Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.
176
votes
13answers
13k views
85
votes
4answers
2k views
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 ...
57
votes
9answers
3k views
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 ...
56
votes
6answers
2k views
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 ...
45
votes
17answers
6k views
Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$?
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$$
Well, can anyone ...
39
votes
3answers
2k views
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 ...
37
votes
2answers
930 views
Is there a bounded function $f$ with $f'$ unbounded and $f''$ bounded?
Is there a $C^{2}$-function $f:\mathbb{R}\to\mathbb{R}$ that is bounded and such that $f'(x)$ is unbounded, but $f''(x)$ is bounded again?
For example, $f(x)=\sin(x^2)$ is bounded and has ...
36
votes
5answers
3k views
Why does $1+2+3+\dots = {-1\over 12}$?
$$\lim_{N\to\infty} \sum_{i=1}^N \, i = +\infty$$
$\displaystyle\sum_{n=1}^\infty n^{-s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$.
Why should analytically continuing to $\zeta(-1)$ ...
35
votes
5answers
4k views
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?
34
votes
6answers
1k views
Why do we negate the imaginary part when conjugating?
For $z=x+iy \in \mathbb C$ we all know the definition for the "conjugate" of $z$, $\bar{z}=x-iy$. Geometrically this is the reflection of $z$ across the $y$ axis.
My question is: couldn't we have ...
33
votes
6answers
2k views
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, ...
32
votes
6answers
2k views
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 ...
32
votes
1answer
452 views
Does there exist $f:(0,\infty)\to(0,\infty)$ such that $f'=f^{-1}$?
Recently the following question was posed: does there exist a differentiable bijection $f:\mathbb R\to\mathbb R$ such that $f'=f^{-1}$? (Here, $f^{-1}$ is the inverse of $f$ with respect to ...
31
votes
5answers
1k views
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 ...
31
votes
3answers
1k views
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 ...
