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 ^¥_-¥   c _n  exp i2px/ L 

And the Fourier coefficient can be written:

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

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

c _n    =  a _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:

c _n  =  1/2L  ò _-L  ^+L   f(x) exp (- inpx/ L )  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:    c _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)}