Monday, April 4, 2022

Revisiting Complex Fourier Series

  The Fourier series can easily be represented in complex form, which I examine in this post.  To begin with, we consider representing the Fourier trig function in complex form, e.g.

cos npx/ L    = [exp inpx/ L  +   exp - inpx/ L] / 2

sin  npx/ L    = [exp inpx/ L  -   exp - inpx/ L] / 2i  

And one complex form for the Fourier series is:

f(x) =     å ¥-¥   n  exp i2px/ L 

And the Fourier coefficient can be written:

n  =  1/2 (n    -   i b n )    For n >  0

n  =  1/2 (n    +    i b n )    For n <   0

n    =  o  / 2  

 The basic complex Fourier series integral can then be written:

ò -L  +L   exp inpx/ L  exp - impx/ L  dx   =    ò -L +L    exp  i(np  -mp)x/ L  dx

And the functions are orthogonal on the interval (-L < x L)  and we obtain for the Fourier coefficient:

n  =  1/2L  ò -L  +L   f(x) exp (- inpxL )  dx


Problems for the math whiz:  

1) For the function:

f(t)  =   {0    for  h <  t <   2p    

         =   {1  for    0  <  t  <   h 

Over 0 < t   <    2 p        h >  0  

Give a simplified expression for the associated Fourier coefficient if:    n  =  1/2 p  ò 2p  0   f(t)  exp (-int) dt  

1/ 2p ò 2p  h    e - int  dt  

2) Let k =  k1 + ik2 be a complex variable.  Let f(x) be a real-valued function of a real variable x and define:

F + (k)  =   Ö (2/  p )    ò ¥ 0    f(x) exp (ikx) dx

Find F + (k)    for:   x N e -x    

(N an integer)

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