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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triple line equals symbol

I keep seeing this symbol $\equiv$ in Mathematical Analysis -1, Zorich. What does it mean? For example: in page 180 we have, Some other pages it occurs in: 117, 139.
Ashwin B
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Show function $u$ is continuously differentiable with integration by parts

Let $u$ be a continuous function. Assume for any $v$ that is continuously differentiable on $[0,1]$ and vanishes at the boundary points, $0$ and $1$, there exist a continuous function $f$ such that $$ \int u v' \ dx = - \int f v \ dx \ . $$ Show…
XIAODUO WANG
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$f \in L^1 \implies \lim_{ |x| \to \infty } f(x) = 0$

Suppose $f\in L^{1}(\mathbb R) $ My Question is: Can we show, $\lim_{ |x| \to \infty } f(x) = 0$? Thanks, (this question is obviously related to $f, f'\in L^{1}(\mathbb R) \implies \lim_{x\to \infty} f(x)=0 ?$ a thing I've already proven.)
Marine Galantin
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Uniform convergence of power series of $\log(1+x)$ on $[0,1]$

Define $$f(x): = \sum_{k = 1}^\infty \frac{(-1)^{k+1}x^k}{k},\;\;f_n(x) = \sum_{k = 1}^n \frac{(-1)^{k+1}x^k}{k}$$ $f(x)$ is a power series with radius of convergence 1, and $f(-1)$ diverges but $f(1)$ converges. I want to show $f$ is left…
mez
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Let $T: C^{1} \rightarrow C^{0}$ be the operator such that $T(f)(x) = f^{1}(x) $, is this operator uniformly continuous?

$C^{1}$ is the space of continuously differentiable functions and $C^{0}$ the space of continuous functions, both defined on the real interval $[0,1]$. Both spaces equipped with $d(f,g) = \sup_{x \in [0,1]|f(x) - g(x)|}$ as metric. $f^{1}(x)$ is the…
nandevers
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True or false? If f is continuous on (a,b), then f(a) and f(b) can be defined so that f isintegrable on [a,b].

True or false? If f is continuous on (a,b), then f(a) and f(b) can be defined so that f is integrable on [a,b]. I know the answer is false but why is it false?
user597188
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What does it mean for two continuous functions to have a common zero?

Suppose $f,h$ are continuous real functions on the unit circle where $f(-x)=-f(x)$ and $g(-x)=-g(x)$ for all $x$. What does it mean for $f$ and $g$ to have a common zero on the unit circle? Does it mean that there exists a $p\in S^1$ where…
mathlover314
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Calculate the number e using Taylor polynomial and Lagrange Remainder

I am supposed to determine the value of number e with exactness (mistake) less than $10^{-8}$, using Taylor polynomial and Lagrange Remainder. My hint is that $T_{n}(e^{x},0)(x)$. I know the formula for Lagrange Remainder, as well as for Taylor…
Shelley
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How can I find such function?

Let $f\in C^1(\mathbb{R})$ be a function with compact support satisfying $f(0)>0$ and $f(a)=0$ for some $a>0$. For each fixed $p\in (2,\infty)$, I want to find a continuous function $g:\mathbb{R}\rightarrow\mathbb{R}$, such that $g(0)=0$, $g$ is…
Tomás
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Is this integral convergent?

If $f\in L^2(\mathbb{R}^3)$, what can I say of the integral $$\int_{\mathbb{R}^3}dx\frac{f(x)}{\vert x\vert}$$? Is it convergent?
Sue
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Constraints on $\alpha, \beta$, if $\alpha f(x)^2 + \beta f(x) + \gamma \equiv 0$

I am dealing with an expression of the form $$\alpha f(x)^2 + \beta f(x) + \gamma = 0,$$ where $f: \mathbb{R} \to \mathbb{R}$ is smooth and not identically zero, and $\alpha$, $\beta$, and $\gamma$ are real coefficients. I strongly suspect that for…
FraGrechi
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Let $a_n$ be the sequence given by $a_n$ = 1/$2^n$ if n is even, and 0 if odd. Find the radius of convergence of $\sum_{i=0}^n a_n x^n$

Let $a_n$ be the sequence given by $a_n$ = 1/$2^n$ if n is even, and 0 if odd. Find the radius of convergence of $\sum_{i=0}^n a_n x^n$ I'm not sure how to do this. I'm pretty sure the radius of convergence is 2 but I don't know the method of…
HistoryFan12
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Does $\int_0^\infty g(y)dy$ converge and $\int_0^\infty f(x)dx=\int_0^\infty g(y)dy$?

Suppose $F(x,y)$ is continuous on $(0,\infty)\times (0,\infty)$. $\forall x\in (0,\infty)$, $f(x)=\int_0^\infty F(x,y)dy$ converges, and $f(x)$ is continuous on $(0,\infty)$. $\forall y\in (0,\infty)$, $g(y)=\int_0^\infty F(x,y)dx$ converges, and…
Born to be proud
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Metric space and distance function on a compact set

Let $(X,d)$ be a metric space, $K \subset X$ a nonempty compact subset and $d(x,K) = \inf\{d(x,y):y \in K\}$. Show that there is a $y \in K$ such that $d(x,K) = d(x,y)$. I have showed that the function $f_{K}(x) = d(x,K), \forall x \in X$ is…
Thiago Alexandre
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Differentiation of Taylor Series

Let $g(x) = e^{-1/x^2}$ for $x$ not equal to zero, and $g(0) = 0$. a) Please Show that $g^{(n)}(0) = 0$, for all $n = 0,1,2,3,4, \ldots$ Can someone please elaborate on the comments below for this one? b) Please Show that the Taylor Series for $g$…
mary
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