Here the recurrence formula is
(m+ 1)h cm+1 +m cm=n cm or cm+1 = (n−m)
(m+ 1)hcm (12.171)for m= 0 , 1 , 2 ,.. .. If y=y(x) = (h+x)n, then y(0) = c 0 =hn.Substitute the values
m= 0, 1 , 2 , 3 ,.. ., n, n + 1,... into the recurrence formula (12.171) to obtain
m=0m=1m=2..
.
..
.
m=n− 1
m=n
m=n+ 1c 1 =n
hc 0 =nh n−^1 =(
n
1)
hn−^1c 2 =(n−1)
2 h c^1 =n(n−1)
2! hn− (^2) =
(
n
2
)
hn−^2
c 3 =(n−2)
3 h
c 2 =n(n−1)(n−2)
3!
hn−^3 =
(
n
3
)
hn−^3
..
.
..
.
cn=nh^1 cn− 1 =nn!!h^0 = 1 =(
n
n)
h^0cn+1 =0
cn+2 =0(12.172)and cn= 0 for all integer values of msatisfying m≥n. In the equations (12.172) the
terms (
n
m)
={ n!
m!(n−m)!, m ≤n
0 , m > n(12.173)are the binomial coefficients. Substituting the values given by equations (12.172)
into the power series (12.166) one obtains the finite series of terms
y= (h+x)n=(
n
0)
hn+(
n
1)
hn−^1 x+(
n
2)
hn−^2 x^2 +···+(
n
n− 1)
hx n−^1 +(
n
n)
xn (12.175)which is the well-known binomial expansion.
The Laplace Transform
Consider the mathematical operator box labeled L{ f(t)} or L{ ;t→s} as il-
lustrated in the figure 12-8. This mathematical operator L is called the Laplace
transform operator and is defined
L{ f(t)}=∫∞0f(t)e−st dt =F(s)or L{ f(t); t→s}= limT→∞
∫T0f(t)e−st dt =F(s), s > 0(12.175)and represents a transformation from a function f(t)in the t-domain to a function
F(s)in the s-domain (frequently called the frequency domain). The Laplace trans-
form^7 has many applications in mathematics, statistics, physics and engineering.
(^7) This operator is named after Pierre Simon Laplace (1749-1857) A famous French mathematician.