Intereting Posts

Showing the metric $\rho=\frac{d}{d+1} $ induces the same toplogy as $d$
Crossing a lane of traffic
row operations, swapping rows
No maximum(minimum) of rationals whose square is lesser(greater) than $2$.
Showing Lipschitz continuity of Sobolev function
Induction and convergence of an inequality: $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \frac{1}{\sqrt{2n+1}}$
Concise proof that every common divisor divides GCD without Bezout's identity?
Real Analysis – A sequence that has no convergent subsequence
Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$
Computing sums of divisors in $O(\sqrt n)$ time?
Is there a name for the “famous” inequality $1+x \leq e^x$?
The group of $k$-automorphisms of $k]$, $k$ is a field
A trigonometric identity for special angles
Multivariable Calculus for GRE
Sufficient conditions for separately measurable functions being jointly measurable.

I notice both wikipedia and mathworld have the great result of $\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx$ that:

$\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx=\sum\limits_{n=0}^m\dfrac{(-1)^n(n+1)^{n-1}}{n!}\Gamma(n+1,-\ln x)+\sum\limits_{n=m+1}^\infty(-1)^na_{mn}\Gamma(n+1,-\ln x)$ , where $a_{mn}=\begin{cases}1&\text{if}~n=0\\\dfrac{1}{n!}&\text{if}~m=1\\\dfrac{1}{n}\sum\limits_{j=1}^nja_{m,n-j}a_{m-1,j-1}&\text{otherwise} \end{cases}$

How does this result derived?

- Integration of sqrt Sin x dx
- what are the possible answers we can get for the below intergral?
- Is there any closed form solution available for following integral?
- What is the integration of $\int 1/(x^{2n} +1)dx$?
- Evaluating $\int \frac{\sec^2 x}{(\sec x + \tan x )^{{9}/{2}}}\,\mathrm dx$
- Tough quadrilogarithm integral

- Indefinite Integral of $\frac{1}{(ax^2+bx+c)^n}$
- How to evaluate $\int 1/(1+x^{2n})\,dx$ for an arbitrary positive integer $n$?
- Demystify integration of $\int \frac{1}{x} \mathrm dx$
- Find the integration of $\sec(x)$ and prove it
- What is the integral of $e^x \tan(x)$?
- Help with $\int \frac 1{\sqrt{a^2 - x^2}} \mathrm dx$
- calculation of $\int\frac{1}{\sin^3 x+\cos^3 x}dx$ and $\int\frac{1}{\sin^5 x+\cos^5x}dx$
- Calculate $\int \limits {x^n \over 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+…+\frac{x^n}{n!}} dx$ where $n$ is a positive integer.
- How to evaluate the trigonometric integral $\int \frac{1}{\cos x+\tan x }dx$
- What is wrong with my integral? $\sin^5 x\cos^3 x$

Somebody posted a nice question on sci.math long time ago about finding the expansion of the function exp(exp(x)) in terms of x. Two people answered correctly: Robert Israel and Leroy Quet. I used Leroy’s result and found that a secondary pattern emerged, which is the series up to m, which is related to the W function, for which I later published (this and some other results as incidental by indcution). In reality Leroy’s name should be on the reference, since he solved the basic step of finding the first expansion. At the time I underestimated the importance of his result and forgot to give him credit for it.

- Understanding the Analytic Continuation of the Gamma Function
- Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $
- prove that $\displaystyle\lim_{x\rightarrow 1}\frac{x^{1/m}-1}{x^{1/n}-1}=\frac{n}{m}$
- A categorical first isomorphism theorem
- How should I understand the $\sigma$-algebra in Kolmogorov's zero-one law?
- When is a metric space Euclidean, without referring to $\mathbb R^n$?
- Where can I find SOLUTIONS to real analysis problems?
- Proof of “continuity from above” and “continuity from below” from the axioms of probability
- Global convergence for Newton's method in one dimension
- Number of permutations which fixes a certain number of point
- how to prove that $ \lim x_n^{y_n}=\lim x_n^{\lim y_n}$?
- Prove $0$ is a partial limit of $a_n$
- Big List of Fun Math Books
- Find the limiting value of $S=a^{\sqrt{1}}+a^{\sqrt{2}}+a^{\sqrt{3}}+a^{\sqrt{4}}+…$ for $0 \leq a < 1$
- Minimum principle in Hilbert space