# Eikonal Blog

## 2010.01.04

### LaTeX in WordPress

Filed under: knowledgeManagement, mathematics, typesetting — Tags: , , — sandokan65 @ 20:46

There are several links talking about embedding and use of LaTeX in WordPress:

### Fourier transform

Filed under: mathematics — Tags: , — sandokan65 @ 20:31

Definition:

• Direct: $F(p) = {\bf F}[f](p) :\equiv \int_{-\infty}^{+\infty} \frac{dt}{(2\pi)^{\mu}} e^{ipt} f(t)$,
• Inverse: ${\bf F}^{-1}[g](t) :\equiv \int_{-\infty}^{+\infty} \frac{dp}{(2\pi)^{1-\mu}} e^{-ipt} g(p)$.

Here some author use $\mu=1$, and some authors $\mu=\frac12$ (e.g. Mitrinović)

Properties:

• ${\bf F}[f(t+a)](p) = e^{iap} F(p)$
• ${\bf F}[e^{iat}f(t)](p) = F(p+a)$
• if $\lim_{t\rightarrow \pm\infty} |f(t)| = 0$, then ${\bf F}[f'(t)](p) = -i p F(p)$
• ${\bf F}[\int_{{\Bbb R}}f(t-u)g(u)\frac{du}{(2\pi)^{\mu}}](p) = F(p) G(p)$

## Table of Fourier transformations

$f(t)$ $F(p)={\bf F}[f](p)$
$\frac{\sin(at)}{t}$ $\frac{1}{2}\sqrt{\frac{\pi}{2}}(\frac{p+a}{|p+a|} - \frac{p-a}{|p-a|})$
$exp(-a t^2)$ $\frac{1}{\sqrt{2a}} exp(-\frac{p^2}{4\pi})$
$\cos(at^2)$ $\frac{1}{\sqrt{2a}} \cos(\frac{p^2}{4a}-\frac{\pi}{4})$
$\sin(at^2)$ $\frac{1}{\sqrt{2a}} \sin(\frac{p^2}{4a}+\frac{\pi}{4})$
$|t|^{-a}$ ($0) \$ $\sqrt{\frac{2}{\pi}} |p|^{1-a} \Gamma(1-a) \sin(\frac{\pi a}{2})$
$(1+t^2)^{-1}$ $\sqrt{\frac{\pi}{2}} e^{-|p|}$

## Sources

• 1) D.S. Mitrinović, J.D. Kečkić: “Jednačine Matematičke Fizike”; 2nd edition; Građevinska Knjiga, Beograd 1978.

### Integral transforms

Filed under: mathematics — Tags: — sandokan65 @ 20:29

General linear integral transformation:

$F(p) = {\bf T}[f](p) :\equiv \int_a^b K(p,t) f(t) dt$

Here we assume that this integral exist, $K$ is a fixed complex function, $p \in {\Bbb C}$ and $a,b\in{\Bbb R}$.

Specific examples:

• Laplace transformation: ${\bf L}$ defined by $a=0$, $b=+\infty$, $K(p,t)=e^{pt}$.
• Fourier transformation: ${\bf F}$ defined by $a=-\infty$, $b=+\infty$, $K(p,t)=\frac1{\sqrt{2\pi}}e^{ipt}$.
• Mellin transformation: ${\bf M}$ defined by $a=0$, $b=+\infty$, $K(p,t)=t^{p-1}$.
• Hankel transformation: ${\bf H}$ defined by $a=0$, $b=+\infty$, $K(p,t)=t \ J_n(pt)$ (a Bessel function of order $n$).

### Schrodinger equation

Filed under: Uncategorized — Tags: — sandokan65 @ 20:26

$\Delta \psi + \frac{2m}{\hbar^2}(E-U)\psi = 0$

### Harmonic oscillator

Filed under: Quantum mechanics — Tags: , , — sandokan65 @ 20:23

Potential function: $U(x)=\frac{m\omega^2 x^2}{2}$

Solutions of the Schr\”odinger equation:

• States: $\psi_n(x) = (\frac{m}{\hbar\omega})^{\frac14} \frac{1}{\sqrt{2^n n! \pi}} H_n(\sqrt{\frac{m\omega}{\hbar}}x) e^{-\frac{m\omega x^2}{\hbar}}$
• Energies: $E_n=\hbar\omega (n+\frac12)$

### Asymptotic expansion

Filed under: mathematics — Tags: , — sandokan65 @ 20:22

Definition:

• a series $\sum_{k=0}^\infty \frac{a_k}{z^k}$ is \underline{an asymptotic expansion} of $f(z)$ iff
• $\lim_{|z|\rightarrow\infty} z^n\{f(z)-\sum_{k=0}^n \frac{a_k}{z^k} \} =0$ for all $n\in{\Bbb N}_0$.

