#16 (10/30/2028)

Period other than 2π

If the period of f(x) is 2l over [−l, l] instead of 2π, it is necessary to change the scaling from x to πx/l as

f(x)
=
∞
∑
m=−∞ 
 cm exp⎛
⎝ 		i m πx
l
⎞
⎠ 		,
(1)
cm
=
1
2l
 ⌠
⌡ 		l

−l 
 f(x) exp⎛
⎝ 		− i m πx
l
⎞
⎠ 		dx.
(2)

Fourier integral (infinite periodicity)

It can be shown that Fourier integrals are derived as a limiting case of Fourier series by letting l (periodicity) tend to ∞. By combining Eq.(1) with Eq.(2), f(x) can be expressed as

f(x)
=
∞
∑
m=−∞ 
 ⎛
⎝ 		1
2l
 ⌠
⌡ 		l

−l 
 f(y) exp⎛
⎝ 		− imπy
l
⎞
⎠ 		dy⎞
⎠ 		exp⎛
⎝ 		imπx
l
⎞
⎠
=
1
2l
 ⌠
⌡ 		l

−l 
 ⎛
⎝ 		∞
∑
m=−∞ 
 exp⎛
⎝ 		imπ(x−y)
l
⎞
⎠ 		⎞
⎠ 		f(y) dy
=
1
2π
 ⌠
⌡ 		l

−l 

⎛
⎝ 		∞
∑
m=−∞ 
 ⎛
⎝ 		π
l
⎞
⎠ 		exp(im [(π)/(l)] (x−y))⎞
⎠
I(x, y)
 
 f(y) dy
=
1
2π
 ⌠
⌡ 		l

−l 
 I(x, y) f(y) dy,
(3)
where

I(x, y)
=
∞
∑
m=−∞ 
 ⎛
⎝ 		π
l
⎞
⎠ 		exp⎛
⎝ 		im π
l
  (x−y)⎞
⎠
=
∞
∑
m=−∞ 
 (∆ω)ei  m  ∆ω  (x−y)
=
∞
∑
m=−∞ 
 (∆ω) em ∆ω  z
=
∆ω{…+ e−z ∆ω+1 +exp(z ∆ω) +exp(2z ∆ω) + …}
 ∼ 
⌠
⌡ 		∞

−∞ 
 ez  ω dω
=
⌠
⌡ 		∞

−∞ 
 ei  (x−y)  ω dω,
(4)
where ∆ω = π/l and z = i  (x−y) were used.
By combining Eq.(3) with Eq.(4) and exchange the order of integrals, one obtains

f(x)
=
1
2π
 ⌠
⌡ 		∞

−∞ 
 ⎛
⎝ 		⌠
⌡ 		∞

−∞ 
 ei x ω e−iyω dω⎞
⎠ 		f(y) dy
=
⌠
⌡ 		∞

−∞ 
 1
2π
 ⎛
⎝ 		⌠
⌡ 		∞

−∞ 
 f(y) e−i y ω dy⎞
⎠ 		ei x ω dω
=
⌠
⌡ 		∞

−∞ 
 F(ω) ei x ω dω,
(5)
where

F(ω) ≡ 1
2π
 ⌠
⌡ 		∞

−∞ 
 f(y) e−i y ω dy.
(6)
So the (non-periodic) function, f(x) can be expressed as

f(x) = ⌠
⌡ 		∞

−∞ 
 F(ω) eiωx dω,
(7)
where

F(ω) = 1
2π
 ⌠
⌡ 		∞

−∞ 
 f(x) e−iωx dx.
(8)

