Differential Equations and The - Braun, Martin 7579 · Problem 2.12.Ex9

Introduction to Applied Nonlinear Dynamical Systems and Chaos — Braun, Martin 7579

Mathematics — Calculus · dirac-delta-generalized-function

Problem

Let

f(t)

be the function which is

\frac{1}{2}

for

t>t_0

0

for

t=t_0

, and

-\frac{1}{2}

for

t<t_0

. Let

K[\phi]

be the linear functional

K[\phi]=\int_{-\infty}^{\infty}\phi(t)f(t)\,dt.

Show that

K'[\phi]\equiv K[-\phi']=\phi(t_0)

. Thus,

\delta(t-t_0)

may be viewed as the derivative of

f(t)