Mathematics Stack Exchange
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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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Convergence of $f(x)e^{-tx}$ implies the convergence of $f(x)$?

Let $f(x) > 0$ on $[a,\infty)$, are integrable on $[a,b]$ whenever $0 0$ $$ \int_{a}^{\infty} f(x)e^{-tx}dx \le M \quad $$ Then how to prove that $\int_{a}^{\infty} f(x)dx$ converges as…
ppphy
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Why difference quotient of convex functions increases in both variables

Let $f: \mathbb R \rightarrow \mathbb R$ be a convex function and $$ g(x,y)=\frac{f(x)-f(y)}{x-y} \textrm{ for } x\neq y. $$ I wish to prove that $g$ is increasing function in both variables. Thanks
Richard
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Which function grows faster? x^0.0001 or ln(x)?

I am looking for a definition of "which function grows faster", I came across this but I'm not sure if it's right: $\rightarrow$ $e^x >> x^k >> \ln(x)$ (for $x \rightarrow \infty$ and $k \in \mathbb{R}$) Is this right? Or should it be $k > 1$? So…
choco_lion
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question about the conditions on the weak derivative of periodic function

Let $f: \mathbb R \rightarrow \mathbb R$ be a periodic, say with the period 1, and locally integrable function. Assume that $g: \mathbb R \rightarrow \mathbb R$ is a weak derivative of $f$ of order $k$, that is $g$ is also locally integrable and…
Alex
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Show that $|f(x)|≤1$ a.e

Let $(\Omega ,A,\mu )$ be a measure space and $f,\,f_1,\ldots,\,f_n,\ldots$ measurable functions such that: $(1)$ $f_n \overset{\mu }{\longrightarrow}f$ and $(2)$ $|f_n(x)|\leq 1$ for a.e. $x$ for all $n$. Show that $|f(x)| \leq 1$ for a.e.…
Bünzli Refinej
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Prove a function is constant over an interval

Suppose $f$ is a function of $\mathbb{R}$ and satisfies $$\forall \ x_0 \in \mathbb{R},\ \exists \ \delta > 0,\ f(x_0) \geq f(x),\ \forall x \in (x_0 - \delta, x_0 + \delta).$$ Prove that there exists a nondegenerate interval $I$, $f$ is constant…
Jingyu Liu
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Inverse polynomial decay of arbitrary derivatives of certain functions

In Sogge's Lectures on Nonlinear Wave equations, Chapter II, Section 1, Proposition 1.1, is about expressing the radial vector field $\partial_r$ as a linear combination of conformal Killing fields in Minkowski spacetime. It is claimed that $$ (t -…
Chris
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Supremum of subsets in $\mathbb{R}$

I'm trying to solve this question: (a) Let $E \subset \mathbb{R}$ be a set bounded from above with supremum $x \in \mathbb{R}$ Prove that for every $\varepsilon>0$ the intersection $E \cap(x-\varepsilon, x]$ is nonempty. (b) Give an example of a…
Zeno cosini
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A set constructed with the rational numbers is dense in a certain set.

Let, $$\mathbb{Q} = \{q_1,\dots,q_n,\dots \}.$$ Let, $$I_{n} = q_n + (-2^{-n},2^{-n}).$$ Let, $B \subseteq \mathbb{R}$ such that $\mu(B^{c})=0$. Then we define $$U := \bigcup_{n=1}^{\infty}I_n.$$ Is $(U \cap B)$ dense in $B$? I'm having trouble…
Johndoe
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Does from $f'(x)= \lim_{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2h}$ follow $f$ is differentiable

Let $f$ be differentiable on $(a,b),$ then $f'(x)= \lim_{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2h}$ I used that $\frac{f(x+h)-f(x-h)}{h}=\frac{f(x+h)-f(x)}{h} + \frac{f(x-h) + f(x)}{h}$ So $$\lim_{h \rightarrow 0}…
John.W
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Question about a comment in my analysis book (differentiation)

In the book I am reading differentiability is defined as: f is differentiable in $x_0\in I$ if $\lim_{x\rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}$ exists After this definition there is a comment that states because $f$ has the derivative $f'(x_0)$…
John.W
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Forming a number by summing up or subtracting 21,19,12,8. Trying to make a mathematical formula for a chess problem.

I am trying to make a mathematical formula that determines the minimal amount of moves that is necessary for a knight to reach a square in a XxX board. I have figured out that the knight moves can be converted to the numbers +-21,+-19,+-12,+-8 by…
XianLiu
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$f(x)= e^{f'(x)}$. Solve the DE for $f(x)$

f differentiable function in R. $f(x)= e^{f'(x)}$ $f(0)=1$ I have proved that $f(x)=1$ for every $x\lt0$. im stuck for $x\gt0 $
Plom
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$L^1$ weak topology

Let $T=[0,1]$, $$P=\{x=(x^1,x^2,…,x^n)\in\mathbb{R}^n \vert \; x^{i}\geq 0,i=1,2,…,n; x^1+x^2+…+x^n=1\}.$$ A $T$-strategy is a measurable function $y$ from $T$ to $P$. Hence $y$ is Lebesgue-integrable and we write $\int y$ for $(\int…
student
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using a bump function

I want to drift a $C^1$ function $g(x):\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ to reach $"0"$ smoothly on some interval $[x_{1},x_{2}]$ and then drift it again to continue it's values on $(x_{2},\infty)$. I've found something about partitions of…
stavros
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