Let $f : [a,b]→\mathbb{R}$ be a function that admits a derivative (not necessarily finite!) at any point of $[a,b]$. Prove…
ZNatox
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Convergence of integral $\int_{1}^{\infty}\frac{\sqrt{x}+2+\cos{x}}{x+2}dx$

Is $$\frac{1}{x^{\frac{3}{4}}}<\frac{\sqrt{x}+2+\cos{x}}{x+2},\forall x>x_0$$ enough to prove that $$\int_{1}^{\infty}\frac{\sqrt{x}+2+\cos{x}}{x+2}dx$$ diverges? What are the standard techniques which can be used in these type of problems?
Juliusz
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Do all perfect subsets of [0,1] contain a sequence such that the ratio of its increments converges?

Fix a perfect set $D\subseteq[0,1]$. Does there necessarily exist a monotone sequence $\{x_m\}_{m\in\mathbb{N}}\subseteq D$ such that $x_m\rightarrow x\in D$ with the property that: \begin{equation} \frac{x_m-x_{m-1}}{x_{m-1}-x_{m-2}}\rightarrow…
Analysis Qs
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Is the set of 'crossing points' of every curve closed?

I want to know if the following statement is true or not: $\gamma : [a, b] \to \mathbb{R}^k$ is continuous then $E := \{ t \vert \exists s \neq t: \gamma (t) = \gamma (s) \}$ is closed. So here, $E$ is just the set of 'crossing points' of the…
Calmadeas
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Help check that function is non-negative

Consider the function: $$f_a(x) = [\phi(a+x)(a+x) + \phi(a-x)(a-x)][\Phi(a-x)-\Phi(-a-x)]+[\phi(a+x)-\phi(a-x)]^2$$ where $a>0$ is a real number and $\phi(\cdot), \Phi(\cdot)$ denote the probability density function and the cumulative distribution…
Barreto
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Did I use Stirling formula incorrectly?

I have to prove that $\frac{\sqrt{2\pi n} * e^{-n}*n^k}{k!} \to e^{-x^2/2}$ when $\frac{k-n}{\sqrt{n}} \to x$ and $n \to \infty$ I used Stirling formula and get that $\frac{\sqrt{2\pi n} * e^{-n}*n^k}{k!} = e^{k-n} * (n/k)^k \frac{\sqrt{2\pi…
Захар Наумець
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How can i write $f(x)=\cos(x)$ as the difference of two monotonically increasing functions?

This is a Question from an Analysis 1 exam. The question is as follows: Decide if the functions $f: \mathbb{R} \longrightarrow \mathbb{R}$ can be written as the difference of two monotonically increasing functions a) $f(x) = \cos(x)$ b) $f(x) =…
fluffy
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Problem with integral

Is there a number $1
user111
  • 447
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Property of convex functions.

I have seen the following argument: If $F: \mathbb{R}^n \to \mathbb{R}$ is a convex function, then there exist a Borel function $\lambda\colon\mathbb{R}^n \to \mathbb{R}^n$ bounded on compact subsets, and such that \begin{equation} F(w) \ge F(z)…
user29999
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For what kind of a subset its sums equal $\mathbb{R}^4$

For short, suppose $a,b$ are real numbers. Let $A=\{(\cos(at), \cos(bt), \sin(at), \sin(bt))\mid t\in \mathbb{R}\}$. Let $B=\sum A=\{\sum_{i=1}^n x_i\mid x_i\in A, n \geq 1\}$. For what values $a,b$, $B$ equals $\mathbb{R}^4$? In general, what…
wxu
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How can I solve this infinite sum?

I calculated (with the help of Maple) that the following infinite sum is equal to the fraction on the right side. $$ \sum_{i=1}^\infty \frac{i}{\vartheta^{i}}=\frac{\vartheta}{(\vartheta-1)^2} $$ However I don't understand how to derive it…
Julian
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Counterexample for a differentiable structure

I'm trying to understand the definition of a differentiable structure. Is it correct that $x\rightarrow x$ and $x \rightarrow x^3$ doesn't form a diffeomorphism, since $x\rightarrow x^{1/3}$ isn't differentiable in $0$?
marco31
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(1)Questions about differentiable functions

1)The functions $f$ and $g$: $\mathbb{R} \rightarrow \mathbb{R} $ shall be 3-times differentiable. Calculate $(f \cdot g)^{(3)}$. 1) $(f \cdot g)'=(f'g+fg')$ $(f'g+fg')'= (f''g+f'g')+(f'g'+fg'')=…
Phil
  • 549
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Would it be possible to skip undergraduate college with self study/online courses

Hi I was wondering if it were possible to skip undergraduate and apply to a graduate school. I have been taking some math courses through CTY a somewhat accredited program offered by Johns Hopkins university. They offer up to real analysis 1 and…
Joe Jameson
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How do I prove $\int_a^b |f(x)|^2 dx \leq \frac{(b-a)^2}{2} \int_a^b |f'(x)|^2 dx$.

Let $f$ be $C^1[a,b]$ continuously differentiable) with $f(a)=0$. [This may be generalized to $f(b)=0$ or $f(a)f(b)=0$] I want to show $$\int_a^b |f(x)|^2\ \mathrm{d}x\ \leq\ \frac{(b-a)^2}{2} \int_a^b |f'(x)|^2\ \mathrm{d}x.$$ Since $f$ is…
phy_math
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