Intereting Posts

Proving that $S_n$ has order $n!$
Find this limit: $ \lim_{n \to \infty}{(e^{\frac{1}{n}} – \frac{2}{n})^n}$
Understanding Gödel's Incompleteness Theorem
Continuous and bounded imply uniform continuity?
The closure of a product is the product of closures?
Where are the transcendental numbers?
If $f(z)=(g(z),h(z))$ is continuous then $g$ and $h$ are as well
Order of a product of subgroups
Bulgarian Solitaire: Size of root loops
General Continued Fractions and Irrationality
${{p^k}\choose{j}}\equiv 0\pmod{p}$ for $0 < j < p^k$
Advice about taking mathematical analysis class
Dimension for a closed subspace of $C$.
Eigenvalues For the Laplacian Operator
How to calculate the percentage of increase/decrease with negative numbers?

One usually gets several definitions of the logarithm along his studies.

- You might be first introduced to the exponential and then told that the logarithm is its inverse.
- You might be given

$$\log x = \int\limits_1^x {\frac{{du}}{u}} $$ - Like Landau does. Let $k = 2^n$, then:

$$\log x =\mathop {\lim }\limits_{n \to \infty } k\left( {\root k \of x – 1} \right)$$ - And last, if you ever read, Euler famously wrote:

$$ – \log x = \frac{{1 – {x^0}}}{0}$$

Landau’s definition (although I find it the most usefull to work with) really baffled me untill just now. Since

$$\int\limits_1^x {\frac{{du}}{{{u^{\alpha + 1}}}}} = \frac{{{x^{ – \alpha }} – 1}}{{ – \alpha }}$$

Then being $\frac{1}{k} = -\alpha$ one hopes to have:

- Question about basis and finite dimensional vector space
- What sets are Lebesgue measurable?
- Localization of the Integer Ring
- Why not defining random variables as equivalence classes?
- A definition of Conway base-$13$ function
- What is a special function?

$$\mathop {\lim }\limits_{\alpha \to 0} \int\limits_1^x {\frac{{du}}{{{u^{\alpha + 1}}}}} = \int\limits_1^x {\frac{{du}}{u}} = \log x = \mathop {\lim }\limits_{k \to \infty } k\left( {\root k \of x – 1} \right)$$

How can one justify taking the limit before integration? Continuity suffices?

- Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$
- $\int_0^{a} x^\frac{1}{n}dx$ without antiderivative for $n>0$
- How to evaluate $\int 1/(1+x^{2n})\,dx$ for an arbitrary positive integer $n$?
- $f \in L^1 ((0,1))$, decreasing on $(0,1)$ implies $x f(x)\rightarrow 0$ as $x \rightarrow 0$
- Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $
- Is this closed-form of $\int_0^1 \operatorname{Li}_3^2(x)\,dx$ correct?
- “sheaf” au sens de Serre
- How to find the integral $\int \tan (5x) \tan (3x) \tan(2x) \ dx $?
- Riemann's Integrals Question
- Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$

You can use the fact that it’s a uniform limit for $u \in [1,x]$, or use Dominated Convergence or Monotone Convergence.

- A question about numbers from Euclid's proof of infinitude of primes
- Surjectivity of a map between a module and its double dual
- Calculating $ \int _{0} ^{\infty} \frac{x^{3}}{e^{x}-1}\;dx$
- Monic and epic implies isomorphism in an abelian category?
- Rank-one perturbation proof
- How to prove that $\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}$ holds for any $(x,y,z)\in^3$
- Triangles area question
- On the element orders of finite group
- Applying derangement principle to drunken postman problem.
- Is it possible to solve $i^2+i+1\equiv 0\pmod{2^p-1}$ in general?
- Show structure of a commutative ring in a tensor product
- Inducing homomorphisms on localizations of rings/modules
- Is the value of $\sin(\frac{\pi}{n})$ expressible by radicals?
- Prove that this function is bounded
- Is normal extension of normal extension always normal?