Note:

• The asymptotic expansions are not unique. For example $f(z)$ and $F(z)+e^{-z}$ have the same asymptotic expansion at $z\rightarrow +\infty$.

Watson Lemma:

• If $F(\tau)$ has following expansion near $\tau =0$: $F(\tau) = \sum_{n=1}^\infty a_n \tau^{\frac{n}{r}-1}$ (for $|\tau| \le a$ and $r>0$),
• and if there exist $K$ and $b$ s/t $|F(\tau)| < K e^{b|\tau|}$ for $|\tau| \ge a$,
• then, for large $|\nu|$ (and for $|\arg \nu| < \frac\pi2$):
• ${\cal F}(\nu) :\equiv \int_0^\infty e^{-\nu \tau} F(\tau) d\tau \sim \sum_{n=1}^\infty a_n \Gamma(\frac{n}{\nu}) \nu^{-\frac{n}{r}}$.

Source:

• “The Functions of Mathematical Physics” by Harry Hochstadt; Ch3. The Gamma Function

### Beta function

Filed under: mathematics — Tags: , — sandokan65 @ 20:19

Definition:

$B(x,y):\equiv \int_0^1t^{x-1} (1-t)^y dt$, ($\Re x, \Re y > 0$).

Properties:

• $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$.
• $B(x,y) = (1-e^{2\pi i x})^{-1}(1-e^{2\pi i y})^{-1} \int_{1+,0+,1-,0-} \zeta^{x-1} (1-\zeta)^{y-1} d\zeta$. where the closed countour goes once around $z=1$ in positive direction, the around $z=0$ in positive direction, the around $z=1$ in negative direction and $z=0$ in negative direction.

Source: “The Functions of Mathematical Physics” by Harry Hochstadt

### Logarithmic Gamma function

Filed under: mathematics — Tags: , — sandokan65 @ 20:17

Definition:

$\psi(z) :\equiv d_z \ln\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)}$.

Properties:

• $\psi(z) = -\gamma + \sum_{n=0}^\infty \frac1{n+1} - \frac1{z+1}$.
• $\psi(1) = -\gamma$.
• $\psi(k+1) = -\gamma +(1+\frac12+\frac13+\cdots+\frac1{k})$.
• $\psi'(z) = \sum_{n=0}^\infty \frac1{(z+n)^2}$.

Source: “The Functions of Mathematical Physics” by Harry Hochstadt

### Gamma Function

Definitions:

• 1) Euler: $\Gamma(z) = \int_0^\infty e^{-t} t^{z-1} dt$, ($\Re z > 0$);
• 2) Gauss: $\Gamma(z) = \lim_{n\rightarrow\infty} \frac{n! n^z}{(z)_{n+1}}$, ($z\not\in{\Bbb N}_0$)
• 3) Weierstrass: $\Gamma(z)^{-1} =z e^{\gamma z} \prod_{n=1}^\infty (1+\frac{z}{n}) e^{-\frac{z}{n}}$, where $\gamma :\equiv \lim_{n\rightarrow \infty} (\sum_{k=1}^n \frac1{k} - \ln(n+1)) \sim 0.577,215,7$ is the Euler-Mascheroni constant.

The Mittag-Leffler expansion:

$\Gamma(z) = \sum_{n=0}^\infty \frac{(-)^n}{n!} \frac1{n+z} + \int_{1}^\infty e^{-t} t^{z-1} dt.$

It indicates that $\Gamma(-n+\epsilon) \sim \frac{(-)^n}{n!} \frac1\epsilon$.

