Created by T.

Madas

FOURIER
TRANSFORM

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 Created by T. Madas

Fourier Transform Summary

Definitions

∫
1
• F  f ( x )  = fˆ ( k ) = f ( x ) e−i kx dx
2π −∞

∫
−1  1
fˆ k  = f ( x ) = f ( k ) ei kx dk
 ( )
• F
2π −∞

Useful Results

• F  f ′ ( x )  = i k fˆ ( k )

d
• F  x f ( x )  = i  fˆ ( k ) 
dk  

Shift Results

• F  f ( x + c )  = ei kc fˆ ( k )

• F −1  fˆ ( k + c )  = e−i cx f ( x )
 

Convolution Theorem

F {[ f ∗ g ] ( x )} = 2π F  f ( x )  F  g ( x ) 

∞
where [ f ∗ g ]( x ) =
∫ −∞
f ( x − y ) g ( y ) dy

Parseval’s Theorem

∞ ∞ ∞ ∞

∫ ∫ ∫ ∫
2 2
h ( y ) g ( y ) dy = hˆ ( k ) gˆ ( k ) dk or h( y) dy = hˆ ( k ) dk
−∞ −∞ −∞ −∞

Created by T. Madas
 Created by T. Madas

FINDING FOURIER
TRANSFORMS
and
INVERSES

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 Created by T. Madas

Question 1

f ( x ) = e− ax , x > 0 ,

where a is a positive constant.

Find the Fourier transform of f ( x ) .

a − ik
fˆ ( k ) =
(a 2
+ k2 ) 2π

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 Created by T. Madas

Question 2

1 x < 1a
 2
f ( x) =  ,
0 1
x > a
2

where a is a positive constant.

Find the Fourier transform of f ( x ) .

2 a
fˆ ( k ) =
k 2π
( )
sin 1 ka =
2 2π
( )
sinc 1 ka
2

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 Created by T. Madas

Question 3

1 0 ≤ x ≤ 2
f ( x) =  .
0 otherwise

Find the Fourier transform of f ( x ) .

2
fˆ ( k ) = e−ik sinc k
π

Question 4

 1 x ≤ω
f ( x) =  ω
 0 x >ω

where ω is a positive constant.

Find the Fourier transform of f ( x ) .

2
fˆ ( k ) = sinc ω
π

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 Created by T. Madas

Question 5
The function f ( x ) is defined in terms of the positive constant a , by

 x
1 − x ≤a
f ( x) =  a
 0 x >a


Find the Fourier transform of f ( x ) .

2 1 a
F  f ( x )  = fˆ ( k ) = 1 − cos ( ak )  =
π ak 2  2π
( )
sinc 2 1 ka
2

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 Created by T. Madas

Question 6

 1
mx x≤
m
f ( x) =  .
0 1
x >
 m

where m is a positive constant.

Find the Fourier transform of f ( x ) .

i 2
fˆ ( k ) =
k π  ( )
m ( )
cos k − sinc k 
m 

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 Created by T. Madas

Question 7

f ( x ) = x e−2 x , x > 0 .

Find, by direct integration, the Fourier transform of f ( x ) .

1
MM3-D , fˆ ( k ) =
( 2 + ik ) 2 2π

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 Created by T. Madas

Question 8
The triangle function Λ n ( x ) is defined as

1
 n2 ( n + x ) −n< x <0

Λn ( x ) =  1
 n2 ( n − x ) 0< x<n

0 otherwise

where n is a positive constant.

a) Sketch the graph of Λ n ( x ) .

b) Show that the Fourier transform of Λ n ( x ) is

1
2π
( )
sinc2 1 kn .
2

proof

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 Created by T. Madas

Question 9
The function f is defined by

−a x
f ( x) = e ,

where a is a positive constant.

Find the Fourier transform of f ( x ) .