Fourier series
Fourier integrals
f(x) = ∞
∑
−∞ 
 cm ei m x
f(x) = ⌠
⌡ 		∞

−∞ 
 F(ω) ei ωx dω
cm = 1
2π
 ⌠
⌡ 		π

−π 
 f(x) e−i m x dx
F(ω) = 1
2π
 ⌠
⌡ 		∞

−∞ 
 f(x)e−iωx dx
(9)
prism.jpg
Example 1: Rectangular pulse.
recpulse.jpg

F(ω)
=
1
2π
 ⌠
⌡ 		∞

−∞ 
 f(x)e−iωx dx
=
1
2π
 ⌠
⌡ 		1

−1 
 e−iωx dx
=
sin ω
πω
 .
(10)
So
f(x)
=
⌠
⌡ 		∞

−∞ 
 sin ω
πω
 eiωx dω
=
2
π
 ⌠
⌡ 		∞

0 
 sin ω
ω
 cos ωx dω.
(11)
By substituting x=0, one obtains
1 = 2
π
 ⌠
⌡ 		∞

0 
 sin ω
ω
 dω,
(12)
or
⌠
⌡ 		∞

0 
 sin ω
ω
 dω = π
2
 .
(13)
sinwoverw.jpg
The figure above shows the Fourier transform of f(x). It is seen that f(x) contains frequencies from the entire interval but the largest contribution is ω = 0.
Example 2: Error function
f(x) = e−x2/2.
gaussian.jpg

F(ω)
=
1
2 π
 ⌠
⌡ 		∞

−∞ 
 e−x2/2 e−i ωx dx
=
1
2 π
 ⌠
⌡ 		∞

−∞ 
 exp⎛
⎝ 		− 1
2
 (x + i ω)2 − ω2
2
⎞
⎠ 		dx
=
1
2 π
 e−ω2/2 ⌠
⌡ 		∞

−∞ 
 exp⎛
⎝ 		− 1
2
 (x + i ω)2⎞
⎠ 		dx
=
1
2 π
 e−ω2/2 ⌠
⌡ 		∞

−∞ 
 exp⎛
⎝ 		− 1
2
 x2⎞
⎠ 		dx
=
1

√
2 π
 e−ω2/2.
(14)
Note that the Fourier integral of the error function is also an error function.
Example 3: White color
f(x) = δ(x).

F(ω) = 1
2 π
 ⌠
⌡ 		∞

−∞ 
 δ(x) e− i ωx dx = 1
2 π
 .
Example 4: Diffusion equation

∂u
∂t
=
∂2 u
∂x2
 ,     (−∞ < x < ∞, 0 < t < ∞)
(15)
u(x,0)
=
f(x)     (−∞ < x < ∞)
(16)
Assume that the solution, u(x, t), is expressed by the Fourier integral as

u(x, t) = ⌠
⌡ 		∞

−∞ 
 U(ω, t) ei ωx dω.
(17)
Noting that
∂u
∂t
=
⌠
⌡ 		∞

−∞ 
 dU
dt
 ei ωx dω,
(18)
∂2 u
∂x2
=
⌠
⌡ 		∞

−∞ 
 (− ω2) U ei ωx dω,
(19)
it follows
dU
dt
 = (− ω2) U,
(20)
which can be solved as
U(ω, t) = Aω e−ω2 t.
(21)
Thus, Equation (17) becomes

u(x, t) = ⌠
⌡ 		∞

−∞ 
 Aω e−ω2 t ei ωx dω.
(22)
From the initial condition, it follows

f(x) = ⌠
⌡ 		∞

−∞ 
 Aω ei ωx dω,
(23)
which implies that Aω is the Fourier transform of f(x) as

Aω
=
1
2π
 ⌠
⌡ 		∞

−∞ 
 f(x) e−iωx dx.
(24)
If the initial temperature is the Dirac delta function with a peak at x=0 at t=0,

f(x) = δ(x),
(25)

Aω
=
1
2π
 ⌠
⌡ 		∞

−∞ 
 δ(x) e−iωx dx
=
1
2π
 .
(26)
Hence, Equation (17) becomes

u(x, t)
=
1
2π
 ⌠
⌡ 		∞

−∞ 
 exp(−tω2) eiωx dω
=
1
2π
 ⌠
⌡ 		∞

−∞ 
 exp( −t(ω− ix
2t
 )2− x2
4t
 ) dω
=
1
2π
 exp(− x2
4t
 ) ⌠
⌡ 		∞

−∞ 
 exp( −t(ω− ix
2t
 )2) dω
=
1
2π
 exp(− x2
4t
 ) ⌠
⌡ 		∞

−∞ 
 exp( −tω2) dω
=
1
2π
 exp(− x2
4t
 )   ⎛
√
π
t
 
=
exp(− x2
4t
 )
2
√
πt
 .
(27)
Here is how the temperature decays:
It is seen that a pulse initially at x=0 propagates to x=±∞ instantly at any small t which is the nature (and flaw) of the diffusion equation.


File translated from TEX by TTH, version 4.03.
On 29 Oct 2024, 19:24.