Properties:

• $\Gamma(z+1)=z\Gamma(z)$.
• $\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$, ($z\not\in{\Bbb Z}$).
• $\Gamma(\frac12)=\sqrt{\pi}$.
• The multiplication theorem: $\prod_{n=0}^{m-1}\Gamma(z+\frac{n}{m}) = (2\pi)^{\frac{m-1}2} m^{\frac12 - mz} \Gamma(mz)$.
• The duplication formula: $\Gamma(x)\Gamma(z+\frac12) = 2^{1-2z} \sqrt{\pi} \Gamma(2z)$.

Hankel representations:

• $\Gamma(z) = \frac1{e^{2\pi i z}-1} \int_C e^{-\zeta}\zeta^{z-1}d\zeta$,
• $\Gamma(z) = -\frac1{2i\sin(\pi z)} \int_C e^{-\zeta}(-\zeta)^{z-1}d\zeta$,
• $\Gamma(z)^{-1} = -\frac1{2\pi} \int_C e^{-\zeta}\zeta^{-z}d\zeta$.

Here the contour $C$ comes from plus infinity narrowly above the $x$ axis, circles once around origin and returns to plus infinity narrowly below the real axis.

Stirling formula:

$\Gamma(z) = \sqrt{\frac{2\pi}{z}} (\frac{z}{e})^z [1+\frac1{12 z}+\cdots]$

Misc:

• $\lim_{n\rightarrow\infty} \int_0^{n} [e^{-t}-(1-\frac{t}{n})^n] t^{z-1} dt = 0$, $\Re z > 0$.
• $|\Gamma(iy)|^2 = \frac{\pi}{y \sinh(\pi y)}$

Source: “The Functions of Mathematical Physics” by Harry Hochstadt

### Trinomial equation related to Mellin transform

Filed under: mathematics — Tags: , , , — sandokan65 @ 20:08

The algebraic equation $y^n + x y^p -1 = 0$ for ($n>p$) has following series solutions:

$y_0(x) = \frac1{n} \sum_{r=0}^\infty \frac{(-)^r}{r!} \frac{\Gamma(\frac{1+pr}{n}) x^r}{\Gamma(\frac{1+pr}{n}+1-r)}.$

Other $n-1$ solutions are given by the complex rotations as $y_k(x) = \omega^k y_0(\omega^{pk} x)$ where $\omega:\equiv e^{i\frac{2\pi}{n}}$, $k=\overline{0,n-1}$.

The solutions satisfy following ODE:

$(-d_x)^n y(x) = p (-x d_x -n) y(x)$.

Source: “The Functions of Mathematical Physics” by Harry Hochstadt

### Mellin-Barnes integrals

Filed under: mathematics — Tags: , — sandokan65 @ 20:02

$\int_{-i\infty}^{+i\infty} \Gamma(\alpha+s) \Gamma(\beta+s) \Gamma(\gamma-s) \Gamma(\delta-s) ds = 2\pi i \frac{\Gamma(\alpha+\gamma)\Gamma(\beta+\gamma)\Gamma(\alpha+\delta)\Gamma(\beta+\delta)}{\Gamma(\alpha \beta+\gamma+\delta)}$

### Integrals with Gamma functions

Filed under: mathematics — Tags: , , — sandokan65 @ 20:00
• $\int_0^\infty \frac{\cosh(2yt)}{(\cosh(t))^{2x}} dt = 2^{2x-2} \frac{\Gamma(x+y)\Gamma(x-y)}{\Gamma(2x)}$, for $\Re x > |\Re y|$.
• $\int_{-i\infty}^{+i\infty} \frac{e^{its}ds}{\Gamma(\mu+s)\Gamma(\nu-s)} = \theta(\pi - |t|) \frac{2(\cos(\frac{t}2))^{\mu+\nu-2} e^{\frac{1}{2} i t (\nu-\mu)}}{\Gamma(\mu+\nu-1)}.$

Sources:

• “The Functions of Mathematical Physics” by Harry Hochstadt

### Binomial symbol

Filed under: mathematics — Tags: , , — sandokan65 @ 19:56
• Binomial theorem: $(1-t)^{-\rho} = \sum_{n=0}^\infty \frac{\Gamma(\rho+n)}{\Gamma(\rho)}\frac{t^n}{n!}$
• “Source: The Functions of Mathematical Physics” by Harry Hochstadt
• $\sum_{r=0}^m(-)^r \binom{n}{r} = (-)^n \binom{n-1}{m}$ for $0 \le m \le n-1$.
• $\sum_{r=k+1}^n (-)^{n-r} \binom{n}{r} = (-)^{n-k-1} \binom{n-1}{k}$.