2 a
F e  = fˆ ( k ) =
−a x
  π a + k2
2

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 Created by T. Madas

Question 10
The function f is defined by

1
f ( x) = , x ≠ 0.
x

a) Determine the Fourier transform of f ( x ) , assuming without proof any

∞

∫
sin ax
standard results about dx .
0 x

−ε x
b) By introducing the converging factor e and letting ε → 0 , invert the
answer of part (a) to obtain f .

1 π
MM3-E , F   = fˆ ( k ) = − i sign ( k )
x 2

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 Created by T. Madas

Question 11
The impulse function δ ( x ) is defined by

∞ x = 0
δ ( x) = 
0 x ≠ 0

a) Determine

i. … F δ ( x )  .

ii. ... F δ ( x − a )  , where a is a positive constant.

iii. ... F −1 δ ( k )  .

b) Use the above results to deduce F [1] and F −1 [1] .

1 1 −i ka 1
F δ ( x )  = , F δ ( x − a )  = e , F −1 δ ( k )  = ,
2π 2π 2π
F [1] = 2π δ ( k ) , F −1 [1] = 2π δ ( x )

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 Created by T. Madas

Question 12
The signum function sign ( x ) is defined by

 1 x>0
sign ( x ) = 
 −1 x<0

−ε x
By introducing the converging factor e and letting ε → 0 , determine the Fourier
transform of sign ( x ) .

i 1
F sign ( x )  = −
k π

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 Created by T. Madas

Question 13
The Unit function U ( x ) is defined by

U( x) = 1 .

−ε x
By introducing the converging factor e and letting ε → 0 , determine the Fourier
transform of U ( x ) .

1  ε 
You may assume that δ ( t ) = lim  2 2  .
π ε →0  ε + t 

F  U ( x )  = 2π δ ( k )

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 Created by T. Madas

Question 14
The Unit function U ( x ) is defined by

U ( x ) = 1.

−ε k
By introducing the converging factor e and letting ε → 0 , find F −1  U ( k )  .

1  ε 
You may assume that δ ( t ) = lim  2 2  .
π ε →0  ε + t 

F −1  U ( k )  = 2π δ ( x )

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 Created by T. Madas

Question 15
The function g ( x ) has Fourier transform given by

gˆ ( k ) = −i sign ( k ) .

−ε k
By introducing the converging factor e and letting ε → 0 , find F −1  ĝ ( k )  .

2 1
MM3-B , F −1  ĝ ( k )  =
π x

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 Created by T. Madas

Question 16
The Heaviside function H ( x ) is defined by

 1 x≥0
H( x) = 
 0 x<0

By introducing the converging factor e−ε x and letting ε → 0 , determine the Fourier
transform of H ( x ) .

1  ε 
You may assume that δ ( t ) = lim  2 2  .
π ε →0  ε + t 

1  i
F [ H( x)] = πδ ( k ) − k 
2π

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 Created by T. Madas

Question 17
The impulse function δ ( x ) is defined by

∞ x = 0
δ ( x) = 
0 x ≠ 0

a) Determine the inverse Fourier transform of the impulse function F −1 δ ( k )  ,

and use it to deduce the Fourier transform of f ( x ) = 1 .

b) Find directly the Fourier transform of f ( x ) = 1 , by introducing the converging

−ε x
factor e and letting ε → 0 .

F [1] = 2π δ ( k )

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 Created by T. Madas

Question 18
The function f is defined by

 1 x>0
f ( x ) = sign( x) = 
 −1 x<0

−ε x
a) By introducing the converging factor e and letting ε → 0 , find the Fourier
transform of f .

−ε x
b) By introducing the converging factor e and letting ε → 0 , find the Fourier
transform of g ( x ) = 1 .

1  ε 
You may assume that δ ( t ) = lim  2 2  .
π ε →0  ε + t 

c) Hence determine the Fourier transform of the Heaviside function H ( x ) ,

 1 x≥0
H( x) = 
 0 x<0

i 1 1  i
F [sign( x)] = − , F [1] = 2π δ ( k ) , F [ H( x)] = πδ ( k ) − k 
k π 2π

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 Created by T. Madas

Question 19
The Fourier transforms of the functions f ( x ) and g ( x ) are

1
fˆ ( k ) = δ ( k ) and gˆ ( k ) = ,
ik

where δ ( x ) denotes the impulse function.

