Jan 7.
Fourier Methods  What, How, and Why?
In the first decade of the 19th century, Jean Baptiste Joseph Fourier invented a technique
using sums of trigonometric functions called ``Fourier Series'' to solve the differential
equations of heat conduction. Today, physicists, astronomers, engineers, and many others
use Fourier series to analyze functions of time and space, solve differential equations, and
compress data, to name just a few applications.
Periodic functions
Definition.
f
(
x
)
is
periodic
with
period
p
if
f
(
x+p
) =
f
(
x
), for all
x
.
For example.
sin
x
and cos
x
both have period 2
π
,
sin
λ
x
and cos (
λ
x+
α
)
both have period 2
π
/
λ
,
pulse functions are also periodic.
Definition.
Given any
f
(
x
) defined for
a<x<b
, we can define its periodic extension to all
x
by
f
(
x+nT
) =
f
(
x
),
n = 0,
±
1,
±
2, ...,
a<x<b.
The extended function is not yet defined at
x=a+nT
; if
f
(
a
)
≠
f
(
b
), it will be
discontinuous at
x=a+nT
. For allowing functions with discontinuities at any point
x= x
0
,
we introduce
righthand limit:
f
(
x
0
+0) = lim
f(x
0
+
ε
2
)
ε→
0
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 Spring '10
 Luu
 Differential Equations, Calculus, Equations, Fourier Series, Limits, Cos, Periodic function, Jean Baptiste Joseph Fourier

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