### Products

Filed under: mathematics — Tags: , — sandokan65 @ 19:55
• $\prod_{k=0}^{m-1} \sin(\pi(z+\frac{k}{m})) = 2^{1-m} \sin(\pi m z)$.

Sources:

• “The Functions of Mathematical Physics” by Harry Hochstadt

### Euler-Mascheroni constant

Filed under: mathematics — Tags: , — sandokan65 @ 19:52
• $\gamma :\equiv \lim_{n\rightarrow \infty} (\sum_{k=1}^n \frac1{k} - \ln(n+1))$.
• $\gamma \sim 0.577,215,664,9...$
• $\gamma = - \int_0^\infty e^{-t} \ln(t) dt$.

For the more precise numerical value see https://eikonal.wordpress.com/2010/03/09/eulers-constant-euler-mascheroni-constant-gamma/.

Also:

• $\psi(1) = \Gamma'(1) = -\gamma$,
• $\psi(1/2) = - \gamma -2\ln(2)$,
• $\gamma = 1 - \Gamma'(2)$.

Inequalities:

• $\frac1{24 (n+1)^2} < R_n - \gamma < \frac1{24 n^2}$, where $R_n :\equiv \sum_{k=1}^n \frac1{k} - \ln(n+\frac12))$. (source: [2])
• $- c_k \le \gamma -1 + \ln(k) \sum_{l=1}^{12k+1} d(k,l) - \sum_{l=1}^{12k+1} \frac{d(k,l)}{l+1} \le c_k$, where $c_k :\equiv \frac{2}{(12k)!} + 2k^2 e^{-k}$ and $d(k,l) = (-)^{l-1} \frac{k^{l+1}}{(l-1)!(l+1}$. (source: [2])

Sources:

1. [1] “The Functions of Mathematical Physics” by Harry Hochstadt
2. [2] “Topics in Special Functions” by G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen – http://arxiv.org/abs/0712.3856

### A math blog with implementation of LaTeX in JavaScript

Filed under: knowledgeManagement, mathematics — Tags: , , — sandokan65 @ 03:54

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Related here: Mathematics sites (go to Books section)- https://eikonal.wordpress.com/2010/03/17/mathematics-sites/ | Physics books online – https://eikonal.wordpress.com/2010/03/28/physics-books-online/ | Infosec books – https://eikonal.wordpress.com/2010/10/19/infosec-books/ | Toward a New Alexandria Library – https://eikonal.wordpress.com/2010/03/13/toward-a-new-alexandria/ | Martial Arts magazines and other sources (see Books) – https://eikonal.wordpress.com/2010/02/05/martial-arts-magazines-and-other-sources/ | Philosophy books – https://eikonal.wordpress.com/2011/03/09/philosophy/ | Emanuel Haldeman-Julius – https://eikonal.wordpress.com/2010/05/04/emanuel-haldeman-julius/ | Download sites – https://eikonal.wordpress.com/2010/01/23/download-sites/

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### 100 Pushups chalenge

Filed under: health, workout — Tags: , , , — sandokan65 @ 01:46

We started this challenge 6 weeks ago. Some hints:

• Repeat the previous week (actually the two-week block) as frequently as necessary
• Give yourself enough pause between sets: if instructions say 1 minute, but you need more rest, take 2 minutes.
• Same is if you feel that you need an extra day in middle of a practice week – it feels that I can do more and easier after such a extra day of rest.
• If you do these in the morning the first several sets will hurt more that if you do the challenge in the evening. I guess it is up to not being warmed-up and stretched enough.
• If you are doing some other challenge, too, you may want to stagger the days, e.g. do the push-ups training on Mon/Wed/Fri and sit-ups training on Tue/Thu/Sat. The only downside of such combination is that towards the end of the week you are really temped to postpone one or another challenge practice, since your body may be aching for a rest day.
• The current combination that works very well for us to do both push-ups and sit-ups challenges the same day, by interleaving their sets. Do the first push-up set, than immediately the first sit-ups set, and do the first 1/2 minutes pause only then, etc.

Links:

Local links: Fitness links – https://eikonal.wordpress.com/2010/02/10/fitness-links/.