By Edgar Asplund; Lutz Bungart
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Extra resources for A first course in integration
8 The factorial powers can be defined for values of x that are not integers. Lotting m = 1 and n = x - 1, in the relation X ' (m+m) = x^m\x - m) (n) , we get x^ — x(x— l)^" 1 ). Setting y ( x ) — x^x\ the previous expression can be written as y(x] = xy(x-l). The solution of this equation is y(x) — F(x + l), where F(x) is the Euler function, which is denned for all real values of x except the negative integers. This allows us to define x^ for x — 0. In fact, we have 0 (0) = T(l) = 1. 9 The function F(x) is also defined as roo F(x) = / Jo e-^-^dt.
2. FUNDAMENTAL THEORY 41 /I 0 ... 0 \ o -, -, ; : '-. 0 ... Q '-. , k, is said to be a canonical base. 1, we have the following result. 12) is a vector space of dimension k. If ai G IRfc, i = 1 , 2 , . . , fc, are linearly independent, the set of solutions y(n,no,0i) can also be used as a base of the space S. , a&) where a^ is the i-th column of the matrix K(HQ). Because they are linearly independent, it follows that detA'(n) ^ 0 for all no' The matrix K(ri) is called the Casorati matrix and in the theory of difference equations it plays the same role the Wronskian matrix does in the theory of linear differential equations.
11 Show that AlogF(x) = logx and A"1 logx = logF(x) +u;(x). 12 Let where F'(x) is the derivate of F(x). Show that A'0(x) = -. x The function //'(x) has many interesting properties. Among them we recall lim x _ o c (0(x) - log(x)) = 0. 13 Show that, if p(x) is a polynomial of degree k. it can be written in the form which is the Newton form. Copyright © 2002 Marcel Dekker, Inc. 7. /V^, one has -\ / ~ / i Note that S(q, 0, n) = (^1*) as established in the previous exercise. 16 Verify that, if yn and zn are two sequences, one has (a) (b) A— = —— —-, (c) JV-1 N-l XI l/nA2 n = 2/ n ^n|n=o ~ S n=() n=0 the last formula is called summation by parts.
A first course in integration by Edgar Asplund; Lutz Bungart