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 [math]\displaystyle{ ax=by\pm 1 }[/math].^[1] This notation should not be confused with the widely prevalent use of square brackets to denote the greatest integer function: [math]\displaystyle{ [x] }[/math] denotes the greatest integer less than or equal to [math]\displaystyle{ x }[/math]. This notation was also invented by Gauss and was used in the third proof of the quadratic reciprocity law. The notation [math]\displaystyle{ \lfloor x \rfloor }[/math], denoting the floor function, is now more commonly used to denote the greatest integer less than or equal to [math]\displaystyle{ x }[/math].^[2]

The notation

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

[math]\displaystyle{ \begin{align} \quad[\,\,] & = 1\\[1mm] [a_1] & = a_1\\[1mm] [a_1, a_2] & = [a_1]a_2 + [\,\,]\\[1mm] & = a_1a_2+1\\[1mm] [a_1, a_2, a_3] & = [a_1, a_2]a_3 + [a_1] \\[1mm] & = a_1a_2a_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_1a_2a_3a_4 + a_1a_2 + a_1a_4 + a_3a_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_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 & \\[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{align} }[/math]

The expanded form of the expression [math]\displaystyle{ [a_1,a_2,\ldots, a_n] }[/math] 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]

[math]\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]} }[/math]

Properties

1. The bracket notation can also be defined by the recursion relation: [math]\displaystyle{ \,\,[a_1,a_2, a_3, \ldots, a_n]=a_1[a_2,a_3, \ldots,a_n] + [a_3,\ldots,a_n] }[/math]
2. The notation is symmetric or reversible in the arguments: [math]\displaystyle{ \,\,[a_1,a_2, \ldots,a_{n-1},a_n]=[a_n,a_{n-1},\ldots, a_2,a_1] }[/math]
3. The Gaussian brackets expression can be written by means of a determinant: [math]\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} }[/math]
4. The notation satisfies the determinant formula (for [math]\displaystyle{ n=1 }[/math] use the convention that [math]\displaystyle{ [a_2,\ldots,a_0]=0 }[/math]): [math]\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 }[/math]
5. [math]\displaystyle{ [-a_1, -a_2, \ldots, -a_n] = (-1)^n[a_1,a_2, \ldots,a_n] }[/math]
6. Let the elements in the Gaussian bracket expression be alternatively 0. Then

[math]\displaystyle{ \begin{align} \,\,\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{align} }[/math]

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

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. doi:10.1364/OE.25.010016. PMID 28468369. Bibcode: 2017OExpr..2510016Z. https://opg.optica.org/abstract.cfm?URI=oe-25-9-10016. Retrieved 24 January 2023. 
• Xiangyu Yuan and Xuemin Cheng (November 2014). "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. doi:10.1117/12.2073422. Bibcode: 2014SPIE.9272E..1LY. 

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