H.3 Generalized Fourier Transform
673
f (t) =
∞
j =−∞
1
2π
T /2
−T /2
exp
iω j (t
− t)
f (t
)dt
(2π/T ) .
(H.7)
where ω j ≡ 2πj/T . As the period T becomes large, the factor 2π/T = ω j +1 − ω j
is small, and becomes the incremental interval dω in the summation over the angular
frequencies ω. Thus, in the limit of an infinite period,
f (t) =
1
2π
∞
−∞
∞
−∞
exp
iω (t
− t)
f (t
)dt
dω .
(H.8)
Separating the operations in each integral, one can define the so-called ‘inverse
Fourier transform’ as
˜
f (ω) =
1
2π
∞
−∞
exp
iω t
f (t
)dt
,
(H.9)
which makes the Fourier transform of f (t) have the expression
f (t) =
1
2π
∞
−∞
exp (−iω t) ˜
f (ω)dω .
(H.10)
The Fourier transform of a Fourier transform is the original function (apart from a
flip of the sign of the argument). Conventionally, for physical applications, the sign
in exponent for the time-transform component wave, exp (−iωt) is taken negative,
while the corresponding the component for a spatial Fourier transform, exp (ikx),
is taken positive, with opposite signs in the inverse Fourier transform. This makes
a pure plane wave moving forward along the x-axis, in the form A exp (ikx − iωt),
the default Fourier component in an expansion over both space and time.
H.3 Generalized Fourier Transform
Expressing a given function as an infinite series of simpler functions is a useful
exercise when the given function is a solution to a linear differential equation. This
will be the case for sound waves obeying a linear wave equation, for light waves, and
for quantum wave functions. If the choice of simpler functions is based on having
them match the same boundary conditions as the given function, then the analysis
of the given function reduces to knowing the properties of the simpler functions.
The expansion of the given function f (x) into simpler functions φ(k, x)
appears as
f (x) =
g(k)φ(k, x)dk .
(H.11)
673
f (t) =
∞
j =−∞
1
2π
T /2
−T /2
exp
iω j (t
− t)
f (t
)dt
(2π/T ) .
(H.7)
where ω j ≡ 2πj/T . As the period T becomes large, the factor 2π/T = ω j +1 − ω j
is small, and becomes the incremental interval dω in the summation over the angular
frequencies ω. Thus, in the limit of an infinite period,
f (t) =
1
2π
∞
−∞
∞
−∞
exp
iω (t
− t)
f (t
)dt
dω .
(H.8)
Separating the operations in each integral, one can define the so-called ‘inverse
Fourier transform’ as
˜
f (ω) =
1
2π
∞
−∞
exp
iω t
f (t
)dt
,
(H.9)
which makes the Fourier transform of f (t) have the expression
f (t) =
1
2π
∞
−∞
exp (−iω t) ˜
f (ω)dω .
(H.10)
The Fourier transform of a Fourier transform is the original function (apart from a
flip of the sign of the argument). Conventionally, for physical applications, the sign
in exponent for the time-transform component wave, exp (−iωt) is taken negative,
while the corresponding the component for a spatial Fourier transform, exp (ikx),
is taken positive, with opposite signs in the inverse Fourier transform. This makes
a pure plane wave moving forward along the x-axis, in the form A exp (ikx − iωt),
the default Fourier component in an expansion over both space and time.
H.3 Generalized Fourier Transform
Expressing a given function as an infinite series of simpler functions is a useful
exercise when the given function is a solution to a linear differential equation. This
will be the case for sound waves obeying a linear wave equation, for light waves, and
for quantum wave functions. If the choice of simpler functions is based on having
them match the same boundary conditions as the given function, then the analysis
of the given function reduces to knowing the properties of the simpler functions.
The expansion of the given function f (x) into simpler functions φ(k, x)
appears as
f (x) =
g(k)φ(k, x)dk .
(H.11)