NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA
1505 Mathematics
I
Revision Notes (Fourier Series and Vector Spaces)
5
Fourier Series
Lecture Notes and Tutorial Question Review
1.
The Formulas of Fourier Series.
There are two kinds of formulas: one is for period 2
π
,theotherper
iod2
L
. The later one is
much more general, and enough for all problems.
f
(
x
)=
a
0
+
∞
±
n
=1
(
a
n
cos
nπ
L
x
+
b
n
sin
nπ
L
x
)
,
and the coeﬃcients are given by the
Euler’s formulas
:
a
0
=
1
2
L
²
L
−
L
f
(
x
)
dx
a
n
=
1
L
²
L
−
L
f
(
x
)cos
nπ
L
xdx
n
=1
,
2
,
···
b
n
=
1
L
²
L
−
L
f
(
x
)sin
nπ
L
n
,
2
,
The diﬃcult part for fourier series is integration.
2.
Trigonometric Functions.
(a) The graph of cosine and sine functions.
(b) Some special values of those function, for example sin
nπ
2
.
(c) Some trigonometric identities.
(d) Integration of these functions. (can use identities or integration by parts)
3.
Parity.
You need to know:
(a) The advantage to consider parity Frst;
(b) When
f
is an even
function, which series will be applied?
4.
Periodicity and Continuity.
By deFnition, it is obvious that only periodic and piecewise continuous functions have their
±ourier series.
However, for some kinds of partial functions, I mean a function is deFned on an interval, but
not deFned on the other parts, then we can write out the ±ourier series of the function by
using
halfrange expansions
. Check details in Tutorial 4 Question 6.
Also check tutorial notes 4 for more details about the advantage of halfrange expansion
compared with other expansions.
121
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Vector Spaces
1.
Arithmetic
I skip the details here, since they are not diﬃcult.
You need to be very familiar with them.
2.
Magnitude (Length)
Given a vector
u
=
u
1
i
+
u
2
j
+
u
3
k
in 3dimensional space, the length of the vector is given
by
±
u
±
=
±
u
2
1
+
u
2
2
+
u
2
3
.
Remark
:
There are two kinds of notations for magnitude:
±±
and

.(Th
ef
o
rm
e
ri
su
s
ed
in your lecture notes; and the latter one is used in your textbook.)
What’s the diFerence? No diFerence for 3dimensional spaces.
However,
is more general, and it can be use in all cases to denote the length.
is
used only for some special spaces, for example, 3dimensional spaces, 4dimensional
spaces,
n
dimensional spaces.
Application
:
Magnitude of a vector (like the module of a real number) does not have some
good properties, for example, the function is not diFerentiable. Check Tutorial 5
Question 6.
Some times we need to remove the magnitude by using the following property:
±
u
±
2
=
u
·
u
.
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 Spring '09
 MA1505
 Fourier Series, Vector Space, Dot Product, 3dimensional space

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