Find simplified expressions for f ( x ) and g ( x ) , and use them to show that

1  1
F [ H( x) ] = π δ ( k ) + i k  ,
2π  

where H( x) denotes the Heaviside function.

1
f ( x) = , g ( x ) = 1 π sgn ( x )
2π 2

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 Created by T. Madas

Question 20
The function f is defined by

sin ax
f ( x) = , a >0.
x

Find the Fourier transform of f ( x ) , stating clearly any results used.

 π
 k <a
 2
 sin ax 
F =  π
 x   k =a
8

 0 k >a

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 Created by T. Madas

Question 21
Given that l is a non zero constant, show that


F
( )
 exp − x2 
l 2  = fˆ k =
 ( )
1  k 2l 2 
exp  − .
l π 2π  4 
  
 

proof

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 Created by T. Madas

Question 22
The Gaussian function f ( x ) is defined by

2
f ( x ) = A e−α x ,

where A and α are positive constants.

Find the Fourier transform of f ( x ) .

 k2 
MM3-A , F  A e −α x  = fˆ ( k ) =
2 A
exp  −
  2α  4α 
 

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 Created by T. Madas

Question 23
The function f is defined by

1
f ( x) = ,
x + a2
2

where a is a positive constant.

Use contour integration to find the Fourier transform of f ( x ) .

 1  ˆ π e−a k
F 2 = f (k ) =
 x + a 2  2 a

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 Created by T. Madas

Question 24
The function f is defined by

2
f ( x ) = x e− x , x ∈ ℝ .

Find the Fourier transform of f ( x ) , stating clearly any results used.

2
F  x e− x  = 1 k 2 e 4
2 −1k
  4

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 Created by T. Madas

Question 25
The function f is defined by

x
f ( x) = ,
x + a2
2

where a is a positive constant.

Use contour integration to find the Fourier transform of f ( x ) .

−a k
 x  ˆ π e sign k
MM3-F , F  2 2
= f (k ) = − i
x +a  2 a

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 Created by T. Madas

Question 26
Find the inverse Fourier transform of

2 2
σ t
gˆ ( k ) = e− k ,

where σ and t are positive constants.

 x2 
F −1 e− k σ t
2 2 1
exp  −
 4tσ 2 
=
  2t σ  

proof

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 Created by T. Madas

Question 27
The Fourier transform fˆ ( k ) , of function f ( x ) is

2 a
fˆ ( k ) = ,
π a + k2
2

where a is a positive constant.

Use contour integration to find an expression for f ( x ) .

ax
f ( x) = e

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 Created by T. Madas

Question 28
The function f is defined by

1
f ( x) = ,
2 2
(x 2
+a )
where a is a positive constant.

Use contour integration to find the Fourier transform of f ( x ) .

  −a k
 1  ˆ π (1 + a k ) e
MM3-C , F  2
= f (k ) =
a3
2
 x + a
2
( ) 
8

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 Created by T. Madas

VARIOUS PROBLEMS
on
FOURIER
TRANSFORMS

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 Created by T. Madas

Question 1
Find the Fourier transform of an arbitrary function f ( x ) if

i. f ( x ) is even.

ii. f ( x ) is odd.

Give the answers as a simplified integral form.

∞ ∞

∫ ∫
2 2
fˆ ( k ) = f ( x ) cos kx dx , fˆ ( k ) = −i f ( x ) sin kx dx
π 0 π 0

Question 2
Use the definition of the Fourier transform, of an absolutely integrable function f ( x ) ,
to show that

F  f ′ ( x )  = i k F  f ( x )  .

proof

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 Created by T. Madas

Question 3
The Fourier transform of an absolutely integrable function f ( x ) , is denoted by fˆ ( k ) .

