Eikonal Blog

2010.07.13

TeX and LaTeX links

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

Tools


(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

Mathematics links

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

Blogs

Tools

Primary tool is, off course, your brain

Numerical calculations

Analytic/Symbolic calculations

Information:

Graphics/Plotting

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}

and following two “sporadic” occurrences:

  • 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}}


Links and sources:


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

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