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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1290 votes
27 answers

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
BBSysDyn's user avatar
  • 16.1k
890 votes
22 answers

The staircase paradox, or why $\pi\ne4$

What is wrong with this proof? Is $\pi=4?$
Pratik Deoghare's user avatar
456 votes
18 answers

To sum $1+2+3+\cdots$ to $-\frac1{12}$

$$\sum_{n=1}^\infty\frac1{n^s}$$ only converges to $\zeta(s)$ if $\text{Re}(s)>1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
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274 votes
32 answers

Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int\limits_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$ Well, can ...
user avatar
220 votes
3 answers

When can a sum and integral be interchanged?

Let's say I have $\int_{0}^{\infty}\sum_{n = 0}^{\infty} f_{n}(x)\, dx$ with $f_{n}(x)$ being continuous functions. When can we interchange the integral and summation? Is $f_{n}(x) \geq 0$ for all $x$ ...
user192837's user avatar
  • 2,211
218 votes
5 answers

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
Jesse Madnick's user avatar
216 votes
13 answers

Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
FireGarden's user avatar
  • 5,805
178 votes
10 answers

How far can one get in analysis without leaving $\mathbb{Q}$?

Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
Alexander Gruber's user avatar
  • 26.9k
175 votes
9 answers

Why is $1^{\infty}$ considered to be an indeterminate form

From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...
user avatar
174 votes
4 answers

Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( 1-\...
Shobhit Bhatnagar's user avatar
163 votes
7 answers

Is non-standard analysis worth learning?

As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides ...
127 votes
9 answers

Continuity of the roots of a polynomial in terms of its coefficients

It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed ...
Harry Altman's user avatar
  • 4,642
118 votes
0 answers

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
Elias Costa's user avatar
  • 14.6k
113 votes
9 answers

Is $[0,1]$ a countable disjoint union of closed sets?

Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?
Kevin Ventullo's user avatar
111 votes
6 answers

Why don't analysts do category theory?

I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects. Recently, ...
gary's user avatar
  • 1,111
110 votes
4 answers

Factorial and exponential dual identities

There are two identities that have a seemingly dual correspondence: $$e^x = \sum_{n\ge0} {x^n\over n!}$$ and $$n! = \int_0^{\infty} {x^n\over e^x}\ dx.$$ Is there anything to this comparison? (I ...
Mitch's user avatar
  • 8,571
107 votes
5 answers

Pointwise vs. Uniform Convergence

This is a pretty basic question. I just don't understand the definition of uniform convergence. Here are my given definitions for pointwise and uniform convergence: Pointwise convergence: Let $X$ be ...
Jeff's user avatar
  • 1,601
105 votes
8 answers

Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations? [closed]

Moderator Notice: I am unilaterally closing this question for three reasons. The discussion here has turned too chatty and not suitable for the MSE framework. Given the recent pre-print of ...
99 votes
3 answers

How to evaluate $\int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x$ using complex analysis?

We were told today by our teacher (I suppose to scare us) that in certain schools for physics in Soviet Russia there was as an entry examination the following integral given $$\int\limits_{0}^{\...
user avatar
99 votes
5 answers

Why are gauge integrals not more popular?

A recent answer reminded me of the gauge integral, which you can read about here. It seems like the gauge integral is more general than the Lebesgue integral, e.g. if a function is Lebesgue ...
Chris Brooks's user avatar
  • 7,404
93 votes
15 answers

Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$

I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$ What do you think? It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm ...
TCL's user avatar
  • 14.2k
92 votes
4 answers

A continuous, nowhere differentiable but invertible function?

I am aware of a few example of continuous, nowhere differentiable functions. The most famous is perhaps the Weierstrass functions $$W(t)=\sum_k^{\infty} a^k\cos\left(b^k t\right)$$ but there are ...
levitopher's user avatar
  • 2,635
92 votes
9 answers

Why is the construction of the real numbers important?

There are a lot of books, specially in Real Analysis and set theory, which define the real numbers by Cauchy sequences or Dedekind cuts. So my question is why don't we simply define the Real numbers ...
user42912's user avatar
  • 23.5k
85 votes
8 answers

What is integration by parts, really?

Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher ...
Elle Najt's user avatar
  • 20.7k
82 votes
6 answers

The Intuition behind l'Hopitals Rule

I understand perfectly well how to apply l'Hopital's rule, and how to prove it, but I've never grokked the theorem. Why is it that we should expect that $$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x \to a}\...
user avatar
80 votes
1 answer

Does every Cauchy sequence converge to *something*, just possibly in a different space?

Question. If I attempt to prove that space $X$ is complete by pursuing the strategy, “Assume $x_n \rightarrow x$; the space $X$ is complete if $x \in X$,” then why is that wrong? Context. I know the ...
1Teaches2Learn's user avatar
77 votes
11 answers

Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$?

It seems as if no one has asked this here before, unless I don't know how to search. The Gamma function is $$ \Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx. $$ Why is $$ \Gamma\left(\frac{1}{2}...
Michael Hardy's user avatar
72 votes
3 answers

What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?

I tried and got this $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an ...
M. Amin's user avatar
  • 721
72 votes
5 answers

Under what condition we can interchange order of a limit and a summation?

Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? ...
zzzhhh's user avatar
  • 839
72 votes
2 answers

Continuity and the Axiom of Choice

In my introductory Analysis course, we learned two definitions of continuity. $(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ satisfies ...
John Gowers's user avatar
  • 24.9k
71 votes
2 answers

Let, $A\subset\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$.

A challenge problem from Sally's Fundamentals of Mathematical Analysis. Problem reads: Suppose $A$ is a subset of $\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is ...
David Bowman's user avatar
  • 5,572
69 votes
6 answers

"Gaps" or "holes" in rational number system

In Rudin's Principles of Mathematical Analysis 1.1, he first shows that there is no rational number $p$ with $p^2=2$. Then he creates two sets: $A$ is the set of all positive rationals $p$ such that $...
Larry's user avatar
  • 723
66 votes
6 answers

How do you show monotonicity of the $\ell^p$ norms?

I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
user1736's user avatar
  • 8,553
66 votes
2 answers

Why doesn't Cantor's diagonal argument also apply to natural numbers?

In my understanding of Cantor's diagonal argument, we start by representing each of a set of real numbers as an infinite bit string. My question is: why can't we begin by representing each natural ...
usul's user avatar
  • 3,694
65 votes
15 answers

Choice of $q$ in Baby Rudin's Example 1.1

First, my apologies if this has already been asked/answered. I wasn't able to find this question via search. My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch ...
Rachel's user avatar
  • 2,874
65 votes
12 answers

Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $

I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges,...
Isaac's user avatar
  • 36.5k
65 votes
5 answers

Why is the Daniell integral not so popular?

The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some ...
gifty's user avatar
  • 2,211
62 votes
3 answers

A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$

How can we prove that: $$\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx=\frac{7\pi^2}{48}\zeta(3)-\frac{25}{16}\zeta(5)$$ where $\zeta(z)$ is the Riemann Zeta Function. The best I could do was ...
Shobhit's user avatar
  • 1,070
60 votes
7 answers

What is the theme of analysis?

It is safe to say that every mathematician, at some point in their career, has had some form of exposure to analysis. Quite often, it appears first in the form of an undergraduate course in real ...
Sandesh Jr's user avatar
60 votes
7 answers

Should I be worried that I am doing well in analysis and not well in algebra? [closed]

I attend a mostly liberal arts focused university, in which I was able to test out of an "Introduction to Proofs" class and directly into "Advanced Calculus 1" (Introductory Analysis I) and I loved it....
Eric's user avatar
  • 1,821
60 votes
1 answer

Baby/Papa/Mama/Big Rudin

Recently, I was looking for the reviews of some Analysis books while encountered terms such as Baby/Papa/Mama/Big Rudin. Firstly, I thought that these are the names of a book! But it turned out that ...
Hosein Rahnama's user avatar
59 votes
7 answers

Does $\zeta(3)$ have a connection with $\pi$?

The problem Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)? Details Several $\...
GarouDan's user avatar
  • 3,418
59 votes
6 answers

How hard is the proof of $\pi$ or $e$ being transcendental?

I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
Sean Tilson's user avatar
  • 4,456
56 votes
6 answers

Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$

While trying some problems along with my friends we had difficulty in this question. True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
user avatar
55 votes
14 answers

Nonobvious examples of metric spaces that do not work like $\mathbb{R}^n$

This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general ...
Samuel Handwich's user avatar
55 votes
4 answers

Better Proofs Than Rudin's For The Inverse And Implicit Function Theorems

I am finding Rudin's proofs of these theorems very non-intuitive and difficult to recall. I can understand and follow both as I work through them, but if you were to ask me a week later to prove one ...
AnonSubmitter85's user avatar
55 votes
4 answers

Prove/disprove $(\int_0^{2 \pi} \!\!\cos f(x) \, d x)^2+(\int_0^{2 \pi}\!\!\! \sqrt{(f'(x))^2+\sin ^2 f(x)} \, dx)^2\ge 4\pi^2$

Let $f(x)$ be a differentiable function on $[0,2\pi]$ s.t. $0\leq f(x)\leq 2\pi$ and $f(0)=f(2\pi)$. Prove or disprove that $$ \left(\int_0^{2 \pi} \cos f(x) \,d x\right)^2+\left(\int_0^{2 \pi} \sqrt{(...
FFjet's user avatar
  • 5,041
54 votes
5 answers

What books are prerequisites for Spivak's Calculus?

For financial reasons, I dropped out my senior year of college as a piano performance major. I will be returning to college to dual major in mathematics and computer science. I've taught myself to ...
user avatar
53 votes
5 answers

Are there periodic functions without a smallest period?

The Wikipedia page for periodic functions states that the smallest positive period $P$ of a function is called the fundamental period of the function (if it exists). I was intrigued by the condition ...
user28375028's user avatar
  • 1,659
53 votes
11 answers

Why is the notion of analytic function so important?

I think I have some understanding of what an analytic function is — it is a function that can be approximated by a Taylor power series. But why is the notion of "analytic function" so important? I ...
Code Complete's user avatar

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