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11: About MathML
, built-in to the browser) support for MathML is growing, (see Browsers supporting MathML). …
Browsers supporting MathML
The Firefox browser has traditionally had the strongest support for MathML and its native MathML is used by default. Recent enhancements to the WebKit engine now provide support for MathML Core. … Most modern browsers support ‘Web Fonts’, fonts that are effectively included with a web site. …
12: 24.13 Integrals
For m , n = 1 , 2 , ,
24.13.6 0 1 B n ( t ) B m ( t ) d t = ( 1 ) n 1 m ! n ! ( m + n ) ! B m + n .
24.13.8 0 1 E n ( t ) d t = 2 E n + 1 ( 0 ) n + 1 = 4 ( 2 n + 2 1 ) ( n + 1 ) ( n + 2 ) B n + 2 ,
For m , n = 1 , 2 , ,
24.13.11 0 1 E n ( t ) E m ( t ) d t = ( 1 ) n 4 ( 2 m + n + 2 1 ) m ! n ! ( m + n + 2 ) ! B m + n + 2 .
13: 26.14 Permutations: Order Notation
The set 𝔖 n 26.13) can be viewed as the collection of all ordered lists of elements of { 1 , 2 , , n } : { σ ( 1 ) σ ( 2 ) σ ( n ) } . …Equivalently, this is the sum over 1 j < n of the number of integers less than σ ( j ) that lie in positions to the right of the j th position: inv ( 35247816 ) = 2 + 3 + 1 + 1 + 2 + 2 + 0 = 11 . It is also equal to the number of permutations in 𝔖 n with exactly k + 1 weak excedances. …
26.14.5 k = 0 n 1 n k ( x + k n ) = x n .
26.14.8 n k = ( k + 1 ) n 1 k + ( n k ) n 1 k 1 , n 2 ,
14: 26.7 Set Partitions: Bell Numbers
B ( n ) is the number of partitions of { 1 , 2 , , n } . …
26.7.1 B ( 0 ) = 1 ,
26.7.4 B ( n ) = e 1 k = 1 k n k ! = 1 + e 1 k = 1 2 n k n k ! .
26.7.6 B ( n + 1 ) = k = 0 n ( n k ) B ( k ) .
26.7.7 B ( n ) = N n e N n 1 ( 1 + ln N ) 1 / 2 ( 1 + O ( ( ln n ) 1 / 2 n 1 / 2 ) ) , n ,
15: 27.14 Unrestricted Partitions
For example, p ( 5 ) = 7 because there are exactly seven partitions of 5 : 5 = 4 + 1 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 . … with p ( 0 ) = 1 . …where the exponents 1 , 2 , 5 , 7 , 12 , 15 , are the pentagonal numbers, defined by … Logarithmic differentiation of the generating function 1 / f ( x ) leads to another recursion: …where σ 1 ( k ) is defined by (27.2.10) with α = 1 . …
16: 24.14 Sums
24.14.4 k = 0 n ( n k ) E k E n k = 2 n + 1 E n + 1 ( 0 ) = 2 n + 2 ( 1 2 n + 2 ) B n + 2 n + 2 .
24.14.7 j = 0 m k = 0 n ( m j ) ( n k ) B j B k m + n j k + 1 = ( 1 ) m 1 m ! n ! ( m + n ) ! B m + n .
Let det [ a r + s ] denote a Hankel (or persymmetric) determinant, that is, an ( n + 1 ) × ( n + 1 ) determinant with element a r + s in row r and column s for r , s = 0 , 1 , , n . …
24.14.11 det [ B r + s ] = ( 1 ) n ( n + 1 ) / 2 ( k = 1 n k ! ) 6 / ( k = 1 2 n + 1 k ! ) ,
24.14.12 det [ E r + s ] = ( 1 ) n ( n + 1 ) / 2 ( k = 1 n k ! ) 2 .
17: 27.18 Methods of Computation: Primes
Two simple algorithms for proving primality require a knowledge of all or part of the factorization of n 1 , n + 1 , or both; see Crandall and Pomerance (2005, §§4.1–4.2). These algorithms are used for testing primality of Mersenne numbers, 2 n 1 , and Fermat numbers, 2 2 n + 1 . …
18: 3.11 Approximation Techniques
If f is continuously differentiable on [ 1 , 1 ] , then with … Moreover, the set of minimax approximations p 0 ( x ) , p 1 ( x ) , p 2 ( x ) , , p n ( x ) requires the calculation and storage of 1 2 ( n + 1 ) ( n + 2 ) coefficients, whereas the corresponding set of Chebyshev-series approximations requires only n + 1 coefficients. … With b 0 = 1 , the last q equations give b 1 , , b q as the solution of a system of linear equations. … Here x j , j = 1 , 2 , , J , is a given set of distinct real points and J n + 1 . … (3.11.29) is a system of n + 1 linear equations for the coefficients a 0 , a 1 , , a n . …
19: 24.11 Asymptotic Approximations
24.11.1 ( 1 ) n + 1 B 2 n 2 ( 2 n ) ! ( 2 π ) 2 n ,
24.11.2 ( 1 ) n + 1 B 2 n 4 π n ( n π e ) 2 n ,
24.11.3 ( 1 ) n E 2 n 2 2 n + 2 ( 2 n ) ! π 2 n + 1 ,
24.11.5 ( 1 ) n / 2 1 ( 2 π ) n 2 ( n ! ) B n ( x ) { cos ( 2 π x ) , n  even , sin ( 2 π x ) , n  odd ,
24.11.6 ( 1 ) ( n + 1 ) / 2 π n + 1 4 ( n ! ) E n ( x ) { sin ( π x ) , n  even , cos ( π x ) , n  odd ,
20: Foreword
In 1964 the National Institute of Standards and Technology11 1 Then known as the National Bureau of Standards. published the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. … Particular attention is called to the generous support of the National Science Foundation, which made possible the participation of experts from academia and research institutes worldwide. …