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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A quick question on Baby Rudin Theorem 2.40: Every k-cell is compact.

I have a quick question on the excerpt of Theorem 2.40 of Baby Rudin. How would I get "If n is so large that $2^{-n}\delta
Daniel
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Local extremum o a three-variable function

Let $f: \mathbb{R}^3 \to \mathbb{R}$, defined by $f(x,y,z)=x^2+4y^2+8z^2+4xy+6xz+12yz+8x+16y+24z$. Find its local extremum points. I find that the critic points are: $(-2a-4,a,0) , \forall a \in \mathbb{R}$. Unfortunately, the Hessian matrix has a…
npatrat
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transforming $\,x^3 + x^2 + 1 \implies x\left(x^2 + x + \dfrac{1}{x}\right) \implies\,$ div by $\,0\,$?!

EDIT: Sorry. I basically was confused by that just valid mathematical transforming could lead into a undefined behavior. I have to admit, my question has not much to do with the zero of a function, i've just used one method of the "zero calculating…
uuu
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choose $M$ (depend on $n$) such that $\frac{\log(1+M^2)}{\log n} \to 2$ and

Is it possible to choose $M$ (depend on $n$) such that $\frac{\log(1+M^2)}{\log n} \to 2$ and $n(1-\frac{2}{\pi}\tan^{-1}(M)) \to 0$ as $n\to\infty$
Yan Wong
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Continuous only at rational pts

Possible Duplicate: Set of continuity points of a real function I'm studying continuous in analysis class. I have a question. That's simple! Is there a function that is continuous only at rational points? I have no idea for find that. If you…
japee
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Fubini theorem question

Let $f$ and $g$ be Lebesgue measurable nonnegative functions on $\mathbb{R}$. Let $A_y=\{x:f(x) \geq y\}$ Let $F(y)=\int_{A_y} g(x)dx$. Prove $\int_{-\infty}^\infty f(x)g(x)dx=\int_0^\infty F(y)dy$. I know this has to do with Fubini's theorem but…
john
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Find the max, min, sup and inf of a sequence

Let $\{a_n\}=\{x\mid x\in\mathbb {Q},x^2 <2\}$, find the max, min, sup and inf of a sequence Clearly, sup is $\sqrt {2} $ and inf is $-\sqrt {2} $, so we have $-\sqrt {2}
Simple
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What does a function of $C^{1+\epsilon}$ mean?

I am starting to read Boundary Behavior of Holomorphic Funtions of Several Complex Varieties by E.M. Stein. I don't know the meaning of the following symbol: The class $C^{1+\epsilon}$ would suffice... Can you tell what does $C^{1+\epsilon}$ mean?…
Jason785
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Use only Archimedean Property of $\mathbb{R}$ to give a direct $\epsilon$-$N$ verification for $\lim\limits_{n\rightarrow\infty}\frac{1}{\sqrt{n}}=0$

Use only Archimedean Property of $\mathbb{R}$ to give a direct $\epsilon$-$N$ verification for $\lim\limits_{n\rightarrow\infty}\frac{1}{\sqrt{n}}=0$ Let $a_n=\{\frac{1}{\sqrt{n}}\}$ and $\epsilon>0$. Consider that there is a…
Simple
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Terminology question

What do you call a set of points with the following property? For any point and any number $\epsilon$, you can find another point in the set that is less than $\epsilon$ away from the first point. An example would be the rationals, because for any…
badatmath
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differentiable function in higher dimensions

$f:\mathbb R^{n}\to\mathbb R^{k}$ is a function with $\|f(x)\| \leq \|x\|^{2}$ , $x$ is an element of $\mathbb R^{n}$. Show that $f$ is differentiable at $0$. I already found that $f(0)=0$. I thought that $f(x)= o(|x|)$ (little o-notation). And f…
bob
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What's the $\sigma$-algebra generated by a function?

I know that $\sigma(\mathcal{G})$ for a family of sets $\mathcal{G}$ is the smallest $\sigma$-algebra which contains $\mathcal{G}$. I'm reading some online notes, and ran into the term $\sigma(T)$ where $T$ is a measurable map from one measurable…
iikao sero
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Reflection of a set

In analysis, we define a reflection of a set, say $E$ such that $E \subseteq \mathbb{R}$, as follows: $-E := \{x : x = -a \ \text{for some} \ a \in E\}$ So for example, $-(1, 2] = [-2, -1)$. My question is why does the definition say "for some"?…
user247618
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Redefining Outer Lesbegue Measure on $\Bbb{R}^{d}$ From Closed Cubes to Rectangles.

UPDATE: I added an answer based off the hints provided by copper.hat. It may, however, need some adjustment. I'm trying to solve another question from Stein and Shakarchi's analysis text. Basically, I'm trying to prove that…
Sargera
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Prove that $\sum_{k=1}^n\frac{1}{k}-\ln(n)$ converge.

I have to prove that the sequence $(x_n)_{n\geq 1}$ define by $$x_n=\sum_{k=1}^n\frac{1}{k}-\ln(n)$$ converge. I have shown that $(x_n)$ is decreasing but I fail to show that it's undervalued. I tried to show that it's undervalued by $0$, but with…
idm
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