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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Big-$O$ confusion

$f(x) = O(g(x))$ means that $\frac{f(x)}{g(x)}$ is bounded. But $x=O(x^2 + 1),\ x\in \mathbb R$ while $x\ne O(x^2)$. Is there a human friendly explanation of what $O$ is? the definitions I saw are as follows: f, g: E → R, a is a limit point of E.…
user1063088
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Ways to define the directional derivative

I am reading a text where directional derivatives of functions $f:E\rightarrow F$, where $E,F$ are Banach spaces, at the point $x_0 \in E$ are defined as $d_v f(x_0)=\lim_{t\rightarrow 0+} \frac{f(x_0+tv)-f(x_0)}{t}$ for any $v\in E$. My question…
resu
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Limiting value of functions - From Tao Analysis II

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces,$E \subseteq X$ and $f:E \rightarrow Y$ a function. If $x_0 \in \overline E$ we define $$ \lim_{x \rightarrow x_0;x \in E} f(x) = L $$ iff $$ \exists L \in Y \forall \epsilon > 0 \exists \delta > 0…
user42761
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Deformation of integration paths

First of all: I'm more of an algebraic person. So happily differentials/integrals are not what I deal with a lot. However, I got this exercise to solve: Let $\Omega \subset \mathbb{R}^n$ open, $K: \Omega \rightarrow \mathbb{R}^n$ a $C^1$ vector…
Louis
  • 3,455
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Proving Cauchy Theorem

This theorem says that a function $f$ is defined in the interval (a, +$\infty$) and it is bounded in every finite interval (a,b). Then it holds that: (assume that the limit of the RHS exists) $$\lim_{x\to \infty} \frac{f(x)}{x} = \lim_{x\to…
ppphy
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Sum of the square harmonic series

I stumbled across the following series reviewing some HW from a few years…
Jimmy TwoCents
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Fractional Part integral $I=\int_{0}^{1}\int_{0}^{1} \left\{\frac{x}{y}\right\}\mathrm{d}x\mathrm{d}y$

Let $$I=\int_{0}^{1}\int_{0}^{1} \left\{\frac{x}{y}\right\}\mathrm{d}x\mathrm{d}y.$$ When I tried computing the integral I seem to be getting a different answer to Wolramaplha, and can't find a similar integral anywhere on MSE or the…
Sarthak Sahoo
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How to show that for every vector field $v$ there exists a constant $M_{ > 0}$ such that $||v(x)|| \leq M$ $\space \forall x \in \gamma([a,b])$?

Given is that the length $L(\gamma)$ of a $C^1$-curve $\gamma : [a, b] \longrightarrow \mathbb{R^n}$ is defined as: $L(\gamma) = \int^b_a||\gamma'(t)||dt$ Now I need to show that for every continuous vector field $v$ defined on $\gamma([a, b])$…
yungCalculator
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how to use union and intersection symbol to denote the limit

For any $x \in X$, $f_n(x) \to \infty$ as $n \to \infty$. Is this equivalent to saying $\cap_{K=1}^{\infty} \cup_{N=1}^{\infty} \cap_{n=N}^{\infty} \{x: f_n(x)\geq K\}$? Why? My understanding: for any a positive integer K, we can find N, such that…
Mariana
  • 1,253
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Is $f:(-1,1)\to \{y\in\mathbb{C}\mid |y| =1\}\setminus\{-1\}:x\mapsto e^{i\pi x} $ a homeomorphism?

Consider $f:(-1,1)\to \{y\in\mathbb{C}\mid |y| =1\}\setminus\{-1\}:x\mapsto e^{i\pi x} $. I think $f$ is a homeomorphism but I still need to show that the inverse $f^{-1}$ is continuous. For the inverse I calculated if $0\leq\theta<\pi$…
marco31
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Open sets in $\Bbb{R}^2$ and $\Bbb{R}^3$

In $\Bbb{R}^2$ and $\Bbb{R}^3$ are all open sets in the form of an open ball of some positive radius? In other words, do open sets look like anything else except balls/spheres? I know any union of open sets is again open, but that doesn't really…
TheRealFakeNews
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A question on differentiability of a variant of Thomae function

Let $$f(x) = \begin{cases} \dfrac{1}{q^2}, & \text{if $x=\dfrac{p}{q} $ is rational and in lowest terms;} \\[2ex] 0, & \text{if $x$ is irrational} \end{cases}$$ Where is f continuous? Is f differentiable anywhere? My attempt: I can prove that…
Mariana
  • 1,253
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Find surjective, continuous function such that diagram commutes

If we have a continuous function $g\colon X\to X$ with $g(Y)\subset Y$ and $h\colon Y\to Y$, where $h$ is the restriction of $g$ to $Y\subset X$, is there a continuous, surjective function $\pi\colon X\to Y$ such that $$ \pi\circ g = h\circ\pi? $$
Corona21
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Need help with difficult extra credit analysis question

I don't really understand this problem from my analysis class: Let $BC([0,1],R)$ be the metric space of functions which are bounded, continuous on $[0,1]$, and in $C^\infty$, meaning that all of their derivatives exist and are continuous. We're…
Andrew Gosling
  • 31
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Proving that a function is not a contraction

I am solving the following problem and its parts. Let (C[0,1],$d_\infty$) be the metric space of continuous functions on [0,1] where the distance function is defined by $d_\infty(f,g)=\sup_{x∈[0,1]}|f(x)−g(x)|. $ Let $T : (C[0, 1], d_\infty)\to…
Lavorizia Vaughn
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