∞  
f (x) = 9
π2
1
n2
sin nπ
3
sin nπLx ; a = c
2L
n=1


∞  
f (x) = 32
3π 2
1
n2
sin nπ
4
sin nπLx ; a = c
2L
n=1


f (x) = 1
π
+ 1
2
sin ωt − 2
π
1
n2 −1
cos nωt
n=2,4,6,...

Extracted from graphs and formulas, pages 372, 373, Differential Equations in Engineering Problems, Salvadori and Schwarz,
published by Prentice-Hall, Inc.,1954.

THE FOURIER TRANSFORMS

For a piecewise continuous function F (x) over a finite interval 0 ≤ x ≤ π; the finite Fourier cosine transform of F (x) is
π
fc (n) = F (x) cos nx dx (n = 0, 1, 2, . . . )
0

If x ranges over the interval 0 ≤ x ≤ L, the substitution x = π x/L allows the use of this definition, also. The inverse transform is
written.
2
x
1
F (x) = fc (0) − fc (n) cos nx (0 < x < π)
π π n=1

where F (x) = F (x+ )+F (x− )

2
. We observe that F (x+) = F (x−) = F (x) at points of continuity. The formula
π
fc(2) (n) = F  (x) cos nx dx
0 (1)
= −n2 fc (n) − F  (0) + (−1) n F  (π)
makes the finite Fourier cosine transform useful in certain boundary value problems. Analogously, the finite Fourier sine transform
of F (x) is
π
fs (n) = F (x) sin nx dx (n = 1, 2, 3, . . . )
0

and
2
∞
F (x) = fs (n) sin nx (0 < x < π)
π n=1

Corresponding to (1) we have

π
fs(2) (n) = F  (x) sin nx dx (2)
0
= −n2 fs (n) − n F (0) − n(−1) n F (π)
x
If F (x) is defined for x ≤ 0 and is piecewise continuous over any finite interval, and if 0
F (x) dx is absolutely convergent, then

2 x
fc (α) = F (x) cos(αx) dx
π 0

A-62
The Fourier Transforms A-63

is the Fourier cosine transform of F (x). Furthermore,

 x
2
F (x) = fc (α) cos(αx) dα.
π 0

If limx→∞ dn F /dxn = 0, then an important property of the Fourier cosine transform is

 x  
2 d2r F
fc(2r ) (α) = cos(αx) dx
π 0 dx2r
 (3)
2
r −1
=− (−1) n a2r −2n−1 α 2n + (−1)r α 2r fc (α)
π n=0

where limx→∞ dr F /dxr = ar, makes it useful in the solution of many problems.
Under the same conditions.
 x
2
fs (α) = F (x) sin(αx) dx
π 0

defines the Fourier sine transform of F (x), and

 x
2
F (x) = fs (α) sin(αx) dα
π 0

Corresponding to (3) we have

 ∞
2 d2r F
fs(2r ) (α) = sin(αx) dx (4)
π 0 dx2r

2
r
= − (−1) n α 2n−1 a2r −2n + (−1)r −1 α 2r fs (α)
π n=1

Similarly, if F (x) is defined for −∞ < x < ∞, and if ∫∞

−∞ F (x)dx is absolutely convergent, then

∞
1
f (α) = √ F (x)eiax dx
2π −∞

is the Fourier transform of F (x), and

∞
1
F (x) = √ f (α)e−iax dα
2π −∞

Also, if

 n 
d F 
lim  n  = 0 (n = 1, 2, . . . , r − 1)
|x|→∞ dx

then

∞
1
f (r ) (α) = √ F (r ) (x)eiαx dx = (−iα)r f (α)
2π −∞
A-64 The Fourier Transforms

Finite Sine Transforms

π fs (n) F (x)
1 fs (n) = 0
F (x) sin nx dx (n = 1, 2, . . . ) F (x)

