Questions tagged [real-analysis]
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For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, the least upper bound property; and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus. This tag can also be used for more advanced topics, like measure theory.
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A general inequality about $\mathbb{R}^n$ metrics
I was revisiting Terence Tao's Analysis II. And noticed the inequality $$(1.1)\,\,d_{l^2}(x,y)\leq d_{l^1}(x,y)$$ So my questions are Is $d_{l^{n+1}}(x,y)\leq d_{l^n}(x,y)$ true? If so then how do ... real-analysis metric-spaces- 41
Geometric mean is to arthithmetic mean as arithmetic mean is to what?
I am interested in a type of "mean" $r$ associated to a set $\{a_1,a_2,\dots,a_n\}$ where $$ e^r=\frac{1}{n}\sum\limits_{i=1}^n e^{a_i}. $$ I will call this the "? mean" for now ... real-analysis arithmetic mathematical-modeling- 43
For $x \in [a, b]$, if $\lim_{x \to b} f'(x) = +\infty$, then (i) $\lim_{x \to b} f''(x) = +\infty$ and (ii) $\lim_{x \to b} f''(x)/f'(x) = +\infty$
$f$ is $C^2$ on $[a, b]$. For (i) I thought I had it by contradiction: Suppose it's not true, so we can find $\bar{b} \in (a, b)$ such that, for $x \in [\bar{b}, b]$ and $B > 0$, $f''(x) < B$, i.... real-analysis calculus limits derivatives- 26
Why is the line integral along the vertical segment zero
I am reading Theorem 11.10 : Green's Theorem for Plane Regions Bounded By Piecewise Smooth Jordan Curves. I have doubt regarding the last line in page 381, which says the integral along each vertical ... real-analysis calculus integration greens-theorem- 5
Counter example: $f: \mathbb R^n \to \mathbb R$ continuous with a unique global minimum does not imply that $f$ is coercive
I want to answer the question whether a continuous function $f: \mathbb R^n \to \mathbb R$ with a unique global minimum has to be coercive. Intuitively, that is, of course, false. I'm imagining a ... real-analysis nonlinear-optimization- 198
Evaluating $\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx$
How to show that $$\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx=\frac{5\pi^3}{64}+\frac{\pi}{16}\ln^2(2)-4\,\text{G}\ln(2)$$ without breaking up the integrand since we already know: $$\int_0^1\... real-analysis calculus integration alternative-proof polylogarithm- 22.1k
Question about definition
I have a very naive question about the definition of the operator $(1+x)\partial_x$ as an operator $C^1 (\mathbb{R})\to C^0(\mathbb{R})$. Is it $A u = (1+x)\partial_xu$ in the sense that we have $Au(-... real-analysis multivariable-calculus derivatives- 127
A linear change of variables in a PDE, basic help
Consider the PDE to be solved for $f$, and for a given vector field $b:\mathbb{R}^n\to\mathbb{R}^n$ $$ \partial_t f(t,x)=\text{div}(b(x) f(t,x)),~\forall~t,x \in \mathbb{R}_+\times \mathbb{R}^n, $$ ... real-analysis multivariable-calculus chain-rule change-of-variable jacobian- 1
Find limit of inverse function
Let $x$ be a function $x(y): \mathbb{R} \to \mathbb{R}$, where it is not possible to find the inverse relation $y(x)$ in a closed form. Is there a way to find the limit \begin{equation} \lim_{x \to \... real-analysis- 37
Alternating infinite series problem please solve [closed]
Please hint me how to solve this. $(1-\frac12+\frac13-\frac14+......) ^2 = 2[\frac12-\frac13(1+\frac12) +\frac14(1+\frac12+\frac13) -\frac15(1+\frac12+\frac13+\frac14) + ......]$ real-analysis calculus sequences-and-series- 1
A clopen one-dimensional set in $\mathbb R^2$?
The set $\{(x,y):y=x^2\}$ seems (to me) a one-dimensional clopen set in $\mathbb R^2$. Formally, the one-dimensional clopen set is a set that is both closed and open considering the topology of $\... real-analysis general-topology terminology- 513
Characterization of linear differential operators in $\mathbb{R}^n$
I was wondering exactly how to characterize a linear differential operators in $\mathbb{R}^n$. Some lecture notes on the internet told me that a differential operator of order $m$ in $\mathbb{R}^n$ is ... real-analysis linear-algebra partial-differential-equations- 1,839
Measurable functions could be injective but not bijective?
I am trying to understand the concept of measurable functions. In several texts I found that if $(\Omega, \Sigma)$ is a measurable space, then $f: \Omega \to \mathbb{R}$ is measurable if and only if ... real-analysis measure-theory- 17
Calculating the integeral $\int_0^{\infty}\frac{{(e^{-ax}-e^{-bx})}{\cos(cx)}}{x}dx$
I want to calculate:$$\int_0^{\infty}\frac{{(e^{-ax}-e^{-bx})}{\cos(cx)}}{x}dx(a,b,c > 0)\tag{1}$$And I want to use:$$\frac{x}{x^2+k^2}=\int_{0}^{\infty}e^{-xy}\cos{ky}dy\tag{2}$$So,I want to think:... real-analysis integration analysis fubini-tonelli-theorems- 11
Matrix norm (submultiplicative constant)
Is it true that we can always find some $k>0$ such that $||XY||\leq k||X||\cdot||Y||$ holds? $||\cdot|| $ is any matrix norm. I haven't learn functional analysis yet. real-analysis linear-algebra matrices numerical-linear-algebra matrix-analysis- 11
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