FUNDAMENTAL SOLUTIONS 93
For our infinite region our orthonormalisation condition becomes
φ(κ, χ)φ(κ x)dx = (φ(κ, x), φ(κ x)
>^
= δ(κ-κ')
J all
space
(
425)
where the delta function is now the classical Dirac delta function
defined for all space, which is zero when κ Φ κ but infinite at κ = κ .
To define δ uniquely we require that
<φ(κ, 30, δ(κ-κ') >^= φ(κ χ) (4.26)
which is the replacement property of the delta function with respect to
our eigenfunctions. One consequence of (4.26) is that
<U(K-K')>^=1
(4.27)
This function is now expressed in terms of our eigenfunctions by,
δ(χ-ξ) = < Φ(κ, x), φ(κ
9
ξ) >. (4.28)
The brackets < >-now denote an integration over all /c space, i.e.
<0(κ,
χφ(κ,
ξ) >^= f φ(κ, ϊ)φ(κ, ξ)άκ (4.29)
J all
space
The expansion of any bounded function will now be given by
u(x) = <α(£), φ(κ
9
χ)}χ (4.30)
where by an argument similar to equations (4.13) to (4.16) we have
a(ie) = <φ(κ,χ), u(x)>
?
(4.31)
Equations (4.30) and (4.31) are a generalised transform pair.
If we now consider the expression corresponding to (4.17) for the
infinite domain
<«(*),
δ{χ-ξ)>; (4.32)
and expand both in terms of eigenfunctions we obtain
< < α(κ), φ(κ, x) >-, (φ(κ x), φ(κ ξ) >^ >^ (4.33)
assuming convergence of
the
integrals we may rearrange the above to
give