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 .[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][4]
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]
Properties
- The bracket notation can also be defined by the recursion relation:
- The notation is symmetric or reversible in the arguments:
- The Gaussian brackets expression can be written by means of a determinant:
- The notation satisfies the determinant formula (for use the convention that ):
- Let the elements in the Gaussian bracket expression be alternatively 0. Then
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
- ^ 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.
- ^ Weisstein, Eric W. "Floor Function". MathWorld--A Wolfram Web Resource. Retrieved 25 January 2023.
- ^ a b Weisstein, Eric W. "Gaussian Brackets". MathWorld - A Wolfram Web Resource. Retrieved 24 January 2023.
- ^ 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.
- ^ 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.
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: 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). "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.