Show that

d
F  x f ( x )  = i  fˆ ( k )  .
dk  

proof

Question 4
Given that c is a constant show that

F  f ( x + c )  = ei kc F  f ( x )  .

proof

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 Created by T. Madas

Question 5
Given that c is a constant show that

F −1  fˆ ( k + c )  = ei cx f ( x ) ,
 

where fˆ ( k ) ≡ F  f ( x ) 

proof

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 Created by T. Madas

Question 6
Given that c is a constant prove the validity of the two shift theorems

a) F  f ( x + c )  = ei kc F  f ( x )  .

b) F −1  fˆ ( k + c )  = ei cx f ( x ) .

Note that fˆ ( k ) ≡ F  f ( x )  .

proof

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 Created by T. Madas

Question 7
The convolution [ f ∗ g ] ( x ) , of two functions f ( x ) and g ( x ) is defined as

∞
[ f ∗ g ]( x ) =
∫
−∞
f ( x − y ) g ( y ) dy .

Show that

F {[ f ∗ g ] ( x )} = 2π F  f ( x )  F  g ( x )  = 2π fˆ ( k ) gˆ ( k ) .

proof

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 Created by T. Madas

Question 8
It is given that c is a constant and fˆ ( k ) ≡ F  f ( x )  .

a) Prove the validity of the inversion shift theorem

F −1  fˆ ( k + c )  = ei cx f ( x ) .
 

b) Hence determine an expression for

 − k −a 2 
F −1 e ( )  ,
 

where a is a positive constant.

 − k −a 2  1 − 14 x2
F −1 e ( )  = e [ cos ax + isin ax ]
  2

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 Created by T. Madas

Question 9
The convolution theorem for two functions f ( x ) and g ( x ) asserts that

F {[ f ∗ g ] ( x )} = 2π F  f ( x )  F  g ( x )  ,

where

∞
[ f ∗ g ]( x ) =
∫ −∞
f ( x − y ) g ( y ) dy .

a) Starting from the convolution theorem prove Parseval’s Theorem

∞ ∞

∫ ∫
2 2
h( y) dy = hˆ ( k ) dk .
−∞ −∞

b) Use Parseval’s Theorem to evaluate

∫
1
dx .
0 x + a2
2

−a x 2 a
You may assume that if f ( x ) = e , then fˆ ( k ) = .
π a + k2
2

π
4a3

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 Created by T. Madas

Question 10
The convolution [ f ∗ g ] ( x ) , of two functions f ( x ) and g ( x ) is defined as

∞
[ f ∗ g ]( x ) =
∫−∞
f ( x − y ) g ( y ) dy .

a) Show that

F {[ f ∗ g ] ( x )} = 2π F  f ( x )  F  g ( x )  = 2π fˆ ( k ) gˆ ( k ) .

b) Hence prove Parseval’s Theorem

∞ ∞

∫ −∞
h ( y ) g ( y ) dy =
∫ −∞
hˆ ( k ) gˆ ( k ) dk .

c) Use Parseval’s Theorem to evaluate

∫(
1
dx .
0
x2 + a2 )( x2 + b2 )
−a x 2 a
You may assume that if f ( x ) = e , then fˆ ( k ) = .
π a + k2 2

π
2ab ( a + b )

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 Created by T. Madas

APPLICATIONS
of
FOURIER
TRANSFORMS

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 Created by T. Madas

Question 1
The function ϕ = ϕ ( x, y ) satisfies Laplace’s equation in Cartesian coordinates

∂ 2ϕ ∂ 2ϕ
+ = 0.
∂x 2 ∂y 2

Use Fourier transforms to convert the above partial differential equation into an
ordinary differential equation for ϕˆ ( k , y ) , where ϕˆ ( k , y ) is the Fourier transform of
ϕ ( x, y ) with respect to x .

d 2ϕˆ
2
− k 2ϕˆ = 0
dx

Created by T. Madas
 Created by T. Madas

Question 2
The function ϕ = ϕ ( x, y ) satisfies Laplace’s equation in Cartesian coordinates,

∂ 2ϕ ∂ 2ϕ
+ = 0,
∂x 2 ∂y 2

in the part of the x-y plane for which y ≥ 0 .

