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1500 questions
81
votes
9 answers
Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?
I have seen the Fresnel integral
$$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$
evaluated by contour integration and other complex analysis methods, and I have found these methods to be the standard way to evaluate this integral. I was…
Argon
- 25,303
81
votes
7 answers
Does associativity imply commutativity?
I used to think that commutativity and associativity are two distinct properties. But recently, I started thinking of something which has troubled this idea:
$$(1+1)+1 = 1+ (1+1)\implies 2+1=1+2$$ Here using associativity of addition operation,…
Aritra Das
- 3,528
81
votes
5 answers
Derivative of the inverse of a matrix
In a scientific paper, I've seen the following
$$\frac{\delta K^{-1}}{\delta p} = -K^{-1}\frac{\delta K}{\delta p}K^{-1}$$
where $K$ is a $n \times n$ matrix that depends on $p$. In my calculations I would have done the following
$$\frac{\delta…
Sara
- 1,017
81
votes
2 answers
Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?
In his gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$:
Proof of equality of square and curved areas is based on another…
VividD
- 15,966
80
votes
10 answers
Why does being holomorphic imply so much about a function?
I haven't yet started my complex analysis course (soon!), but recently (inspired by you guys) I've been looking into holomorphic functions. And wow, they're cool! There's so much stuff that's true about them... But my question is: why is being…
ShakesBeer
- 3,641
80
votes
7 answers
Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra
I am taking a course next term in homological algebra (using Weibel's classic text) and am having a hard time seeing some of the big picture of the idea behind homological algebra.
Now, this sort of question has been asked many times on forums such…
Alex Youcis
- 54,059
80
votes
4 answers
Integral $\int_1^\infty\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}$
Consider the following integral:
$$\mathcal{I}=\int_1^\infty\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}\,,$$
where $\operatorname{arccsc}$ is the inverse…
Vladimir Reshetnikov
- 47,122
80
votes
4 answers
Integrals of $\sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}}$ in elementary functions
Let $f_n(x)$ be recursively defined as
$$f_0(x)=1,\ \ \ f_{n+1}(x)=\sqrt{x+f_n(x)},\tag1$$
i.e. $f_n(x)$ contains $n$ radicals and $n$ occurences of $x$:
$$f_1(x)=\sqrt{x+1},\ \ \ f_2(x)=\sqrt{x+\sqrt{x+1}},\ \ \…
Vladimir Reshetnikov
- 47,122
80
votes
3 answers
Under what conditions the quotient space of a manifold is a manifold?
There are many operations we can do with topological spaces that when we apply to topological manifolds gives us back topological manifolds. The disjoint union and the product are examples of that.
Another common operation is to take the quotient by…
Gold
- 26,547
80
votes
5 answers
(Theoretical) Multivariable Calculus Textbooks
(Note that I have used bold text frequently simply to highlight the key points of my question for those who do not have the time to read through it thoroughly (it is not very long, however); I hope this is not considered offensive.)
There are many…
Amitesh Datta
- 20,779
80
votes
8 answers
How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?
How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$
I found the above interesting identity in the book $\bf \pi$ Unleashed.
Does anyone knows how to prove it?
Thanks.
Neves
- 5,617
80
votes
4 answers
What is the difference between the Frobenius norm and the 2-norm of a matrix?
Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results?
If they are identical, then I suppose the only difference between them…
Ricket
- 1,171
80
votes
6 answers
How do you compute negative numbers to fractional powers?
My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure if the number stays negative or turns into a…
Kot
- 3,273
80
votes
8 answers
Koch snowflake paradox: finite area, but infinite perimeter
The Koch snowflake has finite area, but infinite perimeter, right?
So if we make this snowflake have some thickness (like a cake or something), then it appears that you can fill it with paint like this ($\text{finite area} \times \text{thickness}…
ArtisanRtasin
- 743
80
votes
18 answers
List of interesting integrals for early calculus students
I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely challenging like Putnam problems or anything). For…
GeoffDS
- 11,270