Gaussian brackets explained
In mathematics, Gaussian brackets are a special notation invented by Carl Friedrich Gauss to represent the convergents of a simple continued fraction in the form of a simple fraction. Gauss used this notation in the context of finding solutions of the indeterminate equations of the form
.
[1] This notation should not be confused with the widely prevalent use of square brackets to denote the greatest integer function:
denotes the greatest integer less than or equal to
. This notation was also invented by Gauss and was used in the third proof of the quadratic reciprocity law. The notation
, denoting the
floor function, is now more commonly used to denote the greatest integer less than or equal to
.
[2] The notation
The Gaussian brackets notation is defined as follows:[3]
\begin{align}
[]&=1\\[1mm]
[a1]&=a1\\[1mm]
[a1,a2]&=[a1]a2+[]\\[1mm]
&=a1a2+1\\[1mm][a1,a2,a3]&=[a1,a2]a3+[a1]\\[1mm]
&=a1a2a3+a1+a3\\[1mm]
[a1,a2,a3,a4]&=[a1,a2,a3]a4+[a1,a2]\\[1mm]
&=a1a2a3a4+a1a2+a1a4+a3a4+1\\[1mm]
[a1,a2,a3,a4,a5]&=[a1,a2,a3,a4]a5+[a1,a2,a3]\\[1mm]
&=a1a2a3a4a5+a1a2a3+a1a2a5+a1a4a5+a3a4a5+a1+a3+a5\\[1mm]
\vdots&\\[1mm]
[a1,a2,\ldots,an]&=[a1,a2,\ldots,an-1]an+[a1,a2,\ldots,an-2]
\end{align}
The expanded form of the expression
can be described thus: "The first term is the product of all
n members; after it come all possible products of (
n -2) members in which the numbers have alternately odd and even indices in ascending order, each starting with an odd index; then all possible products of (
n-4) members likewise have successively higher alternating odd and even indices, each starting with an odd index; and so on. If the bracket has an odd number of members, it ends with the sum of all members of odd index; if it has an even number, it ends with unity."
[4] With this notation, one can easily verify that[3]
\cfrac{1}{a1+\cfrac{1}{a2+\cfrac{1}{a3+ …
{an-1+
} } }}} = \frac
Properties
- The bracket notation can also be defined by the recursion relation:
[a1,a2,a3,\ldots,an]=a1[a2,a3,\ldots,an]+[a3,\ldots,an]
- The notation is symmetric or reversible in the arguments:
[a1,a2,\ldots,an-1,an]=[an,an-1,\ldots,a2,a1]
- The Gaussian brackets expression can be written by means of a determinant:
[a1,a2,\ldots,an]=\begin{vmatrix}a1&-1&0&0& … &0&0&0\\[1mm]1&a2&-1&0& … &0&0&0\\[1mm]
0&1&a3&-1& … &0&0&0\\[1mm]
\vdots&&&&&&&\\[1mm]
0&0&0&0& … &1&an-1&-1\\[1mm]
0&0&0&0& … &0&1&an
\end{vmatrix}
- The notation satisfies the determinant formula (for
use the convention that
):
\begin{vmatrix}[a1,\ldots,an]&[a1,\ldots,an-1]\\[1mm][a2,\ldots,an]&[a2,\ldots,an-1]\end{vmatrix}=(-1)n
[-a1,-a2,\ldots,-an]=
2,\ldots,an]
- Let the elements in the Gaussian bracket expression be alternatively 0. Then
\begin{align}
[a1,0,a3,0,\ldots,a2m+1]&=a1+a3+ … +a2m+1\\[1mm]
[a1,0,a3,0,\ldots,a2m+1,0]&=1\\[1mm]
[0,a2,0,a4,\ldots,a2m]&=1\\[1mm]
[0,a2,0,a4,\ldots,a2m,0]&=0\end{align}
Applications
The Gaussian brackets have been used extensively by optical designers as a time-saving device in computing the effects of changes in surface power, thickness, and separation of focal length, magnification, and object and image distances.[4] [5]
Additional reading
The following papers give additional details regarding the applications of Gaussian brackets in optics.
- Chen Ma, Dewen Cheng, Q. Wang and Chen Xu. Optical System Design of a Liquid Tunable Fundus Camera Based on Gaussian Brackets Method . Acta Optica Sinica . November 2014 . 34 . 11 . 10.3788/AOS201434.1122001.
- Yi Zhong, Herbert Gross . Initial system design method for non-rotationally symmetric systems based on Gaussian brackets and Nodal aberration theory . Opt Express . May 2017 . 25 . 9 . 10016–10030 . 10.1364/OE.25.010016 . 28468369 . 2017OExpr..2510016Z . 24 January 2023. free .
- Xiangyu Yuan and Xuemin Cheng . Yongtian . Chunlei . José . Kimio . Wang . Du . Sasián . Tatsuno . Lens design based on lens form parameters using Gaussian brackets . Proc. SPIE 9272, Optical Design and Testing VI, 92721L . Optical Design and Testing VI . November 2014 . 9272 . 92721L . 10.1117/12.2073422. 2014SPIE.9272E..1LY . 121201008 .
Notes and References
- Book: Carl Friedrich Gauss (English translation by Arthur A. Clarke and revised by William C. Waterhouse) . Disquisitiones Arithmeticae . 1986 . Springer-Verlag . New York . 0-387-96254-9 . 10–11.
- Web site: Weisstein, Eric W. . Floor Function . MathWorld--A Wolfram Web Resource. . 25 January 2023.
- Web site: Weisstein, Eric W. . Gaussian Brackets . MathWorld - A Wolfram Web Resource . 24 January 2023.
- M. Herzberger . Gaussian Optics and Gaussian Brackets . Journal of the Optical Society of America . December 1943 . 33 . 12 . 10.1364/JOSA.33.000651.
- Book: Kazuo Tanaka . Paraxial theory in optical design in terms of Gaussian brackets . Progess in Optics . Progress in Optics . 1986 . XXIII . 63–111 . 10.1016/S0079-6638(08)70031-3. 1986PrOpt..23...63T . 9780444869821 .