18.03 Lecture #20 Oct. 23, 2009: notes
Topic for today is an introduction to Fourier series.
What Fourier series do is express a (complicated) general function as an (infinite) sum of simple
functions. In order to talk about the philosophical underpinnings this idea, begin with a theory
that you have seen before: Taylor series.
So suppose
g
is a function of the real variable
x
. The theory of Taylor series is concerned with
the following pieces of information about
g
:
0) the value
g
(0) of
g
at the origin;
1) the first derivative
g
′
(0) of
g
at the origin;
2) the second derivative
g
′′
(0) of
g
at the origin;
and so on. Taylor series use a collection of very special functions:
0) the constant function 1, whose value at the origin is 1, and whose higher derivatives are all 0
at the origin;
1) the function
x
, whose value at the origin is 0, whose first derivative at the origin is 1, and
whose higher derivatives at the origin are all 0;
2) the function
x
2
/
2!, whose value and first deriviative at the origin are 0; whose second derivative
at the origin is 1; and whose higher derivatives at the origin are all 0;
and so on. The
Taylor series of
g
is the infinite series
g
(0)
·
1 +
g
′
(0)
·
x
+
g
′′
(0)
·
x
2
/
2! +
· · ·
(Taylor)
If you compare the two lists above, you’ll see that
the Taylor series of
g
is designed to have the
same values and derivatives as
g
at the origin
.
Of course it often (but not always!) happens that the Taylor series of
g
converges, and that the
sum is equal to
g
. You may like to think of Taylor series as a tool for computing
g
, but that isn’t
always the best point of view. What is more fundamental is that replacing
g
by its Taylor series
(or by a few terms of that series) focuses attention on the derivatives of
g
at the origin, throwing
away extraneous complications.
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 Fall '09
 unknown
 Derivative, Taylor Series, Fourier Series, Periodic function

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