Mathematical Methods for Physics and Engineering - Matematica.NET

INTEGRAL TRANSFORMSequals zero. This leads immediately to two further useful results:and∫ b−aδ(t) dt = 1 for all a, b > 0 (13.13)∫δ(t − a) dt =1, (13.14)provided the range of integration includes t = a.Equation (13.12) can be used to derive further useful properties of the Diracδ-function:δ(t) =δ(−t), (13.15)δ(at) = 1 δ(t), (13.16)|a|tδ(t) =0. (13.17)◮Prove that δ(bt) =δ(t)/|b|.Let us first consider the case where b>0. It follows that∫ ∞∫ ∞( ) t′f(t)δ(bt) dt = f δ(t ′ ) dt′−∞−∞ b b = 1 b f(0) = 1 ∫ ∞f(t)δ(t) dt,b −∞where we have made the substitution t ′ = bt. Butf(t) is arbitrary and so we immediatelysee that δ(bt) =δ(t)/b = δ(t)/|b| for b>0.Now consider the case where b = −c