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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 .

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Rank theorem in Rudin's Principles of Mathematical Analysis

I have a question about the rank theorem in Rudin's PMA book in page 229. Roughly speaking, it's a statement that for a continuous differentiable map (which can be linearly approximated), certain part of the map can be made precisely linear by a…
liyiontheway
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How to find a function that is normally equidistant to a given other function?

Just a little background, I'm a maths teacher in highschool, and this is not a problem I deal with on the daily. I just came across it trying to create graphics for class and I've been stuck. So, my goal is to find a function (not just the graph, a…
Ellon Feyneth
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A function that satisfies Cauchy Riemann Equations but is not differentiable

I am quoting the solution of this post (https://math.stackexchange.com/a/2924208/603303) I have no idea why the function $f(z)$ is broken down to $f(x)$ and $f(iy)$ to find the partial derivatives. Assuming we can do this then the rest is clear,…
Jonathan Valenzuela
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Use of Newton method to find the value of $x$

A segment of a circle is the region enclosed by an arc and its chord (See figure below). If $r$ is the radius of the circle and $x$ the angle subtended at the center of the circle, find the value of $x$ (correct to $4$ decimal places) for which the…
YuYu Huang
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The classification of differential equations

In textbook, it says that If a differential equation (DE) contains only ordinary derivatives of one or more unknown functions with respect to a single independent variable, then it is called an ODE; An equation involving partial derivatives of one…
Chasel Weng
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interchange summation and iterated integration

A math article makes use of the following equality, without proof. $$ \int_{0}^{1}\int_{0}^{1} \sum_{n\ge 0} (xy)^n dx\, dy = \sum_{n\ge 0} \int_{0}^{1}\int_{0}^{1}x^n y^n dx\, dy$$ But $s_k(x,y) = \sum_{n=0}^{k} (xy)^n$ does not converges…
Robin
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Sobolev space and norm

Is it true that we have equivalence between these two norms? $\|u\|_{H^{k}}^{2}\sim \|u\|_{L^{2}}^{2}+\sum_{\lvert \alpha\rvert=k} \|D^{\alpha}u\|_{L^{2}}^{2}$ Is it also true that we have this inequality and if it is true how can I prove…
Ama NI
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Proving $\|u\|_{H^k}\sim\sum_{n=1}^\infty\lambda_k\lvert a_n\rvert^2$, where $u$ solves heat equation $u_t=\Delta u$ and $u=\sum_{n=1}^\infty a_ne_n$

In Claude Zuily's book Distribution and partial differential equations, it's mentioned that: $$\|u\|_{H^{k}}\sim\sum_{n=1}^{\infty}\lambda_{k}\lvert a_{n}\rvert^{2} \tag1$$ where $u$ is the solution of heat equation: $u_{t}=\Delta u$ and…
Ama NI
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Generalized hypergeometric function 1F2 , special values

I have two hypergeometric functions $\ _{1}\mathcal{F}_{2}[\frac{1}{2}+q;\frac{1}{2},\frac{3}{2}+q;-X^{2}] $ and $\ _{1}\mathcal{F}_{2}[1+2q;\frac{3}{2},2+2q;-X^{2}] $. For fixed integer positive $q$ Mathematica gives me some trigonometric…
Katja
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Walter Rudin Exercise 1.16: computing yields counterintuitive results.

Principles of Mathematical Analysis, Exercise 1.16 (here is a semi-duplicate, same question, but different emphasis): Suppose $k \ge 3, x,y \in \mathbb{R}^k, | x-y |=d>0,$ and $r>0.$ Prove that if $ 2r>d,$ there are infinitely many $ z \in…
C_Arietta_C
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Circle touching the graph of function y = |x|. Determine the function on which the centers of the circles lie for all values of t

I'm curios if I solved this correctly, or if there are any other ways to do it. The problem goes: The point P(t, |t|) lies on the graph of the function $f(x) = |x|$. A circle with a radius of $√2·|t|/3$ touches the graph of the function f from above…
Mixoftwo
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How can I prove the definition of open set

A set $M \subseteq\mathbb R$ is called open if $R \setminus M$ is closed. Show that for a set $M \subseteq \mathbb R$ holds: $$M\text{ open}\iff \forall x\in M \exists \epsilon>0 : \{y\in\mathbb R\mid|x-y|<\epsilon\}\subseteq M$$ Approach: Well the…
xx33xx44
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Analysis question about points on sphere

I'm having trouble with a question: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ a differentiable function. If $\dfrac{\partial f}{\partial u}(u)>0$ for all $u \in S^{n-1}$, there exists a $a\in \mathbb{R}^n$ such that $\dfrac{\partial f}{\partial…
TDg1
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Why does $\epsilon$ become a continuous function of $\Delta x$?

This Proof of Chain Rule comes from the textbook of Stewart I'm stuck here: " If we define $\epsilon$ to be $0$ when $\Delta x=0$, then $\epsilon$ becomes a continuous function of $\Delta x$". How come $\epsilon$ becomes a continuous function of…
Andrew Li
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Formal derivation of the chain rule for partial derivatives

I want to derive the chain rule for partial derivatives, that is if $\gamma (t) = (x(t),y(t))$, then \begin{align*} \frac{\partial f\circ \gamma}{\partial t} (t) = \frac{\partial f}{\partial x} (x(t),y(t)) \frac{\partial x}{\partial t} (t) +…
Maths Wizzard
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