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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Taking inequalities when dealing with sums

I have from a problem im solving and so far I have that $|f_i(x) - f_i(y)| < \epsilon / \sqrt{n}$ where $\epsilon , n >0$ I'm now trying to use this inequality here: $$\|f(x) - f(y) \|_2 = [ \sum_i^n (f_i(x) - f_i(y))^2 ] ^{1/2} < \left ( \sum_i^n…
FACEIT
  • 369
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Show $f$ is locally invertible if $f = L + g$ and $|g(x)| \le M|x|^2$

Let $f, g, L: \mathbb{R}^n \to \mathbb{R}^n$ and $L$ is a linear isomorphism, and let $|g(x)| \le M|x|^2$ on $\mathbb{R}^n$ for some $M > 0$. Prove that $f$ is locally invertible at $0$, i.e. $f$ is invertible on some open neighborhood of $0$. I…
Terry
  • 135
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3 answers

Proving two sets are equal

Suppose we have a claim: $A = \bigcup_{k \in \mathbb N} [-k, \frac{1}{k}) = (-\infty,1)$ = B. I aim to prove this by showing that the two sets are subsets of each other (i.e. $A \subset B$ and $B \subset A$). As for my strategy of showing this, I am…
user247618
4
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About the remainder of Taylor expansion and Riemann-Liouville integral

Integral form of Taylor expansion looks like this:$$f(x)=\sum_{i=0}^k\frac{f^{(i)}(a)}{i!}(x-a)^i+\int_a^x\frac{f^{(k+1)}(t)}{k!}(x-t)^kdt$$ Riemann-Liouville integral is $$I^{\alpha}f=\frac{1}{\Gamma(\alpha)}\int_a^x{f(t)(x-t)^{(\alpha-1)}}dt$$ Q1:…
Tim
  • 123
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Prove $A= \mathbb{R} $

It can be seen easy but i'm really stack.I don't know how to start. Let $A \neq \emptyset $ is a subset of $ \mathbb{R} $.If for every real $x$,$y$ sum $x+y$ belongs to $A$,then $xy$ also belongs to the set $A$.Then prove that $A= \mathbb{R}…
perelman
  • 51
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Ramanujan Infinity sum functional equations

i was reading about the mellin transform ans i found the following $$\sum _{k=1}^{\infty } \left(\frac{e^{-k x}}{e^{-2 k x}+1}-\frac{\pi \text{sech}\left(\frac{\pi ^2 k}{x}\right)}{2 x}\right)=\frac{\pi }{4 x}-\frac{1}{4}$$ but i do not how to…
user167276
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2 answers

When does $\sum_{n=2}^\infty n^\alpha (\log n)^\beta$ converge?

I need to find out for what $\alpha, \beta$ the following sum converges: $$\sum_{n=2}^\infty n^\alpha (\log n)^\beta$$ I thought I'd do that with the help of the integral criterion, that is to say I considered (after I had substituted $x =…
Huy
  • 6,674
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$\int_a^b |f|=0\implies f=0$

Let $f:[a,b]\to\mathbb R$ continuous. I want to show that $\int_a^b |f|=0\implies f=0$. By contradiction, suppose $f\neq 0$ and denote $A=\{x\mid f(x)\neq 0\}$. We have that $$0=\int_{[a,b]}|f|=\int_{[a,b]\backslash A}|f|+\int_A|f|\implies…
idm
  • 11,824
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1 answer

Integrability of $(x+y) ^{-3}$.

I'm asked to determine for what positive values of $\alpha$ is $(x+y)^{-3}$ integrable in the region where $0
Philip Hoskins
  • 2,055
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3 answers

Representing any number in $R_+$ by means of two numbers

I have the following question: Let $R_+$ denote the nonnegative reals. Let $00$, and set $p = a^n b^m$. Is it possible to find a $n,m\in N$ such that $p$ can approximate any number in $R_+$ arbitrarily close?
Nithin Govindarajan
  • 303
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Give an example of a function $f:[0,1] \rightarrow \mathbb{R}$ that is...

Give an example of a function $f:[0,1] \rightarrow \mathbb{R}$ such that... (a) $f$ is bounded, but not Riemann integrable on $[0,1]$. $$ f(x) := \begin{cases} 2x & \text{if $x$ is rational}\\ x & \text{if $x$ is irrational.} \end{cases} $$ (b) $f$…
Mark
  • 171
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Prove that a function is not Riemann integrable.

Suppose $f: [-2,3] \longrightarrow \mathbb{R}$ is defined by $$ f(x) = \left\{ \begin{array}{l l} 2|x| + 1, & \text{if $x$ is rational}, \\ 0, & \text{if $x$ is irrational}. \end{array} \right.$$ Prove that $f$ is not Riemann integrable. We know…
Mark
  • 171
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3 answers

How to prove the elementary inequality?

The inequality is the following: $$\frac{(1+x)^q-1}{x+x^q} \leq C(q),$$ where $q\in [1,+\infty)$ and $x > 0$, and the constant $C$ depends only on $q$. It's very nice if someone can provide the minimal value of $C(q)$, I guess the minimal value is…
bigheadliao
  • 103
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Find a sequence $\{a_n\}$ of real numbers such that $\sum a_n$ converges but $\prod (1+ a_n)$ diverges.

Find a sequence $\{a_n\}$ of real numbers such that $\sum a_n$ converges but $\prod (1+ a_n)$ diverges. The converse is trivial, just make all the $a_n=-1$.
Steven-Owen
  • 5,556
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Find an example of a sequence not in $l^1$ satisfying certain boundedness conditions.

This question is about getting a concrete example for this question on bounded holomorphic functions posed by @user122916 (something that he really expected as explained in the comments). Give an example of a sequence of complex numbers…
orangeskid
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