A function of two variables may be expanded in a double Fourier series, as a function of one variable is expanded in an ordinary Fourier series. Purpose that the function
f
(
x, y
) possesses a double Lebesgue integral over the square (–
π
<
π
; –
π
<
y
<
π
). Then the general term of the double Fourier series of this function is given by cos =
є
mn
{
a
mn
cos
mx
cos
ny
+
b
mn
sin
mx
sin
ny
+
c
mn
cos
mx
sin
ny
+
d
mn
sin
mx
cos
ny
} There
є
00
= ¼,
є
m0
= ½ (
m
> 0),
є
0n
= ½ (
n
> 0),
є
ms
= 1 (
m
> 0,
n
>0). the coefficients are given by the formulæ
a
mn
= 1/
π
2
∫
π
-π
∫
π
-π
f
(
x, y
) cos
mx
cos
ny
dx dy
, obtained by term-by-term integration, as in an ordinary Fourier series. Ti sum of a finite number of terms of the series may also be found as in the ordinary theory. Thus ∫
ms
= Σ
m
μ = 0
Σ
n
v
= 0
A
μ
v
= 1/π
2
∫
π
-π
∫
π
-π
f
(s, t) sin(
m
+½) (
s
-
x
) sin (
n
+ ½) (
t
-
y
)/2 sin ½ (
s
-
x
) 2 sin ½ (
t
-
y
) if
f
(
s
,
t
) is defined outside the original square by double periodicity, we have sub S
ms
= 1/π
2
∫
π
0
∫
π
0
f
(
x
+
s
,
y
+
t
) +
f
(
x
+
s
,
y
-
t
) +
f
(
x
-
s
,
y
+
t
) +
f
(
x
-
s
,
y
-
t
) sin (
m
+ ½)
s
/ 2 sin ½
s
sin (
n
+ ½)
t
/ 2 sin ½
t
ds dt
.
The double Fourier series of a discontinuous function