Search found 59 matches

by r9m
Sat Dec 19, 2015 10:38 am
Forum: Calculus
Topic: my first post and a cool and challenging integral
Replies: 3
Views: 2941

Re: my first post and a cool and challenging integral

Maybe I am missing something simple or direct :? \begin{align*}I&=\int_0^{\infty} \frac{(1-\sin ax)(1-\cos bx)}{x^2}\,dx\\&= \int_0^{\infty} \frac{1-\cos bx - \sin ax + \frac{1}{2}\sin (a+b)x + \frac{1}{2}\sin (a-b)x}{x^2}\,dx\\&= 2\int_0^{\infty} \frac{\sin^2 \frac{bx}{2}}{x^2}\,dx - \i...
by r9m
Tue Dec 15, 2015 2:29 pm
Forum: Calculus
Topic: An arctan series
Replies: 2
Views: 2454

Re: An arctan series

\begin{align*}S&=\sum\limits_{n=1}^{\infty} \arctan \frac{10n}{(3n^2+2)(9n^2-1)} \\&= \sum\limits_{n=1}^{\infty} \arg \left(1+\frac{10in}{(3n^2+2)(9n^2-1)}\right)\\&= \arg \prod\limits_{n=1}^{\infty}\left(1+\frac{10in}{(3n^2+2)(9n^2-1)}\right)\\&= \arg \prod\limits_{n=1}^{\infty}\lef...
by r9m
Mon Dec 14, 2015 8:57 pm
Forum: Linear Algebra
Topic: Identity with matrix exponential
Replies: 2
Views: 2977

Re: Identity with matrix exponential

Hi Papapetros Vaggelis, Nice solution! :D :clap2: Looking at the Eigenvalues indeed is the simplest approach! You derived the characteristic equation to be: $k(k^2+r^2) = 0$, so by Cayley-Hamilton Theorem we have the identity $A^3+r^2A = 0$. Thus in general we have: $$\begin{align*}A^{2n}=(-1)^{n-1}...
by r9m
Sat Dec 12, 2015 4:45 pm
Forum: Calculus
Topic: Computation of integral
Replies: 1
Views: 1912

Re: Computation of integral

We have: $$\begin{align*}\frac{\sin \pi \alpha}{\cosh \pi x + \cos \pi \alpha} &= \frac{1}{1+e^{-\pi(x+i\alpha)}}-\frac{1}{1+e^{-\pi(x - i\alpha)}}\\&= 2\sum\limits_{n=1}^{\infty} (-1)^{n}e^{-n\pi x}\sin n\pi\alpha\end{align*}$$ Thus, we have: \begin{align*}\int_{-\infty}^{\infty} \frac{e^{-...
by r9m
Fri Dec 11, 2015 9:56 pm
Forum: Calculus
Topic: Integral with fractional part
Replies: 2
Views: 2487

Re: Integral with fractional part

\begin{align*}\int_1^{\infty} \frac{\{x\}}{x^n}\,dx &= \sum\limits_{k=1}^{\infty}\int_k^{k+1} \frac{\{x\}}{x^n}\,dx\\ &= \sum\limits_{k=1}^{\infty} \int_k^{k+1} \frac{x - k}{x^n}\,dx\\ &= \sum\limits_{k=1}^{\infty} \left[\frac{x^{2-n}}{2-n} - k\frac{x^{1-n}}{1-n}\right]_k^{k+1}\\ &= ...
by r9m
Fri Dec 11, 2015 2:15 pm
Forum: Linear Algebra
Topic: Identity with matrix exponential
Replies: 2
Views: 2977

Identity with matrix exponential

If $a,b,c$ are real numbers (not all $0$) and denote by $\displaystyle r = \sqrt{a^2+b^2+c^2}$, Show that: $$\exp{\left( \begin{matrix} 0 & a & c \\ -a & 0 & b \\ -c & -b & 0 \end{matrix} \right)} = \cos{r \left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &...
by r9m
Fri Dec 11, 2015 12:35 am
Forum: Real Analysis
Topic: An integral with square roots and logarithms
Replies: 1
Views: 1956

Re: An integral with square roots and logarithms

We make the change of variable $x \mapsto x^{1/k}$ and denote $\dfrac{n+1}{k} = a$ (say). \begin{align*}&\int_0^1 x^n\log \left(\sqrt{1+x^k} - \sqrt{1-x^k}\right)\,dx\\=& \frac{1}{k}\int_0^1 x^{a-1}\log \left(\sqrt{1+x} - \sqrt{1-x}\right)\,dx\\=& \frac{\log 2}{ak} - \frac{1}{2ak}\int_0^...
by r9m
Thu Dec 10, 2015 5:13 pm
Forum: Calculus
Topic: $\int_0^\infty \frac{\log x \sin x}{x}\, dx$
Replies: 2
Views: 2423

Re: $\int_0^\infty \frac{\log x \sin x}{x}\, dx$

We might begin with the observation that, for $\mathfrak{Re}(s) < 1$, the integral $\displaystyle \int_{\gamma_r} z^{s-1}e^{iz}\,dz$ vanishes over the contour $\gamma_r = [0,r] \cup re^{i[0,\pi/2]}\cup [ir,0]$. Since, \begin{align*}\left|\int_{[re^{i[0,\pi/2]}]} z^{s-1}e^{iz}\,dz\right| &\le \in...
by r9m
Thu Dec 10, 2015 2:26 pm
Forum: Calculus
Topic: Improper integral with log.
Replies: 2
Views: 2438

Re: Improper integral with log.

\begin{align*}\int_0^{\infty} \log^n \left(\frac{e^{x}}{1-e^{x}}\right)\,dx&= (-1)^{n}\int_0^{\infty} \log^n (1-e^{-x})\,dx\\&= (-1)^{n}\int_0^1 \frac{\log^n (1-x)}{x}\,dx \qquad \text{ change of variable } e^{-x} \mapsto x \\&= (-1)^{n}\int_0^1 \frac{\log^n x}{1-x}\,dx = n!\zeta(n+1)\en...