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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Does $|\textbf{x}-\textbf y|<\delta$ imply $|x_1- y_1|<\delta$ and $|x_2- y_2|<\delta$

I want to say that $|\textbf{x}-\textbf y|<\delta$ implies $|x_1- y_1|<\delta$ and $|x_2- y_2|<\delta$ for a proof I am working on. This is assuming that $\textbf{x}=(x_1,x_2) \in \text R^2$ and $\textbf{y}=(y_1,y_2) \in \text R^2$. If true, I'd…
rioneye
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Method of steepest descent

Generally, the method of steepest descent describes the asymptotic behavior of integrals of the form $$\int_{-\infty}^\infty h_t(x)\exp(-tg(x)) \,dx$$ in terms of $t$. As long as $h_t(x)$ is controlled nicely as $t\to\infty$ and $g(x)$ has a global…
Ivan
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Show that if $ \lim_{n \to \infty} a_{n}=g$ then $ \lim_{x\to \infty} \frac{ \sum_{n = 0}^\infty a_{n}\frac{x^n}{n!}}{e^{x}} = g $

For starters I've noticed that in this case we can apply L'Hôpital's rule. Due to the fact that we get $\left[\frac{\infty}{\infty}\right]$ symbol (or at least I think that we can do this here) $f(x) =$ $ \sum_{n = 0}^\infty a_{n}\frac{x^n}{n!}$ the…
Bartek Żamojtel
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Proving that Complete Ordered Fields are Archimedean

Proposition. Complete ordered fields are Archimedean. Proof. Let $\mathbb{F}$ be a complete ordered field and consider $x\in\mathbb{F}$. We must show that there is an integer $N$ such that $x
Wooyong Park
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Anything known about the infinite product $ \prod (1 - \frac{x^2}{i^2}) $

I was thinking about constructing functions that have arbitrary values at some integers. One way to do that is by taking the functions $$ f_n = \sum_{0 \leq i \leq m, i \not = n} \frac{x - i}{n-i} $$ This is $ 1 $ at $ x = n $ and $ 0 $ at all other…
Tempestas Ludi
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Show that $ U $ is an open surrounding of $ M $

Define the two sets $$ U:=\left\{(x,y)\in \mathbb{R}^2:\frac{1}{\sqrt{2}}
hallo007
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Equivalent definitions of convergent sequences in metric spaces

I am currently reading A course in point set topology by John B. Conway. The definition he give is standard: A sequence $\{x_n\}$ in X converges to X if for all $\epsilon>0$, there is an integer N such that $d(x,x_n)<\epsilon$ when $n\geq…
Shuyi Leo
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Is my proof that $\sqrt{6} \notin \mathbb{Q}$ correct?

We will attempt a proof by contradiction. Assume there exists two integers $p$ and $q$ such that $\left(\frac{p}{q}\right)^2 =64$. Hence, $p^2 = 6 \cdot q^2$. Thus, $6 | p$ and $3 | p$. Let $3 \cdot r=p$, where $r$ is an integer. Then, $9 \cdot…
philip
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Proof help of backwards implication: A subset K of a metric space $X_d$ is compact iff every infinite subset of K has a limit point in K.

I'm currently reading Rudin's Principles of Mathematical Analysis. At page 38, Theorem 2.37 says that Every infinite subset of a compact set K has a limit point in K. I wonder whether the converse still hold(I guess it holds too), but I don't know…
Shuyi Leo
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Computing Hilbert Transforms of Certain Periodic Functions

I am computing the Hilbert transform of a $2\pi$-periodic function on the interval $[-\pi, \pi]$ of the form $f(x) = |x|^a$, where $a \in \mathbb{R}$. By definition, we know that the Hilbert transform of this function $f$ is given by $$Hf(x) =…
HtmlProg
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Functions without an elementary primitive

$Li(x)$ Is a non elementary function being the primitive of $\frac{1}{log(x)}$. I know also that the primitive of $e^{x^2}$ Is not elementary and I know that Gauss calculated some definite integral involving that function. Are there other examples…
Enzo Creti
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Does L'Hospital's rule still hold when derivatives are w.r.t different variables?

This question is provoked by an (in fact standard) solution to one of Baby Rudin’s exercises. Suppose $f$ is defined in a neighborhood of $x$, and suppose $f^{\prime\prime}(x)$ >exists. Show that $$ \lim_{h \to 0} \frac{ f(x+h)+f(x-h)-2f(x)}{h^2} =…
C_Arietta_C
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Use of the nested interval thereom to prove the non-completness of the real space ]0;1[

I was reading up on some analysis and came across the incompleteness of the real space ]0;1[ ( or (0;1) in amaerican notations). It's easy to see that it's incomplete if you consider that any sequence that is of Cauchy must converge on an element of…
Hongo
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Behavior at infinity in $L^2$ with mixed second derivatives in $L^2$

If $f$, $\nabla_x \cdot \nabla_y f \in L^2(\mathbb{R}^d_x\times \mathbb{R}^d_y)$, what can be said about vanishing at infinity of $\nabla_x f$, $\nabla_y f$ or if they even belong to $L^2_{x,y}$? It is clear to me that $(\nabla_x^2 + \nabla_y^2) f…
Jakob Elias
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least upper bound greatest lower bound theorem

I am trying to understand the following theorem: I can't understand how the author gets to the conclusion that $\alpha = \sup L$ is $\in L$ I'm ok until the "Our hypothesis about $S$ implies therefore that $L$ has a supremum in $S$, call it…
Yossi
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