It is further given that

• ϕ ( x, y ) → 0 as x2 + y 2 → ∞

 1 x <1
• ϕ ( x,0 ) =  2
0 x >1

Use Fourier transforms to show that

∫
1 1 − ky
ϕ ( x, y ) = e sin k cos kx dk ,
π 0 k

and hence deduce the value of ϕ ( ±1,0 ) .

MM4-B , ϕ ( ±1, 0 ) = 1
4

Created by T. Madas
 Created by T. Madas

Question 3
The Airy function Ai ( x ) satisfies the differential equation

d2y
− xy = 0 .
dx 2

Use Fourier transforms to show that

∫
1
Ai ( x ) =
π 0
(3 )
cos 1 t 3 + xt dt ,

for suitable boundary conditions.

d
You may assume that F  x f ( x )  = i
dk
{
F  f ( x )  . }

proof

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 Created by T. Madas

Question 4
The function ψ = ψ ( x, y ) satisfies Laplace’s equation in Cartesian coordinates,

∂ 2ψ ∂ 2ψ
+ =0,
∂x 2 ∂y 2

in the part of the x-y plane for which y ≥ 0 .

It is further given that

• ψ ( x,0 ) = δ ( x )

• ψ ( x, y ) → 0 as x2 + y 2 → ∞

Use Fourier transforms to convert the above partial differential equation into an
ordinary differential equation and hence show that

1 y 
ψ ( x, y ) =  2 .
π  x + y2 

MM4-C , proof

Created by T. Madas
 Created by T. Madas

Question 5
The function u = u ( x, t ) satisfies the partial differential equation

∂u 1 ∂ 3u
+ = 0.
∂t 3 ∂x3

It is further given that

• u ( x,0 ) = δ ( x )

• u ( x, t ) → 0 as x → ∞

Use Fourier transforms to convert the above partial differential equation into an
ordinary differential equation and hence show that

1  x 
u ( x, t ) = Ai  1 ,
1  
t3  t3 

where the Ai ( x ) is the Airy function, defined as

∫
1
Ai ( x ) = cos  1 k 3 + kx  dk .
π 3 
0

proof

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 Created by T. Madas

Question 6
The function ϕ = ϕ ( x, y ) satisfies Laplace’s equation in Cartesian coordinates,

∂ 2ϕ ∂ 2ϕ
+ = 0,
∂x 2 ∂y 2

in the part of the x-y plane for which x ≥ 0 and y ≥ 0 .

It is further given that

1
• ϕ ( x, 0 ) =
1 + x2

• ϕ ( x, y ) → 0 as x2 + y 2 → ∞

∂
• ϕ ( x,0 )  = 0
∂x 

Use Fourier transforms to convert the above partial differential equation into an
ordinary differential equation and hence show that

y +1
ϕ ( x, y ) = 2
.
2
x + ( y + 1)

proof

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 Created by T. Madas

Question 7
The function Φ = Φ ( x, y ) satisfies Laplace’s equation in Cartesian coordinates,

∂ 2Φ ∂ 2Φ
+ = 0,
∂x 2 ∂y 2

in the part of the x-y plane for which y ≥ 0 .

It is further given that

• Φ ( x,0 ) = δ ( x )

• Φ ( x, y ) → 0 as x2 + y 2 → ∞

Use Fourier transforms to find the solution of the above partial differential equation
and hence show that

  −1 
 1 y2  
δ ( x ) = lim 1 + 2  .
α →0  πα  α  
  

proof

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 Created by T. Madas

Question 8
The function y = y ( x ) satisfies the differential equation

dy
+ λ y = f ( x) ,
dx

where f ( x ) is a given function and λ is a real constant.

