Fourier Integrals
Fourier Transforms
F OURIER T RANSFORMS
G. Ramesh
28th Sep 2015
Fourier Integrals
Fourier Transforms
O UTLINE
1
F OURIER I NTEGRALS
2
F OURIER T RANSFORMS
Fourier Integrals
Fourier Transforms
F OURIER I NTEGRAL
Let f : R → R be a function. The representation
Z ∞
f (x) =
A(w) cos(wx) + B(w) sin(wx) dw,
0
where
Z
1 ∞
f (v ) cos(wv )dv
π −∞
Z
1 ∞
B(w) =
f (v ) sin(wv )dv
π −∞
A(w) =
is called the Fourier Integral representation of f .
(1)
Fourier Integrals
Fourier Transforms
Let f : R → R be such that
• piecewise cont in every finite interval
• absolutely integrable
Z
∞
|f (x)|dx < ∞
−∞
• f has left and right derivative at every point in the finite
interval.
Then f (x) can be represented by Fourier integral.
(x0 −)
If f is discontinuous at x0 , then f (x0 ) = f (x0 +)+f
.
2
Fourier Integrals
Fourier Transforms
Find theFourier integral representation of
1, if |x| < 1
f (x) =
0, if |x| > 1.
Fourier Integrals
Fourier Transforms
Find theFourier integral representation of
1, if |x| < 1
f (x) =
0, if |x| > 1.
Sol: We have
2 sin(w)
πw
B(w) = 0
Z
2 ∞ cos(wx) sin(w)
f (x) =
dw
π 0
w
f (1+) + f (1−)
1
f (1) =
= .
2
2
A(w) =
Fourier Integrals
Fourier Transforms
F OURIER C OSINE I NTEGRAL
Z
∞
f (x) =
A(w) cos(wx)dw
(2)
0
where
2
A(w) =
π
Z
∞
f (v ) cos(wv )dv
0
is called the Fourier cosine integral of f .
(3)
Fourier Integrals
Fourier Transforms
F OURIER SINE I NTEGRAL
Z
∞
f (x) =
B(w) sin(wx)dw
(4)
0
where
2
B(w) =
π
Z
∞
f (v ) sin(wv )dv
0
is called the Fourier sine integral of f .
(5)
Fourier Integrals
Fourier Transforms
Find the Fourier cosine and sine integrlas of
f (x) = e−kx , x > 0, k > 0.
Fourier Integrals
Fourier Transforms
Find the Fourier cosine and sine integrlas of
f (x) = e−kx , x > 0, k > 0.
2k
,
Ans: A(w) =
2
π(k + w 2 )
Fourier Integrals
Fourier Transforms
Find the Fourier cosine and sine integrlas of
f (x) = e−kx , x > 0, k > 0.
2k
,
Ans: A(w) =
2
π(k + w 2 )
Z ∞
π −kx
cos(wx)
e
=
dw
2k
k2 + w2
o
(6)
Fourier Integrals
Fourier Transforms
Find the Fourier cosine and sine integrlas of
f (x) = e−kx , x > 0, k > 0.
2k
,
Ans: A(w) =
2
π(k + w 2 )
Z ∞
π −kx
cos(wx)
e
=
dw
2k
k2 + w2
o
B(w) =
2w
and
π(k 2 + w 2 )
Z ∞
π
w sin(wx)
dw = e−kx
2
2
2
k +w
0
The integrals in Equations (6) and (7) are called as Laplace
integrals.
(6)
(7)
Fourier Integrals
Fourier Transforms
Let f : R → R be a function. Then the
q cosine integral is given by
Equations (2) and (3). Let A(w) = π2 f̂c (w), where
r Z
2 ∞
f̂c (w) =
f (x) cos(wx)dx
π 0
is called the Fourier cosine transform of f and
r Z
2 ∞
f̂c (w) cos(wx)dw
f (x) =
π 0
is called the inverse Fourier Cosine transform of f .
Fourier Integrals
Fourier Transforms
Let f : R → R be a function. Then the
q sine integral is given by
Equations (4) and (5). Let B(w) = π2 f̂s (w), where
Z
f̂s (w) =
∞
f (x) sin(wx)dx
0
is called the Fourier Sine transform of f and
r Z
2 ∞
f̂s (w) sin(wx)dw
f (x) =
π 0
is called the inverse Fourier Sine transform of f .
Fourier Integrals
Fourier Transforms
Find the Fourier Cosine and sine transforms of the function

x if 0 < x < a
f (x =
.
0 if x > a
r
Ans: f̂c (w) =
2 k sin(w)
.
π
w
Fourier Integrals
Fourier Transforms
Find the Fourier Cosine and sine transforms of the function

