Continuously Distributed Time Lags - Nonlinear Oligopolies: Stability and Bifurcations - page 302

Notice that the integral converges to .m C 1/ D mŠ, so the right hand side
converges to 1.
(b) For m D 0, weights are exponentially declining with the most weight given to
the most current data. For m 1, zero weight is given to the most current data,
rising to maximum at s D t T and declining exponentially thereafter.
For m D 0, the weighting function is a declining exponential function of .t s/.
For m 1,

m
T

mC1

d
ds w .t s;T;m/ D

1
mŠ

C .t s/ m m
T

e m.t s/ m.t s/ m1

T

e m.t s T .t s/ m1 t s

T 1

m
T

mC1

1
.m 1/Š

D

which is negative for t s<T, positive for t s>T, and zero if t s D T .
(c) As m increases, the weighting function becomes more peaked around t s D T ,
and as m !1 , the weighting function converges to the Dirac delta function
centered at s D t T .
This property can be easily proved by examining the ratio

w .t s;T;m C 1/= w .t s;T;m/

with fixed T and t s.
(d) As T ! 0, the weighting function tends to the Dirac delta function with all
m 0.
Notice that the weighting function is the product of a polynomial and a decreas-
ing exponential function of

1
T

unless t s D T .

Figures D.1 and D.2 show the plots of the weighting function with changing
values of m and T .
In analyzing continuous time systems with continuously distributed time lags,
integrals of the form

e t Z t

1

w .t s;T;m/e s ds

(D.3)

0

often arise. Notice first that by introducing the new variable x D t s, (D.3) can be
simplified as

e t Z t

Z t

1

w .x;T;m/e .t x/ dx D w .x;T;m/e x dx:

0 0

If m D 0,thenwehave

" e x.C T /
C

# t
0 D .1 C T/ 1 .1 e t.C T / /;

Z t

1
T e T e x dx D

1
T

1
T

0