Advanced High-School Mathematics

262 CHAPTER 5 Series and Differential Equations

provided the improper integral exists. Assuming that f ∗g(x)

exists for allx, show that “∗” is commutative, i.e., that

f∗g(x) =g∗f(x), for allx.

We shall meet the convolution again in our study of statistics

(page 370).

6. Leta >0 be a constant, and set

f(x) =







e−at ifx≥ 0

0 ifx < 0.

Show that ifg(x) is defined for allx∈R, then

f∗g(x) = e−ax

∫x

−∞g(t)e

atdx

provided the improper integral exists.

Now computef∗g(x),wheregis as above and where

(a) f(x) = sinbx, whereb >0 is a constant.

(b) f(x) =x^2.

(c) f(x) =







1 ifx≥ 0

0 ifx < 0.

(d) f(x) =







sinbx ifx≥ 0

0 ifx < 0

whereb >0 is a constant.

7. (Convolution and the Low-Pass Filter) In electrical engineering
   one frequently has occasion to study theRClow-pass filter, whose
   schematic diagram is shown below. This is a “series” circuit with
   a resistor having resistance R Ω (“ohms”) and a capacitor with
   capacitance C F (“Farads”). An input voltage of x(t) volts is
   applied at the input terminals and the voltagey(t) volts is observed
   at the output. The variabletrepresents time, measured in seconds.