173
votes
Pointwise vs. Uniform Convergence
Comparison
Pointwise convergence means at every point the sequence of functions has its own speed of convergence (that can be very fast at some points and very very very very slow at others).
...
- 15.2k
169
votes
Accepted
How to evaluate $\int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x$ using complex analysis?
$\phantom{}$ Dear MSE users, this is the new episode of Mister Feynman and Monsieur Laplace versus contour integration.
Tonight we have a scary integral, but we may immediately notice that
$$ \...
- 340k
167
votes
Accepted
A continuous, nowhere differentiable but invertible function?
Interestingly, there are no such examples! For a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ to be invertible, it must be either monotone increasing or decreasing. A famous classical ...
- 4,403
140
votes
Why is the construction of the real numbers important?
First of all, mathematics is based on intuition and on concrete (imaginary, but concrete) objects, often inspired by reality. Let's stop the formalities for a moment and speak freely: we don't think ...
- 26.3k
105
votes
What is integration by parts, really?
I've always found it helpful to think about it like this: (picture source)
The area of the gray areas combined is $u_2v_2 - u_1 v_1$, which is where the $uv$ term comes from.
- 12.3k
100
votes
Accepted
"Gaps" or "holes" in rational number system
It depends on what you consider a “gap” in the rational numbers. As long as this is not a formally defined concept, we’re just talking about our everyday, geometrically informed ...
- 216k
95
votes
Accepted
What is the theme of analysis?
I think that I'd say that one of the underlying themes of analysis is, really, the limit. In pretty much every subfield of analysis, we spend a lot of time trying to control the size of certain ...
- 11.6k
89
votes
Accepted
Does every Cauchy sequence converge to *something*, just possibly in a different space?
You are correct in the narrow sense that every Cauchy sequence does converge in some space.
To be precise, let $(X;d_1)$ be any metric space with at least two points, let $Y$ be the set of Cauchy ...
- 7,586
86
votes
Accepted
Are there periodic functions without a smallest period?
For a nontrivial example, consider the Dirichlet function, which has $$\delta(x) = \begin{cases}0 & \text{ if $x$ is rational}\\1 & \text{ if $x$ is irrational}\end{cases}$$
Then $\delta(x)$ ...
84
votes
78
votes
How to evaluate $\int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x$ using complex analysis?
$\newcommand{\Res}{\text{Res}}$
Firstly define:
\begin{align}
f(z) = \frac{1}{z^2+bz+1}
\end{align}
Secondly define:
\begin{align}
g(z)= (-z)^{-ia}f(z)
\end{align}
We use the Principal Log to define $(...
- 8,438
69
votes
Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
Of course it is a ratio.
$dy$ and $dx$ are differentials. Thus they act on tangent vectors, not on points. That is, they are functions on the tangent manifold that are linear on each fiber. On the ...
- 1,212
58
votes
A continuous, nowhere differentiable but invertible function?
Invertible implies bijective by set theory, and bijective together with continuity implies strictly increasing or decreasing, which imply differentiability almost everywhere! (This is known as ...
- 16.2k
54
votes
Accepted
A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$
Let the considered integral be denoted by $I$. Our starting point is to reduce the number of logarithms of different arguments in the integrand. Thus, using the fact
that $6ab^2=(a+b)^3-2a^3+(a-b)^3$ ...
- 28k
54
votes
Accepted
Why doesn't the nested interval theorem hold for open intervals?
Consider the family $A_n = (0, \frac{1}{n})$. We have $A_{n+1} \subset A_n$ for every $n \in \mathbb{N}$, and the length of $A_n$ approaches zero as $n$ approaches infinity, but $\bigcap_{n=1}^{\infty}...
- 91.3k
50
votes
"Gaps" or "holes" in rational number system
There is a difference between a thing not existing in some set, and the existence of "gap" corresponding to that thing. For example, there is no rational number $p$ such that $p > q$ for all ...
- 24.1k
49
votes
Accepted
Baby/Papa/Mama/Big Rudin
In order to sum up the above comments, the corresponding books for these nick names are
$1$. Baby = Principles of Mathematical Analysis;
$2$. Papa/Big = Real and Complex Analysis;
$3$. Grandpa = ...
49
votes
Why does Rudin say "the rational number system is inadequate as a field"?
The author means that $\mathbb Q$ as an ordered field is incomplete, i.e. not every Cauchy sequence in $\mathbb Q$ converges in $\mathbb Q$, or equivalently not every nonempty subset of $\mathbb Q$ ...
- 4,043
47
votes
What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
For given $n\geq2$ one has
$$e\cdot n!=n!\sum_{k=0}^\infty{1\over k!}=n!\left(\sum_{k=0}^n{1\over k!}+\sum_{k=n+1}^\infty{1\over k!}\right)=m_n+r_n$$
with $m_n\in{\mathbb Z}$ and
$${1\over n+1}<r_n=...
- 217k
46
votes
Addition is to Integration as Multiplication is to ______
If $f\colon [a,b] \to (0,\infty)$, then the closest analogue to a multiplicative integral of $f$ is
$$
{\prod}_a^b \,f(x)^{dx} \;\underset{\mathrm{def}}{=}\; \exp\left(\int_a^b \ln f(x)\,dx\right).
$$
...
- 46.5k
44
votes
What is integration by parts, really?
Integration by parts is a corollary of the product rule:
$(uv)' = uv' + u'v$
Take the integral of both sides to get $uv = \int u \ dv + \int v \ du$.
If you were supposed to remember it separately ...
- 19.2k
44
votes
The staircase paradox, or why $\pi\ne4$
Intuitive Response (for those who don't understand the more analytical responses)
The answer is easy. We just have to zoom in.
We can see at low zoom how the (purple) staircase hugs circle, but ...
- 956
44
votes
Why is the notion of analytic function so important?
Analytic functions have several nice properties, including but not limited to:
They are $C^\infty$ functions.
If, near $x_0$, we have$$f(x)=a_0+a_1(x-x_0)+a_2(x-x_0)^2+a_3(x-x_0)^3+\cdots,$$then$$f'...
- 399k
43
votes
Why is the construction of the real numbers important?
Imagine a mathematical paper that starts with: "Let $\mathbb G$ be a countable complete ordered field." We know that such an object doesn't exist, so anything that the paper has to say about it is ...
- 60.3k
42
votes
How to know if a term is divisible by 10
Your expression can be written as $16^{500}-6^{500}$. Recall that $a^n-b^n$ is divisible by $a-b$ for $n \in \mathbb{N}$.
- 5,321
42
votes
Accepted
Meaning of the backslash operator on sets
It’s set theoretic complement and in this case it denotes the set of all reals which are not natural:
$$ℝ \setminus ℕ = \{x ∈ ℝ;~x \notin ℕ\}$$
- 17.7k
42
votes
Why does $1+2+3+\cdots = -\frac{1}{12}$?
This infinite series is ultimately divergent because
$$
1+2+3+4+\cdots=\sum\limits_{k=1}^{\infty} k$$
$$ = \lim\limits_{n\to\infty} \sum\limits_{k=1}^{n} k = \lim\limits_{n\to\infty} \frac{n(n+1)}{2} ...
- 8,591
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