x if 0 < x < a
f (x =
.
0 if x > a
r
2 k sin(w)
.
π
w
Find the Fourier cosine transform of f (x) = e−x , x ∈ R.
Ans: f̂c (w) =
Fourier Integrals
Fourier Transforms
P ROPERTIES
Let Fc and Fs denote the Fourier Cosine and Sine transforms
of f respectively. Then
1
Fc (af + bg) = aFc (w) + bFc (w)
2
Fs (af + bg) = aFs (w) + bFs (w)
Fourier Integrals
Fourier Transforms
T HEOREM
Let f : R → R be such that
1
f is continuous
2
f is absolutely integrable on R
3
f 0 is piecewise continuous on each finite interval
4
f (x) → 0 as x → ∞.
Then
r
( A ) Fc
(f 0 )
= wFs (f ) −
( B ) Fs (f 0 ) = −wFc (f ).
2
f (0)
π
Fourier Integrals
Fourier Transforms
T HEOREM
Let f : R → R be such that
1
f is continuous
2
f is absolutely integrable on R
3
f 0 is piecewise continuous on each finite interval
4
f (x) → 0 as x → ∞.
Then
r
( A ) Fc
(f 0 )
= wFs (f ) −
2
f (0)
π
( B ) Fs (f 0 ) = −wFc (f ).
Similarly we can prove the following:
Fourier Integrals
Fourier Transforms
r
2 0
f (0)
π
r
2
wf 0 (0).
( B ) Fs (f 00 ) = −w 2 Fs (f ) +
π
( A ) Fc (f 00 ) = −w 2 Fc (f ) −
Fourier Integrals
Fourier Transforms
r
2 0
f (0)
π
r
2
wf 0 (0).
( B ) Fs (f 00 ) = −w 2 Fs (f ) +
π
( A ) Fc (f 00 ) = −w 2 Fc (f ) −
Find the Fourier cosine transform of f (x) = e−ax , (a > 0)
Fourier Integrals
Fourier Transforms
r
2 0
f (0)
π
r
2
wf 0 (0).
( B ) Fs (f 00 ) = −w 2 Fs (f ) +
π
( A ) Fc (f 00 ) = −w 2 Fc (f ) −
Find the Fourier
transform of f (x) = e−ax , (a > 0)
r cosine
2
a
Ans: Fc (f ) =
.
π a2 + w 2
Fourier Integrals
Fourier Transforms
Let f : R → R be a function. The Fourier transform of f is
defined by
Z ∞
1
f̂ (w) = √
e−iwx f (x)dx.
2π −∞
(8)
Fourier Integrals
Fourier Transforms
Let f : R → R be a function. The Fourier transform of f is
defined by
Z ∞
1
f̂ (w) = √
e−iwx f (x)dx.
2π −∞
The inverse Fourier transform of f̂ (w) is defined by
Z ∞
1
eiwx f̂ (w)dw.
f (x) = √
2π −∞
(8)
(9)
Fourier Integrals
Fourier Transforms
E XISTENCE OF F OURIER TRANSFORM
Let f : R → R be a function such that
1
f is piecewise cont. on R
2
f is absolutely integrable.
Then f̂ exists.
Fourier Integrals
Fourier Transforms
E XISTENCE OF F OURIER TRANSFORM
Let f : R → R be a function such that
1
f is piecewise cont. on R
2
f is absolutely integrable.
Then f̂ exists.

k
Find the Fourier transform of f (x) =
0
0<x <a
x > a.
Fourier Integrals
Fourier Transforms
E XISTENCE OF F OURIER TRANSFORM
Let f : R → R be a function such that
1
f is piecewise cont. on R
2
f is absolutely integrable.
Then f̂ exists.

k
Find the Fourier transform of f (x) =
0
2
0<x <a
x > a.
Find the Fourier trans of f (x) = e−ax , a > 0
Fourier Integrals
Fourier Transforms
P ROPERTIES
Let f , g : R → R be such that F(f ) and F(g) exists. Then
1
F(f + g) = F(f ) + F(g)
2
F(αf ) = αF(f )
Fourier Integrals
Fourier Transforms
P ROPERTIES
Let f , g : R → R be such that F(f ) and F(g) exists. Then
1
F(f + g) = F(f ) + F(g)
2
F(αf ) = αF(f )
Let f be a continuous fucntion such that f (x) → 0 as |x| → ∞.
Let f 0 be absolutely integrable. Then
F(f 0 ) = iwF(f ).
Fourier Integrals
Fourier Transforms
P ROPERTIES
Let f , g : R → R be such that F(f ) and F(g) exists. Then
1
F(f + g) = F(f ) + F(g)
2
F(αf ) = αF(f )
Let f be a continuous fucntion such that f (x) → 0 as |x| → ∞.
Let f 0 be absolutely integrable. Then
F(f 0 ) = iwF(f ).
If f 00 is absolutely integrable, we can show that
F(f 00 ) = −w 2 F(f ).
Fourier Integrals
Fourier Transforms
2
Let f (x) = xe−x , x ∈ R. Find F(f ).
Fourier Integrals
Fourier Transforms
2
Let f (x) = xe−x , x ∈ R. Find F(f ).
w2
−iw
Ans: F(f ) = √ e− 4 .
2 2
Fourier Integrals
Fourier Transforms
2
Let f (x) = xe−x , x ∈ R. Find F(f ).
w2
−iw
Ans: F(f ) = √ e− 4 .
2 2
Let f and g are piecewise continuous and absolutely integrable.
Then
√
F(f ∗ g) = 2π F(f )F(g).
Fourier Integrals
Fourier Transforms
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