Next: Dirac's delta function and Up: No Title Previous: Borel Summation

# Convergence of the Fourier series

Here we shall show why the Fourier series (4-1) converges to . Let us consider the sum

 (68)

where An and Bn are defined by eqs. (4-2). Inserting these definitions we get

 = = (69)

The sum over the cosines can be performed,

 = = = (70)

Inserting this result we get by use of the substitution

 (71)

that the sum SN reduces to

 (72)

. Taking we can rewrite this integral as
 = + (73)

Remembering that and taking N to be large, we arrive at the result

 (74)

Here we used that for as well as the value obtained in eq. (3-11) in the book.

The result is thus that the sum is equal to , as mentioned in the book.

Next: Dirac's delta function and Up: No Title Previous: Borel Summation
Mette Lund
2000-04-27