An Etymological Dictionary of Astronomy and Astrophysics

One of the following conditions for a → Fourier series to converge:

1. The function f(x) is defined and single valued, except possibly at a finite number of points in the interval -π, +π.
   

2. f(x) has a period of 2π.
   

3. f(x) and f’(x) are → piecewise continuous functions on -π, +π.
   

Then, the Fourier series converges to:

(a) f(x) if x is a point of continuity.

(b) (f(x + 0) + f(x - 0))/2, if x is a point of discontinuity.

See also: Named after Peter Gustav Lejeune Dirichlet (1805-1859), German mathematician who made valuable contributions to → number theory, → analysis, and → mechanics;
→ condition.