2 (−1) n+1
fs (n) F (π − x)
1 π−x
3 n π

(−1) n+1 x
4 n π

1−(−1) n
5 n
1
⎧
⎪
⎪ when 0 < x < π/2
⎨x
2
6 sin nπ
n2 ⎪
⎩π − x when π/2 < x < π
2
⎪
(−1) n+1 x(π 2 −x2 )
7 n3 6π

1−(−1) n x(π−x)
8 n3 2

π 2 (−1) n−1 2[1−(−1) n ]

9 n
− n3
x2
 2

π
10 π(−1) n 6
n3
− n
x3

11 n
n2 +c2
[1 − (−1) n ecπ ] ecx
n sinh c(π−x)
12 n2 +c2 sinh cπ

fs (n) F (x)
13 n
n2 −k2
(k = 0, 1, 2, . . . ) sin k(π−x)
sin kπ
⎧
⎨ π2 when n = m
14 (m = 1, 2, . . . ) sin mx
⎩
0 when n = m
n
15 [1 − (−1) n cos kπ] cos kx
n2 − k2
⎧ (k = 1, 2, . . . )
⎪ n
⎨ n2 −m2 [1 − (−1) n+m]
16 when n = m = 1, 2, . . . cos mx
⎪
⎩
0 when n = m
π sin kx
17 n
(n −k )
2 2 2 (k =
 0, 1, 2, . . .) 2k sin2 kπ
− x cos k(π−x)
2k sin kπ
bn
18 n
(|b| ≤ 1) 2
π
b sin x
arctan 1−b cos x
1−(−1) n n
19 n
b (|b| ≤ 1) 2
π
arctan 2b sin x
1−b 2
The Fourier Transforms A-65

Finite Cosine Transforms

π fc (n) F (x)
1 fc (n) = 0 F (x) cos nx dx (n = 0, 1, 2, . . . ) F (x)
2 (−1) n fc (n) F (π − x)
3 0 when n = 1, 2, · · · ; fc (0) = π 1
#
1 when 0 < x < π/2
4 2
sin nπ ; fc (0) = 0
n 2
−1 when π/2 < x < π
n 2
5 − 1−(−1)n2
; fc (0) = π2 x
(−1) n 2 x2
6 n2
; fc (0) = π6 2π
(π−x) 2 π
7 n12 ; fc (0) = 0 2π
− 6
n n 4
8 3π 2 (−1) n2
− 6 1−(−1)
n4
; fc (0) = π4 x3
(−1) n ec π −1 1 cx
9 n2 +c2 c
e
1 coshc(π−x)
10 n2 +c 2 csinhcπ
k
11 [( − 1) n cos π k − 1](k = 0, 1, 2, · · · ) sin kx
n2 − n+m
k2
−1
12 (−1) n2 −m2
; fc (m) = 0 (m = 1, 2, · · · ) 1
m
sin mx
13 # 1
n2 −k2
(k = 0, 1, 2, . . . ) − cosk sin
k(π−x)
kπ
0 for n = 1, 2, · · · ; n = m
14 π
cos mx for m = 1, 2, 3, . . .
2
for n = m

Fourier Sine Transforms

# F (x) fs (α)
1 (0 < x < a) 2
 1−cos α 
1 π α
0 (x > a)
( p) pπ
2 x p−1 (0 < p < 1) 2
sin
# π αp 2
sin x (0 < x < a)  
sin[a(1−α)] sin[a(1+α)]
3 √1 −
0 (x > a) 2π 1−α 1+α
 
α
4 e−x 2
π 1+α 2
2 /2 2
5 xe−x αe  −α /2
 2  2 ∗
x2
√ 2 2
6 cos 2 sin α2 C α2 − cos α2 S α2
√   2  2 ∗
2
x2 2 2
7 sin 2
2 cos α2 C α2 + sin α2 S α2

Here C( y) and S( y) are the Fresnel integrals:

y y
1 1 1 1
C( y) = √ √ cos t dt, S( y) = √ √ sin t dt
2π 0 t 2π 0 t
*More extensive tables of the Fourier sine and cosine transforms can be found in Fritz Oberhettinger, Tabellen zur-Fourier Trans-
formation, Springer, 1957.
Fourier Cosine Transforms

