By Alexander Craig Aitken
This publication starts with a short account of matrices and matrix algebra, and derives the speculation of determinants by means of assistance from matrix notation, in an order urged via a clearly alternating improvement of either matters.
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Then we have what is known as an impulsive force, and for m(t) = 1 (unit force) we have t0 f (t) dt = 1. 1), and one such function is f (t) = F (t) = 1 . Using this function, let 2t0 0 if t0 < |t|, 1 2t0 if t0 ≥ |t|. 2) This function F (t) is an ordinary function which belongs to the class C0∞ (I), where I = [−t0 ≤ t ≤ t0 ]. However, F (t) → δ(t) as t0 → 0; that is, in the limit as t0 → 0 the function F (t) converges to the Dirac delta-function (henceforth written δ-function)1 which is not a function in the ordinary sense; it is called a generalized function.
3! 4! + + 3 + 4 + · · · ; (b) + 2 + 3 + 4 · · · . 3 32 3 3 3 3 3 3 22 + 1 32 + 1 42 + 1 (c) 1 + 3 + + + ···. 2 + 1 33 + 1 43 + 1 n n+1 n! Hint. (a) sn = n , sn+1 = n+1 ; Ans. Converges; (b) sn = n , sn+1 = 3 3 3 (n + 1)! n2 + 1 (n + 1)2 + 1 . Ans. Diverges; (c) s = , s = ; Ans. Test n n+1 3n+1 n3 + 1 (n + 1)3 + 1 fails. (a) 2 The Concept of Green’s Functions Green’s function for a differential equation is its solution when the forcing term is the Dirac delta function due to a unit point source (or sink) in a given domain.
1), but their areas remain constant, since fε (t) dt = 1 for any ε > 0. This means that the δ-function can be defined as the limit δ(t) = lim fε (t). Obviously, this limit is zero everywhere ε→0 except at the point t = 0. 8) subject to the additional normalization condition that the area under this function is equal to 1. fε(t ) 0 ε t Fig. 1. Delta Function as a Limit of Rectangular Functions. Properties of the δ-function are: (i) δ(x) = 0 for x = 0, (ii) (iii) ∞ δ(x) dx = 1 −∞ ∞ f (x)δ(x) dx = f (0).