Use Fourier transforms to show that

∞
y ( x) =
∫
0
eλt f ( x − t ) dt .

proof

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 Created by T. Madas

Question 9
The function ϕ = ϕ ( x, y ) satisfies Laplace’s equation in Cartesian coordinates,

∂ 2ϕ ∂ 2ϕ
+ = 0,
∂x 2 ∂y 2

in the semi-infinite region of the x-y plane for which y ≥ 0 .

It is further given that

• ϕ ( x,0 ) = f ( x )

• ϕ ( x, y ) → 0 as x2 + y 2 → ∞

Use Fourier transforms to convert the above partial differential equation into an
ordinary differential equation and hence show that

∞
f ( x − u)
∫
y
ϕ ( x, y ) = du .
π −∞ u2 + y2

MM4-E , proof

Created by T. Madas
 Created by T. Madas

Question 10
The function ϕ = ϕ ( x, y ) satisfies Laplace’s equation in Cartesian coordinates,

∂ 2ϕ ∂ 2ϕ
+ = 0,
∂x 2 ∂y 2

in the semi-infinite region of the x-y plane for which y ≥ 0 .

It is further given that for a given function f = f ( x )

∂ ∂
• ϕ ( x,0 )  =  f ( x ) 
∂y ∂x

• ϕ ( x, y ) → 0 as x2 + y 2 → ∞

Use Fourier transforms to convert the above partial differential equation into an
ordinary differential equation and hence show that

∞
f (u )
∫
1
ϕ ( x, 0 ) = du .
π −∞ x −u

proof

[ solution overleaf ]

Created by T. Madas
Created by T. Madas

Created by T. Madas
 Created by T. Madas

Question 11
The function ϕ = ϕ ( x, y ) satisfies Laplace’s equation in Cartesian coordinates,

∂ 2ϕ ∂ 2ϕ
+ = 0 , −∞ < x < ∞ , y ≥ 0 .
∂x 2 ∂y 2

It is further given that

ϕ ( x, y ) → 0 as x2 + y 2 → ∞

ϕ ( x,0 ) = H ( x ) , the Heaviside function.

Use Fourier transforms to show that

1 1 x
ϕ ( x, y ) = + arctan   .
2 π  y

You may assume that

1  1
F [ H( x) ] = π δ ( k ) + i k  .
2π  

proof

Created by T. Madas
 Created by T. Madas

Question 12
The function u = u ( x, y ) satisfies Laplace’s equation in Cartesian coordinates,

∂ 2u ∂ 2u
+ =0, −∞ < x < ∞ , 0 < y < 1.
∂x 2 ∂y 2

It is further given that

u ( x,0 ) = 0

u ( x,1) = f ( x )

where f ( − x ) = f ( x ) and f ( x ) → 0 as x → ∞

a) Use Fourier transforms to show that

∫
2 fˆ ( k ) cos kx sinh ky
u ( x, y ) = dk , fˆ ( k ) = F  f ( x )  .
π sinh k
−∞

b) Given that f ( x ) = δ ( x ) show further that

sin π y
u ( x, y ) = .
2 [ cosh π x + cos π y ]

You may assume without proof

∞
π  sin ( Bπ / C ) 
∫
cos Au sinh Bu
du =  , 0≤B<C.
0 sinh Cu 2C  cosh ( Aπ / C ) + cos ( Bπ / C ) 

proof

Created by T. Madas
 Created by T. Madas

Question 13
The function ψ = ψ ( x, y ) satisfies Laplace’s equation in Cartesian coordinates,

∂ 2ψ ∂ 2ψ
+ =0,
∂x 2 ∂y 2

in the part of the x-y plane for which y ≥ 0 .

It is further given that

ψ ( x,0 ) = f ( x )

ψ ( x, y ) → 0 as x2 + y 2 → ∞

c) Use Fourier transforms to convert the above partial differential equation into
an ordinary differential equation and hence show that

∫
y f (u )
ψ ( x, y ) = du .
π
−∞
( x − u )2 + y 2

d) Evaluate the above integral for …

i. … f ( x ) = 1 .

ii. … f ( x ) = sgn x

iii. … f ( x ) = H ( x )

commenting further whether these answers are consistent.

