15.6 Fourier transforms 437
We define the function
(15.78)
It is shown in Example 15.9 that equation (15.69) can then be written as
(15.79)
with equations (15.70) and (15.71) replaced by
(15.80)
EXAMPLE 15.9Derivation of the exponential form
Substitution of the Euler relations (15.77) into (15.69) gives
or, replacingyby−yin the second integral on the right,
From equations (15.70) and (15.71), we have
u(−y) 1 = 1 u(y), v(−y) 1 = 1 −v(y)
Therefore,
where
wv()yuyiy() () fxedy()
ixy=−
=
−+−1
2
1
2 π
Z
∞∞=
−+Z
∞∞w()ye dy
ixyfx uy i y e dy
ixy()=−() ()
−1
2
Z
∞∞v
fx uy i y e dy u y
ixy()=−() () (
+−
−1
2
1
2
0ZZ
∞∞0v ))()+−
iyedy
ixyv
=−
++
1
2
1
2
00ZZ
∞∞uy i y e dy uy i y
ixy() ()vv() ())
−edy
ixyfx uy e e i y e e
ixy ixy ixy ixy()=+−−()( ) ()(
−−1
2
0Z
∞v ))
dy
w()yfxed() x
ixy=
−+−1
2 π
Z
∞∞fx ye dy
ixy()= ()
−+Z
∞∞w
wv()yuyiy=−() ()
1
2