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]
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.
ISBN0-387-96254-9.
^Weisstein, Eric W.
"Floor Function". MathWorld--A Wolfram Web Resource. Retrieved 25 January 2023.
^
abWeisstein, Eric W.
"Gaussian Brackets". MathWorld - A Wolfram Web Resource. Retrieved 24 January 2023.
^
abcM. Herzberger (December 1943). "Gaussian Optics and Gaussian Brackets". Journal of the Optical Society of America. 33 (12).
doi:
10.1364/JOSA.33.000651.
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)
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.
S2CID121201008.