Английская Википедия:Gaussian brackets

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

The notation

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

<math>\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>[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> \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>\,\,[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>\,\,[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>\,\,[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>

  1. The notation satisfies the determinant formula (for <math>n=1</math> use the convention that <math>[a_2,\ldots,a_0]=0</math>): <math>\,\, \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>
  2. <math>[-a_1, -a_2, \ldots, -a_n] = (-1)^n[a_1,a_2, \ldots,a_n]</math>
  3. Let the elements in the Gaussian bracket expression be alternatively 0. Then
<math>

\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

Шаблон:Reflist

Additional reading

The following papers give additional details regarding the applications of Gaussian brackets in optics.

Шаблон:Carl Friedrich Gauss