By Philip J. Davis
Final paragraph to First version Preface This booklet provides what we predict are the foremost tools of numerical integration. we've got attempted to supply a balanced paintings that's either worthy to the programmer and stimulating to the theoretician. There are parts of the ebook the place deep result of research are derived or are alluded to;yet, it's been our wish that the majority of the ultimate effects were expressed in a fashion that's available to these with a historical past simply in calculus
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Extra resources for Methods of Numerical Integration
Example text
L)/2 a n d — π < arg(z Τ 1) < π. (iii) α// ζ except for the neighborhood of ζ = 1: Defining ζ = e'* cosh 2ζ, then for β > 0, where fc = n + (a + /? + l)/2 a n d 0 < arg(z Τ 1) < 2π. Γ(α + 1) 12 e" , χ > 0, - 1 < α < 0 ZEROS: X nm 2 48fcn 4kn\ where ic„ = η + (a + l)/2 and function J,(x). ASYMPTOTIC EXPRESSION: (i) (η -* J + K ' is the mth positive zero of the Bessel oo) ζ bounded, α > 0 : where (c = η + (α + l)/2 and 0 < arg ζ < 2π. 13 41 S H O R T G U I D E T O T H E O R T H O G O N A L POLYNOMIALS VII NAME: WEIGHT: Hermite e~ x2 Hn(x) SYMBOL: INTERVAL: STANDARDIZATION: = 2V Hn(x) (-00,00) + ··· NORM: .
F o r the function f(x) = sin πχ, f(0)=f(\) so that the results are identical with those given in the next example to which the reader is referred. Some further remarks, which apply to the numerical examples in this book, should be made here. In the first place, in order to be in a position to compare the computed value with an exact value, we must take integrals that can be evaluated in an alternative way, for example, through indefinite integrals, transformations, infinite series, tables of special functions, and so on.
K;j = k — i. 6), Hâvie gives an algorithm for evaluating the D^. However, in two particular and important cases, αι explicit expressions for the are known. If φι(ή) = η~ for some fixed a, a then Du = ( M i +y n f ) , while if φ^η) = where 0 < y1 < y2 < * * * < ym and Vi nk = n0 p \ ρ > 1, then D 0- = p . ·Γ*\ /c = 0 , l , . . , m . 13) i= 0 If we assume that the fc(y) and the rk are known, then we can approximate A and the ß k i by solving the system of linear equations m A Myô = nj - η* Σ ΦΑ*) Σ Ακ-νΓ"*.