# Eikonal Blog

## 2010.07.13

Filed under: TeX — Tags: , , , , , , , , , , — sandokan65 @ 16:03

## (to be a) Cheatsheet

### Lists

 \begin{list}{label code}{body code} \item Item one \item Item two \item Item three \end{list} 

Example:

 \begin{list}{*}{} \item Item one \item Item two \item Item three \end{list} 

### Error: “extra alignment tab has been changed to \cr” in pmatrix environment

Trying to compile the matrix with 16 columns:

 $$\begin{pmatrix} 1&0&0&0& 0&0&0&0& 0&1&0&0& 0&0&0&0 \\ 0&0&0&0& 1&0&0&0& 0&0&0&0& 0&1&0&0 \\ 0&0&1&0& 0&0&0&0& 0&0&0&1& 0&0&0&0 \\ 0&0&0&0& 0&0&1&0& 0&0&0&0& 0&0&0&1 \end{pmatrix}$$ 

produced following error:

 ./document.tex:750: Extra alignment tab has been changed to \cr. \endtemplate l.750 1&0&0&0& 0&0&0&0& 0&1& 0&0& 0&0&0&0 \\ 

It appears that LaTeX has some internal limitation on default maximal number of columns. This can be changed (accoring to http://newsgroups.derkeiler.com/Archive/Comp/comp.text.tex/2006-06/msg01298.html) by setting

 \setcounter{MaxMatrixCols}{16} 

in the preamble of the document.

## 2010.03.11

Filed under: mathematics — Tags: , , , , , , , , , , — sandokan65 @ 14:02

## Tools

Primary tool is, off course, your brain

Information:

## 2010.03.09

### Computational tools online

Filed under: mathematics — Tags: , , , , — sandokan65 @ 13:22

## 2010.01.22

### BCS formula on computer

Filed under: mathematics — Tags: , , — sandokan65 @ 16:06

Def: $e^C :\equiv e^A e^B$

$C = A + \int_0^1 dt \psi(e^{a}e^{t b}) B$,
where $a:\equiv ad(A)$, $b:\equiv ad(B)$ and
$\psi(z):\equiv \frac{z \ln(z)}{z-1} = 1 + \frac{w}{1\cdot 2} - \frac{w^2}{2\cdot 3} + \frac{w^3}{3\cdot 4} - \cdots$ for $w:\equiv z-1$.

Richtmyer and Greenspan obtained the 512 initial terms of that expansion, but here I present only several of them:

$C = A + B + \frac12 a B + \frac1{12} a^2 B - \frac1{12} b a B + \frac1{12,096} a^4 b^2 a B - \frac1{6,048} a^3 b a^3 B - \frac1{3,780} a^3 b a b a B + ...$

Not all terms are independent, due to Jacobi identities.

Sources:

• T1267: R.D. Richtmyer and S. Greenspan: “Expansion of the Campbell-Baker-Hausdorff formula by Computer”; Communications on Pure and Applied Mathematics, Vol XVIII 107-108 (1965).

### Guenin on commutators

Filed under: Uncategorized — Tags: , — sandokan65 @ 15:46

$A B^n = \sum_{j=0}^n \binom{n}{j} (-)^j B^{n-j} \ ad(B)^j A.$

$B^n A = \sum_{j=0}^n \binom{n}{j} (-)^j \ (ad(B)^j A) B^{n-j}.$

$\partial_s f(F(s)) = \sum_{j=0}^\infty \frac{(-)^j}{(j+1)!} f^{(j+1)}(F(s)) \ ad(F(s)^j \partial_s F(s) = \sum_{j=0}^\infty \frac{1}{(j+1)!} \ (ad(F(s))^j \partial_s F(s) ) f^{(j+1)}(F(s)).$

$[H, f(A)] = \sum_{j=0}^\infty \frac{(-)^j}{j!} f^{(j)}(A) \ ad(A)^j H = - \sum_{j=0}^\infty \frac1{j!}\ (ad(A)^j H) f^{(j)}(A).$
—-
Sources:

• T1266: M. Guenin: “On the Derivation and Commutation of Operator Functionals”; Helv. Phys. Act. 41, 75-76, 1968.

## 2010.01.05

### Squares with just two different decimal digits

Filed under: mathematics, number theory, puzzles — Tags: — sandokan65 @ 16:00

I used to state here

There exist only finitely many squares with just two decimal digits” providing a really short list

• $38^2 = 1,444$
• $88^2 = 7,744$
• $109^2 = 11,881$
• $173^2 = 29,929$
• $212^2 = 44,944$
• $235^2 = 55,255$
• $3,114^2 = 9,696,996$

plus infinite classes:

• $10^{2n}$,
• $4\cdot 10^{2n}$,
• $9\cdot 10^{2n}$.

This is not correct. Thanks to Bruno Curfs for pointing this out, as well as for finding the reference [2] cited below.

