# 18.03 Topic22 Sine and cosine series; calculation tricks. Read

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```					18.03 Topic 22: Sine and cosine series; calculation tricks.
Read: EP §8.3 (up to subsection ’Fourier Series Solutions of DE’s’).

Calculation shortcuts
1. Use even-odd as discussed before.
2. Make new series from old ones.
3. Use diﬀerentiation and integration.
Even and odd functions
Example 1: Used this last time to compute the Fourier series for the period 2π
square wave and period 2π continuous sawtooth
4            1          1
π           = f (t) =     sin(πt) + sin(3πt) + sin(5πt) + · · · (square
π            3          5
wave)
cc     c     c
c  cc  cc 
c  cc  cc 
c         
π   4               cos 3t cos 5t
= g(t) =     −       cos t +         +       + · · · . (sawtooth or
−2π       2π                      2 π                   32     52
triangular wave)

New series from old ones –shifting and scaling
1                          4         sin nt
f (t) = square wave =              π         t    =                     .
2π             π           n
n odd

2

˜                                    ˜                       4             sin nt
f (t) =            π        t      ⇒ f (t) = 1 + f (t) = 1 +                      .
2π                                    π               n
n odd

2

˜                                    ˜                8            sin nt
f (t) =            π        t      ⇒ f (t) = 2f (t) =                     .
2π                             π              n
n odd
−2
1

˜                                    ˜      1             1 2                     sin nt
f (t) =            π        t      ⇒ f (t) = (1 + f (t)) = +                             .
2π                   2             2 π                       n
n odd

shift and scale time:
1                                           4            sin nπt
˜
f (t) =                     t        ˜
⇒ f (t) = f (πt) =                      .
1    2                               π               n
−1                                               n odd

(continued)

1
18.03 topic 22                                                                                                2

1
˜                                                      ˜                     4                   sin n(t + π/2)
f (t) =                                       t      ⇒ f (t) = f (t + π/2) =                                    .
−π/2         π/2                                         π             n odd
n
−1
˜       4                            sin(3t + 3π/2)                    4              cos 3t
⇒ f (t) =            sin(t + π/2) +                   + ...          =        cos t −          + ... .
π                                   3                          π                3
Diﬀerentiation and integration
Can diﬀerentiate term-by-term.
Example: Let f (t) be the continuous sawtooth from example 1 it’s derivative is the
square wave.
π   4           cos 3t cos 5t
f (t) = −      cos t +        +      + ...
2 π               32      52
4          sin 3t sin 5t
⇒ f (t) =     sin t +        +      + ... .
π             3       5
Note f (t) has a corner and its coeﬃcients decay like 1/n2 . f (t) has a jump and and
its coeﬃcients decay like 1/n.
Can also integrate term-by-term.
We will do examples of integration and diﬀerentiation of discontinuous functions next
time.

Even and odd extensions, sine and cosine series
Suppose f (t) is deﬁned on the interval [0, L], we give a graph as an example. We will
need functions like this when we study the wave and heat equations.
R
oo
o o RRR
RR
f (t) =                t
L
Emphatically: this is not periodic since it is only deﬁned on an interval.
So, it doesn’t have a Fourier series. But, it does have periodic ’extensions.
Periodic extension:              period = L, agrees with f (t) on [0, L].
mU         mU         mU
mm UU      mm UU      mm UU
mm     UU mm      UU mm      UU
UU         UU         UU
U          U          U
t
−L                      L         2L
Even periodic extension:                 period = 2L, even, agrees with f (t) on [0, L].
      mUU                  mUU                  mUU
§§ mmmm UU          §§ mmmm UU          §§ mmmm UU
§§§              UU     §§              UU     §§              UU
UU §§                  UU §§                  UU
˜                 §§§                     §§                     §§
fe (t) =                                                                                        t
−L                      L             2L

(continued)
18.03 topic 22                                                                                          3

Odd periodic extension:               period = 2L, odd, agrees with f (t) on [0, L].
mU                       mU                       mU
mm UU                    mm UU                    mm UU
mm     UU                mm     UU                mm     UU
UU                       UU                       UU
U                        U                        U
˜
fo (t) =     UU                      UU                       UU                       UU       t
UU                      UU                       UU                       UU
UU               −L U                      L U          2L                UU
UU mmm                 UU     m                 UU     m                  UU mmm
U mmm                    U mmm
mm                      m                        m                        mm

˜                             ˜       A0                  π
fe is periodic and even ⇒ fe (t) =         +       An cos    nt .
2                  L
˜                            ˜                     π
fo is periodic and odd ⇒ fo (t) =         Bn sin      nt .
L
˜        ˜
On [0, L] they all agree: f (t) = fe (t) = fo (t).
A0                  π           ˜                 π
⇒ on [0, L], f (t) =       +     An cos      nt . = fo (t) =      Bn sin   nt .
2                   L                             L
These are the Fourier cosine and Fourier sine series for f (t). (Emphatically, the
series only agree with f (t) on [0, L].)
Computing sine and cosine series
Directly from the deﬁnition of Fourier coeﬃcients and the integration of even and
odd functions we get:
L                                           L
2                      π                    2                      π
An =                f (t) cos( nt) dt,    Bn =                  f (t) sin( nt) dt
L    0                 L                    L    0                 L

Important:
1. Sine and cosine series are about functions deﬁned on an interval.
2. The sine and cosine series agree with f on (0, L) (assuming f is continuous).
Since f is only deﬁned on [0, L] this is usually what we want.
˜       ˜
3. Repeating (2): fe (t) = fo (t) for 0 < t < L.
4. Computing An and Bn only depends on f .
5. We will make use of sine and cosine series when we do the wave equation.
Example: (Computing sine and cosine series.) Let f (t) = sin t on [0, π]. Find its
Fourier sine and cosine series.
Sine series: f (t) = sin t on [0, π].
 4
 π        for n = 0
2 π                    
Cosine series: L = π, An =            sin t cos nt dt =   0        for n = odd
π 0                      −4

π(n2 −1)
for even n > 0
2   4      cos 2t cos 4t cos 6t                             2   4               cos nt
⇒ sin(t) =     −              +      +       + ...                  =      −                        .
π π          3     15     35                                π π       n>0, even
n2 − 1
(Only on [0, π].)

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