2  x 1 1 x
ψ ( x, y ) = 1 , ψ ( x, y ) = arctan   , ψ ( x, y ) = + arctan  
π  y 2 π  y

[ solution overleaf ]

Created by T. Madas
Created by T. Madas

Created by T. Madas
 Created by T. Madas

Question 14
The function θ = θ ( x, t ) satisfies the heat equation in one spatial dimension,

∂ 2θ 1 ∂θ
2
= , −∞ < x < ∞ , t ≥ 0 ,
∂x σ 2 ∂t

where σ is a positive constant.

Given further that θ ( x,0 ) = f ( x ) , use Fourier transforms to convert the above partial
differential equation into an ordinary differential equation and hence show that

∞
 u2 
∫
1
θ ( x, t ) = f ( x − u ) exp 
 4tσ 2 
du .
2σ π t −∞  

proof

Created by T. Madas
 Created by T. Madas

Question 15
The function u = u ( x, y ) satisfies Laplace’s equation in Cartesian coordinates,

∂ 2u ∂ 2u
+ =0,
∂x 2 ∂y 2

in the part of the x-y plane for which x ≥ 0 and y ≥ 0 .

It is further given that

u ( 0, y ) = 0

u ( x, y ) → 0 as x2 + y 2 → ∞

u ( x,0 ) = f ( x ) , f ( 0 ) = 0 , f ( x ) → 0 as x → ∞

Use Fourier transforms to show that

∫
y  1 1 
u ( x, y ) = f ( w)  2
− 2
 dw .
π  y 2 + ( x − w ) y 2 + ( x + w ) 
0

proof

[ solution overleaf ]

Created by T. Madas
Created by T. Madas

Created by T. Madas
 Created by T. Madas

Question 16
The function T = T ( x, t ) satisfies the heat equation in one spatial dimension,

∂ 2θ 1 ∂θ
2
= , x ≥ 0 , t ≥ 0,
∂x σ ∂t

where σ is a positive constant.

It is further given that

• T ( x,0 ) = f ( x )

• T ( 0, t ) = 0

• T ( x, t ) → 0 as x → ∞

Use Fourier transforms to convert the above partial differential equation into an
ordinary differential equation and hence show that

∫
1  ( x − u )2 
T ( x, t ) = f ( u ) exp   du .
4πσ t  4tσ 
−∞

2
1 4k a
You may assume that F e ax  =
2
e .
  2a

proof

Created by T. Madas
 Created by T. Madas

Question 17
The function f = f ( x ) satisfies the integral equation

∫
f (t ) 1
2
dt = 2
,
−∞ (x −t) +1 x +4

where f ( x ) → 0 as x → ∞

Use Fourier transforms to find the solution of the above integral equation.

 1  1 π −a k
You may assume that F  2 = e .
 x + a 2  a 2

1
f ( x) =
(
2π 1 + x 2 )

Created by T. Madas
 Created by T. Madas

Question 18
The function f = f ( x ) satisfies the integral equation

∫
1
f ( x − u ) f ( u ) du = ,
−∞ 1 + x2

where f ( x ) → 0 as x → ∞

Use Fourier transforms to find the solution of the above integral equation.

You may assume that

∫
cos kx
dx = 1 π e .
k
2
x +1 2
0

2
f ( x) =
(1 + 4 x2 ) π

Created by T. Madas
 Created by T. Madas

Question 19
The function f = f ( x ) satisfies the integral equation

∫
− 12 x 2 − x −u
e =1 e f ( u ) du ,
2
−∞

where f ( x ) → 0 as x → ∞

Use Fourier transforms to find the solution of the above integral equation.

You may assume that

2
1 4k a
F e ax  =
2
• e .
  2a

• F e
ax = 2 a
.
  π a + k2
2

− 12 x 2
( )
f ( x ) = 2 − x2 e

Created by T. Madas