• Namely, one can see that squares of all single digit numbers satisfy targeted condition:
• $1^2 = 01$,
• $2^2 = 04$,
• $3^2 = 09$,
• $4^2 = 16$,
• $5^2 = 25$,
• $6^2 = 36$,
• $7^2 = 49$,
• $8^2 = 64$,
• $9^2 = 81$,
• Then, there are squares of following two-digit numbers:
• $10^2 = 100$,
• $11^2 = 121$,
• $12^2 = 144$,
• $15^2 = 225$,
• $20^2 = 400$,
• $21^2 = 441$,
• $22^2 = 484$,
• $26^2 = 676$,
• $30^2 = 900$,
• $38^2 = 1,444$,
• $88^2 = 7,744$,
• For squares of 3-digit numbers one finds:
• $100^2 = 10,000$,
• $109^2 = 11,881$
• $173^2 = 29,929$
• $212^2 = 44,944$
• $235^2 = 55,255$
• $264^2 = 69,696$ (thanks to Bruno Curfs)

Of the higher cases, there are known only infinitely long classes:

• $10^{2n}$
• $4\cdot 10^{2n}$
• $9\cdot 10^{2n}$

• $3,114^2 = 9,696,996$
• $81,619^2 = 6,661,661,161$.

Author of page “1. Squares of 2 different digits” (http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math02/three01.htm) provides an implementation of the algorithm to perform exhaustive check. According to his numerical experiments, there are no other sporadic occurrences up to at least $y=10^{48}$ (i.e. $x=10^{24}$).

Reference:

Similar here: More simple math wonders – https://eikonal.wordpress.com/2012/03/14/more-simple-math-wonders/ | Mental calculation of cube root of a six-digit number – https://eikonal.wordpress.com/2010/01/14/mental-calculation-of-cube-root-of-a-two-digit-number/ | Squares with just two different decimal digits – https://eikonal.wordpress.com/2010/01/05/squares-with-just-two-different-decimal-digits/ | Number theory finite concidental sums – https://eikonal.wordpress.com/2010/01/05/number-theory-finite-considental-sums/

### Number theory finite concidental sums

Filed under: mathematics, number theory — Tags: — sandokan65 @ 15:43

$1+3+3^n+3^{n+1}+3^{2n} = (2+3^n)^2$

$1+3^n+3^{n+1}+3^{2n} +3^{2n+1}= (1+2 \cdot 3^n)^2$

$1+7+7^2+7^3=20^2$

$1+12^2+12^3+12^4+12^5=521^2$

Reference: T1277

Similar here: More simple math wonders – https://eikonal.wordpress.com/2012/03/14/more-simple-math-wonders/ | Mental calculation of cube root of a six-digit number – https://eikonal.wordpress.com/2010/01/14/mental-calculation-of-cube-root-of-a-two-digit-number/ | Squares with just two different decimal digits – https://eikonal.wordpress.com/2010/01/05/squares-with-just-two-different-decimal-digits/ | Number theory finite concidental sums – https://eikonal.wordpress.com/2010/01/05/number-theory-finite-considental-sums/

### Pi calculations

Filed under: mathematics — Tags: , , , — sandokan65 @ 12:00

Bailey-Borwein-Plouffe formula:
$\pi = \sum_{n=0}^\infty \frac1{16^n} \left( \frac{4}{8n+1} - \frac{2}{8n+4} - \frac1{8n+5} - \frac1{8n+6} \right)$. (Sources: [2], [3]).

Plouffe:
$\frac{\pi}{8} = \sum_{n=0}^\infty \frac1{n 2^{\left[\frac{n+1}2\right]}} \left( \left[\frac{n+7}8\right] - \left[\frac{n+6}8\right] + \left[\frac{n+1}8\right] - \left[\frac{n+4}8\right]\right)$. (Source: [3])

Belard:
$\pi = \frac1{2^6} \sum_{n=0}^\infty \frac{(-)^n}{2^{10n}} \left( -\frac{2^5}{4n+1} - \frac1{4n+3} + \frac{2^8}{10n+1} - \frac{2^6}{10n+3} - \frac{2^2}{10n+5} - \frac{2^2}{10n+7} + \frac1{10n+9} \right)$. (Source: [1])

Belard:
$\pi = \frac1{740,025}\left(\sum_{n=1}^\infty \frac{3P(n)}{\binom{7n}{2n} 2^{n-1}} - 20,379,280 \right)$ where:
$P(n) = - 885,673,181 n^5 + 3,125,347,237 n^4 - 2,942,969,225 n^3 + 1,031,962,795 n^2 - 196,882,274 n + 10,996,648$. (Source: [1])

Chudnovsky:
$\frac1\pi = \frac{\sqrt{10005}}{4270934400} \sum_{k=0}^\infty (-)^k \frac{(6k)!}{(k!)^3 (3k)!} \frac{(13591409+545140134 k)}{640320^{3k}}$

Related at this blog: Calculating e – https://eikonal.wordpress.com/2010/08/06/calculating-e/

## 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:

### 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

### 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