Gaussian brackets

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 ${\displaystyle ax=by\pm 1}$.[1]

This notation should not be confused with the widely prevalent use of square brackets to denote the greatest integer function: ${\displaystyle [x]}$ denotes the greatest integer less than or equal to ${\displaystyle x}$. This notation was also invented by Gauss and was used in the third proof of the quadratic reciprocity law. The notation ${\displaystyle \lfloor x\rfloor }$, denoting the floor function, is now more commonly used to denote the greatest integer less than or equal to ${\displaystyle x}$.[2]

The notation

The Gaussian brackets notation is defined as follows:[3][4]

${\displaystyle {\begin{aligned}\quad [\,\,]&=1\\[1mm][a_{1}]&=a_{1}\\[1mm][a_{1},a_{2}]&=[a_{1}]a_{2}+[\,\,]\\[1mm]&=a_{1}a_{2}+1\\[1mm][a_{1},a_{2},a_{3}]&=[a_{1},a_{2}]a_{3}+[a_{1}]\\[1mm]&=a_{1}a_{2}a_{3}+a_{1}+a_{3}\\[1mm][a_{1},a_{2},a_{3},a_{4}]&=[a_{1},a_{2},a_{3}]a_{4}+[a_{1},a_{2}]\\[1mm]&=a_{1}a_{2}a_{3}a_{4}+a_{1}a_{2}+a_{1}a_{4}+a_{3}a_{4}+1\\[1mm][a_{1},a_{2},a_{3},a_{4},a_{5}]&=[a_{1},a_{2},a_{3},a_{4}]a_{5}+[a_{1},a_{2},a_{3}]\\[1mm]&=a_{1}a_{2}a_{3}a_{4}a_{5}+a_{1}a_{2}a_{3}+a_{1}a_{2}a_{5}+a_{1}a_{4}a_{5}+a_{3}a_{4}a_{5}+a_{1}+a_{3}+a_{5}\\[1mm]\vdots &\\[1mm][a_{1},a_{2},\ldots ,a_{n}]&=[a_{1},a_{2},\ldots ,a_{n-1}]a_{n}+[a_{1},a_{2},\ldots ,a_{n-2}]\end{aligned}}}$

The expanded form of the expression ${\displaystyle [a_{1},a_{2},\ldots ,a_{n}]}$ 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]

${\displaystyle {\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+\cdots {\frac {\ddots }{\cfrac {1}{a_{n-1}+{\frac {1}{a_{n}}}}}}}}}}}}={\frac {[a_{2},\ldots ,a_{n}]}{[a_{1},a_{2},\ldots ,a_{n}]}}}$

Properties

1. The bracket notation can also be defined by the recursion relation: ${\displaystyle \,\,[a_{1},a_{2},a_{3},\ldots ,a_{n}]=a_{1}[a_{2},a_{3},\ldots ,a_{n}]+[a_{3},\ldots ,a_{n}]}$
2. The notation is symmetric or reversible in the arguments: ${\displaystyle \,\,[a_{1},a_{2},\ldots ,a_{n-1},a_{n}]=[a_{n},a_{n-1},\ldots ,a_{2},a_{1}]}$
3. The Gaussian brackets expression can be written by means of a determinant: ${\displaystyle \,\,[a_{1},a_{2},\ldots ,a_{n}]={\begin{vmatrix}a_{1}&-1&0&0&\cdots &0&0&0\\[1mm]1&a_{2}&-1&0&\cdots &0&0&0\\[1mm]0&1&a_{3}&-1&\cdots &0&0&0\\[1mm]\vdots &&&&&&&\\[1mm]0&0&0&0&\cdots &1&a_{n-1}&-1\\[1mm]0&0&0&0&\cdots &0&1&a_{n}\end{vmatrix}}}$
4. The notation satisfies the determinant formula (for ${\displaystyle n=1}$ use the convention that ${\displaystyle [a_{2},\ldots ,a_{0}]=0}$): ${\displaystyle \,\,{\begin{vmatrix}[a_{1},\ldots ,a_{n}]&[a_{1},\ldots ,a_{n-1}]\\[1mm][a_{2},\ldots ,a_{n}]&[a_{2},\ldots ,a_{n-1}]\end{vmatrix}}=(-1)^{n}}$
5. ${\displaystyle [-a_{1},-a_{2},\ldots ,-a_{n}]=(-1)^{n}[a_{1},a_{2},\ldots ,a_{n}]}$
6. Let the elements in the Gaussian bracket expression be alternatively 0. Then

${\displaystyle {\begin{aligned}\,\,\quad [a_{1},0,a_{3},0,\ldots ,a_{2m+1}]&=a_{1}+a_{3}+\cdots +a_{2m+1}\\[1mm][a_{1},0,a_{3},0,\ldots ,a_{2m+1},0]&=1\\[1mm][0,a_{2},0,a_{4},\ldots ,a_{2m}]&=1\\[1mm][0,a_{2},0,a_{4},\ldots ,a_{2m},0]&=0\end{aligned}}}$

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]

References

1. Carl Friedrich Gauss (English translation by Arthur A. Clarke and revised by William C. Waterhouse) (1986). Disquisitiones Arithmeticae. New York: Springer-Verlag. pp. 10–11. ISBN 0-387-96254-9.
2. Weisstein, Eric W. "Floor Function". MathWorld--A Wolfram Web Resource. Retrieved 25 January 2023.
3. Weisstein, Eric W. "Gaussian Brackets". MathWorld - A Wolfram Web Resource. Retrieved 24 January 2023.
4. M. Herzberger (December 1943). "Gaussian Optics and Gaussian Brackets". Journal of the Optical Society of America. 33 (12). doi:10.1364/JOSA.33.000651.
5. Kazuo Tanaka (1986). II Paraxial Theory in Optical Design in Terms of Gaussian Brackets. Progress in Optics. Vol. XXIII. pp. 63–111. Bibcode:1986PrOpt..23...63T. doi:10.1016/S0079-6638(08)70031-3. ISBN 9780444869821.

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 (November 2014). "Optical System Design of a Liquid Tunable Fundus Camera Based on Gaussian Brackets Method". Acta Optica Sinica. 34 (11). doi:10.3788/AOS201434.1122001.
• Yi Zhong, Herbert Gross (May 2017). "Initial system design method for non-rotationally symmetric systems based on Gaussian brackets and Nodal aberration theory". Opt Express. 25 (9): 10016–10030. Bibcode:2017OExpr..2510016Z. doi:10.1364/OE.25.010016. PMID 28468369. Retrieved 24 January 2023.
• Xiangyu Yuan and Xuemin Cheng (November 2014). "Lens design based on lens form parameters using Gaussian brackets". In Wang, Yongtian; Du, Chunlei; Sasián, José; Tatsuno, Kimio (eds.). Optical Design and Testing VI. Vol. 9272. pp. 92721L. Bibcode:2014SPIE.9272E..1LY. doi:10.1117/12.2073422. S2CID 121201008.