1.3. Heat equation 25
Also,
(1.18)
N−1
x=1
sin2
πjx
N
=
N
2
.

The Nth roots of unity, ζ1, . . . , ζN are the N complex numbers ζ such
that ζN = 1. They are given by
ζk = cos
2kπ
N
+ i sin
2kπ
N
, j = 1, . . . , N.
The roots of unity are spread evenly about the unit circle in C; in particular,
ω1 + ω2 + · · · + ωN = 0,
which implies that
N
j=1
cos
2kπ
N
=
N
j=1
sin
2kπ
N
= 0.
The double angle formula for sine gives
N−1
j=1
sin2
jxπ
N
=
N
j=1
sin2
jxπ
N
=
1
2
N−1
j=0
1 cos
2jxπ
N
=
N
2

1
2
N
j=1
cos
2jxπ
N
.
If x is an integer, the last sum is zero. This gives (1.18).
In particular, if we choose the solution with initial condition
f(x) = 1; f(z) = 0,z = x we can see that
P{Sn∧TA = y | S0 = x} =
2
N
N−1
j=1
φj(x) cos

N
n
φj(y).
It is interesting to see what happens as n ∞. For large n, the
sum is very small but it is dominated by the j = 1 and j = N 1
Previous Page Next Page