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 a x = b y ± 1 {\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: [ x ] {\displaystyle [x]} denotes the greatest integer less than or equal to x {\displaystyle x} . This notation was also invented by Gauss and was used in the third proof of the quadratic reciprocity law. The notation ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } , denoting the floor function, is now more commonly used to denote the greatest integer less than or equal to x {\displaystyle x} .[2]

The notation

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

[ ] = 1 [ a 1 ] = a 1 [ a 1 , a 2 ] = [ a 1 ] a 2 + [ ] = a 1 a 2 + 1 [ a 1 , a 2 , a 3 ] = [ a 1 , a 2 ] a 3 + [ a 1 ] = a 1 a 2 a 3 + a 1 + a 3 [ a 1 , a 2 , a 3 , a 4 ] = [ a 1 , a 2 , a 3 ] a 4 + [ a 1 , a 2 ] = a 1 a 2 a 3 a 4 + a 1 a 2 + a 1 a 4 + a 3 a 4 + 1 [ 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 ] = 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 ⋮ [ a 1 , a 2 , … , a n ] = [ a 1 , a 2 , … , a n − 1 ] a n + [ a 1 , a 2 , … , a n − 2 ] {\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 [ a 1 , a 2 , … , a n ] {\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]

1 a 1 + 1 a 2 + 1 a 3 + ⋯ ⋱ 1 a n − 1 + 1 a n = [ a 2 , … , a n ] [ a 1 , a 2 , … , a n ] {\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: [ a 1 , a 2 , a 3 , … , a n ] = a 1 [ a 2 , a 3 , … , a n ] + [ a 3 , … , a n ] {\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: [ a 1 , a 2 , … , a n − 1 , a n ] = [ a n , a n − 1 , … , a 2 , a 1 ] {\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: [ a 1 , a 2 , … , a n ] = | a 1 − 1 0 0 ⋯ 0 0 0 1 a 2 − 1 0 ⋯ 0 0 0 0 1 a 3 − 1 ⋯ 0 0 0 ⋮ 0 0 0 0 ⋯ 1 a n − 1 − 1 0 0 0 0 ⋯ 0 1 a n | {\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 n = 1 {\displaystyle n=1} use the convention that [ a 2 , … , a 0 ] = 0 {\displaystyle [a_{2},\ldots ,a_{0}]=0} ): | [ a 1 , … , a n ] [ a 1 , … , a n − 1 ] [ a 2 , … , a n ] [ a 2 , … , a n − 1 ] | = ( − 1 ) n {\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. [ − a 1 , − a 2 , … , − a n ] = ( − 1 ) n [ a 1 , a 2 , … , a n ] {\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
[ a 1 , 0 , a 3 , 0 , … , a 2 m + 1 ] = a 1 + a 3 + ⋯ + a 2 m + 1 [ a 1 , 0 , a 3 , 0 , … , a 2 m + 1 , 0 ] = 1 [ 0 , a 2 , 0 , a 4 , … , a 2 m ] = 1 [ 0 , a 2 , 0 , a 4 , … , a 2 m , 0 ] = 0 {\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. ^ a b Weisstein, Eric W. "Gaussian Brackets". MathWorld - A Wolfram Web Resource. Retrieved 24 January 2023.
  4. ^ a b c 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). "Paraxial theory in optical design in terms of Gaussian brackets". Progress in Optics. XXIII: 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.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  • 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). Wang, Yongtian; Du, Chunlei; Sasián, José; Tatsuno, Kimio (eds.). "Lens design based on lens form parameters using Gaussian brackets". Proc. SPIE 9272, Optical Design and Testing VI, 92721L. Optical Design and Testing VI. 9272: 92721L. Bibcode:2014SPIE.9272E..1LY. doi:10.1117/12.2073422. S2CID 121201008.
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