# 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).

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### Does the airy group “commute” with time-compactly supported functions?

I am currently reading the book "Introduction to nonlinear dispersive PDEs" by Felipe Linares and Gustavo Ponce and found an equality that really kept my attention. At the top of page 173 (...
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### Showing that Lebesgue Dominated convergence theorem is false in case of Riemann integration.

I was reading Tom Apostol book called "Mathematical Analysis" and I read this statement: the Lebesgue Dominated convergence theorem is false in case of Riemann integration. Here is the ...
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### What is area using a double integrals between the curves?

What is area enclosed between curve/coordinate axes? $$(\frac{x}{a})^3+(\frac{y}{b})^3=\;(\frac{x}{h})^2+(\frac{y}{k})^2\;,(x=0,y=0);$$ $(a,b,h,k)$ are constants. And $x,y \geq 0$
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### Integrability of $\frac{|xy|^a}{nz^3}$ over $\sqrt{x^2+y^2}<z<\sqrt{n^2-x^2-y^2}$

Let $a>0,$ $$f_a(x,y,z)= \frac{|xy|^a}{nz^3}$$ on $$A_n=\{(x,y,z):\sqrt{x^2+y^2}<z<\sqrt{n^2-x^2-y^2}\}$$ I want to find all $a>0$ such that $f_a \in L^1(A_n)$ for $n \geq1.$ Looking at ...
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### Alternative reading materials for Baby rudin after chap.8

Next semester I will study chapter 9, 10, and 11 of Baby Rudin. However, from Chapter 9, I was told that understanding the part of analysis (like functions of several variables, integration of ...
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### Finding the local extrema of $y = \frac{\ln x} {\sqrt{x}}$

I'm given a function: $$y = \frac{\ln x} {\sqrt{x}}$$ I've found the first derivative, which is $$y' = \frac{2-\ln(x)}{2x\sqrt(x)}$$ and got that the domain is $x > 0$, but I just don't know how to ...
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### Perturbing the Initial Condition of an ODE within a compact

Let $\gamma: (a,b) \rightarrow \mathbb{R}^d$ be the maximal solution of the differential equation $x' = F(t,x)$ with initial condition $\gamma(t_0)=x_0$. Show that given any compact $K \subset I$ and ...
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### Integrability of $f(x,y)=\frac{(x+y)\log(y/x)}{(xy)^{a}(1+x^2+y^2)}$ on $O=\{(x,y):x>0,y>0\}.$

Let $$f_a(x,y)=\frac{(x+y)\log(y/x)}{(xy)^{a}(1+x^2+y^2)}=\frac{1}{x^a}\frac{(x+y)\log(y/x)}{y^a(1+x^2+y^2)}$$ defined on $$O=\{(x,y):x>0,y>0\}.$$ I need to find all $a\in \mathbb{R}$ such that ...
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### Prove that the integral $\int_a^b \frac{\sin(x)}{x}dx$ is bounded uniformly.

How do I show that exists a constant $M>0$ such that, for all $0\leq a \leq b < \infty$, $$\left|\int_a^b \frac{\sin(x)}{x}dx\right| \leq M.$$ I just read on Richard Bass book's that is enough ...