# F (x) fc (α)
1 (0 < x < a) 2 sin aα
1 π α
0 (x > a)
2 ( p) pπ
2 x p−1 (0 < p < 1) cos
# π αp 2
cos x (0 < x < a)  
sin[a(1−α)] sin[a(1+α)]
3 √1 +
0 (x > a) 2π 1−α 1+α
 
4 e−x π
2 1
1+α 2
2 /2 −α 1 /2
5 e−x e  
x2 α2 π
6 cos cos −
2
 2 4

x2 α2 π
7 sin 2
cos 2
+ 4
 Fourier Transforms
F (x) !
π f (α)
sin ax 2
|α| < a
1
!
x 0 |α| > a
eiwx ( p < x < q) √i e
i p(w+α) −eiq(w+α)
2
! 0−cx+iwx (x < p, x > q) 2π (w+α)

e (x > 0)
3 (c > 0) √ i
0 (x < 0) 2π(w+α+ic)

√1 e−α
2 2 /4 p
4 e− px R( p) > 0
2p  
√1 cos α2 π
5 cos px2 4p
− 4
2p  
√1 cos α2 π
6 sin px2 4p
+ 4
2p
(1− p) sin pπ
7 |x|− p (0 < p < 1) 2
π |α|(1− p)
2

√
√(a +α )+a
−a|x| 2 2
e√
8 |x| a 2 +α 2
a α
2 cos 2 cosh 2
9 cosh ax
cosh π x
(−π < a < π) π cosh α+cos a
10 sinh ax
(−π < a < π) √1 sin a
#
sinh π x 2π cosh α+cos a
√1 (|x| < a)
π
11 2 a −x
2
J 0 (aα)
2
0 (|x| > a) !
√
0
√ (|α| > b)
√ a +x ]
2 2
12
sin[b

π
! a 2 +x2 2
J 0 (a b2 − α 2 ) (|α| < b)
pn (x) (|x| < 1) in
13 √ J n+ 1 (α)
⎧ 0 (|x| > 1) α 2
√
⎨ cos[b a 2 −x2 ]
√
π √
(|x| < a)
14 a −x
2 2 J 0 (a a 2 + b2 )
⎩ 0 (|x| > a)
2
⎧ √
⎨ cosh[b a −x ]
√
2 2

π √
(|x| < a)
15 2 a −x
2 J 0 (a α 2 − b2 )
⎩ 0 (|x| > a)
2

*More extensive tables of Fourier transforms can be found in W. Magnus and F . Oberhettinger, Formulas and Theorems of the
Special Functions of Mathematical Physics. Chelsea, 1949, 116–120.

SERIES EXPANSION
The expression in parentheses following certain of the series indicates the region of convergence. If not otherwise indicated it is to
be understood that the series converges for all finite values of x.

Binomial Series
n(n − 1) n−2 2 n(n − 1)(n − 2) n−3 3
(x + y) n = xn + nxn−1 y + x y + x y + · · · ( y2 < x2 )
2! 3!
n(n − 1)x2 n(n − 1)(n − 2)x3
(1 ± x) n = 1 ± nx + ± + · · · (x2 < 1)
2! 3!
n(n + 1)x2 n(n + 1)(n + 2)x3
(1 ± x) −n = 1 ∓ nx + ∓ + · · · (x2 < 1)
2! 3!
(1 ± x) −1 = 1 ∓ x + x2 ∓ x3 + x4 ∓ x5 + · · · (x2 < 1)
(1 ± x) −2 = 1 ∓ 2x + 3x2 ∓ 4x3 + 5x4 ∓ 6x5 + · · · (x2 < 1)

Reversion of Series
Let a series be represented by
y = a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + · · ·
with a1 = 0. The coefficients of the series
x = A1 y + A2 y2 + A3 y3 + A4 y4 + · · ·

A-66