18.03 Topic22 Sine and cosine series; calculation tricks. Read

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18.03 Topic22 Sine and cosine series; calculation tricks. Read Powered By Docstoc
					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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