∫ ba ƒ(x)dx = Σbr=0 ∫ ba ƒr(x)dx,
or a series which converges uniformly may be integrated term by term.
(6) If Σ∞r=0 ƒ′r(x) converges uniformly in an interval, then Σ∞r=0 ƒr(x) converges in the interval, and represents a continuous differentiable function, φ(x); in fact we have
φ′(x) = Σ∞r=0 ƒ′r(x),
or a series can be differentiated term by term if the series of derived functions converges uniformly.
A series whose terms represent functions which are not continuous throughout an interval may converge uniformly in the interval. If Σ∞r=0 ƒr(x) = ƒ(x), is such a series, and if all the functions ƒr(x) have limits at a, then ƒ(x) has a limit at a, which is Σ∞r=0 Lt x=a ƒr(x). A similar theorem holds for limits on the left or on the right.
23. Fourier’s Series.—An extensive class of functions admit of being represented by series of the form
| a0 + Σ∞n=1 ( an cos | nπx | + bn sin | nπx | ), |
| c | c |
and the rule for determining the coefficients an, bn of such a series, in order that it may represent a given function ƒ(x) in the interval between −c and c, was given by Fourier, viz. we have
| a0 = | 1 | ∫ c−c ƒ(x)dx, an= | 1 | ∫ c−c ƒ(x)cos | nπx | dx, bn= | 1 | ∫ c−c sin | nπx | dx. |
| 2c | c | c | c | c |