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

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] denotes the greatest integer less than or equal to x . This notation was also invented by Gauss and was used in the third proof of the quadratic reciprocity law. The notation \lfloor x \rfloor , denoting the floor function, is now more commonly used to denote the greatest integer less than or equal to x .^[2]

The notation

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

\begin{align} [] & = 1 \\[ 1 mm] [a _{1]} & = a _{1\\[1mm] [a} _{1,} a _{2]} & = [a _{1]a} _{2} + []\\[ 1 mm] & = a _{1a} _{2+1\\[1mm]} [a _{1,} a _{2,} a _{3]} & = [a _{1,} a _{2]a} _{3} + [a _{1]} \\[ 1 mm] & = a _{1a} _{2a} _{3} + a _{1} + a _{3} \\[ 1 mm] [a _{1,a} _{2,a} _{3,a} _{4]} & = [a _{1,a} _{2,a} _{3]a} _{4} + [a _{1,a} _{2]\\[1mm]} & = a _{1a} _{2a} _{3a} _{4} + a _{1a} _{2} + a _{1a} _{4} + a _{3a} _{4} + 1 \\[ 1 mm] [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 _{1a} _{2a} _{3a} _{4a} _{5} + a _{1a} _{2a} _{3} + a _{1a} _{2a} _{5} + a _{1a} _{4a} _{5} + a _{3a} _{4a} _{5} + a _{1+a} _{3+a} _{5\\[1mm] \vdots} & \\[ 1 mm] [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{align}

The expanded form of the expression

[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]

\cfrac{ 1 }{a _{1} + \cfrac{ 1 }{ a _{2} + \cfrac{ 1 }{a _{3} + … (\ddots)/(\cfrac{ 1) {a _{n-1} + (1)/(a _{n}) } } }}} = \frac

Properties

1. The bracket notation can also be defined by the recursion relation:

[a _{1,a} _{2,} a _{3,} \ldots, a _{n]=a} _{1[a} _{2,a} _{3,} \ldots,a _{n]} + [a _{3,\ldots,a} _{n]}

1. The notation is symmetric or reversible in the arguments:

[a _{1,a} _{2,} \ldots,a _{n-1} ,a _{n]=[a} _{n,a} _{n-1} ,\ldots, a _{2,a} _{1]}

1. The Gaussian brackets expression can be written by means of a determinant:

[a _{1,a} _{2,\ldots,a} _{n]} = \begin{vmatrix} a _{1} & -1 & 0 & 0 & … & 0 & 0 & 0 \\[ 1 mm] 1 & a _{2} & -1 & 0 & … & 0 & 0 & 0 \\[ 1 mm] 0 & 1 & a _{3} & -1 & … & 0 & 0 & 0 \\[ 1 mm] \vdots & & & & & & & \\[ 1 mm] 0 & 0 & 0 & 0 & … & 1 & a _{n-1} & -1 \\[ 1 mm] 0 & 0 & 0 & 0 & … & 0 & 1 & a _{n \end{vmatrix}}

1. The notation satisfies the determinant formula (for

n =1 use the convention that [a _{2,\ldots,a} _{0]=0} ): \begin{vmatrix} [a _{1,\ldots,a} _{n]} & [a _{1,\ldots,a} _{n-1} ]\\[ 1 mm] [a _{2,} \ldots, a _{n} ] & [a _{2,\ldots,} a _{n-1} ]\end{vmatrix} = ( -1 ) ^{n} [ - a _{1,} - a _{2,} \ldots, - a _{n]} = ( -1 ) ^{n[a} _{1 ,a} _{2,} \ldots,a _{n]}

1. Let the elements in the Gaussian bracket expression be alternatively 0. Then

\begin{align} [a _{1,0,a} _{3,0,\ldots,a} _{2m+1} ] & = a _{1+a} _{3+ …} + a _{2m+1} \\[ 1 mm] [a _{1,0,a} _{3,0,\ldots,a} _{2m+1} , 0 ] & = 1 \\[ 1 mm] [ 0 , a _{2,} 0 , a _{4,} \ldots, a _{2m} ] & = 1 \\[ 1 mm] [ 0 , a _{2,} 0 , a _{4,} \ldots, a _{2m} , 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

1. 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.
2. Web site: Weisstein, Eric W. . Floor Function . MathWorld--A Wolfram Web Resource. . 25 January 2023.
3. Web site: Weisstein, Eric W. . Gaussian Brackets . MathWorld - A Wolfram Web Resource . 24 January 2023.
4. M. Herzberger . Gaussian Optics and Gaussian Brackets . Journal of the Optical Society of America . December 1943 . 33 . 12 . 10.1364/JOSA.33.000651.
5. 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 .