(1)

This method of determining the coefficients in the series would not be valid without the assumption that the series is in general uniformly convergent, for in accordance with a known theorem the sum of the integrals of the separate terms of the series is otherwise not necessarily equal to the integral of the sum. This assumption being made, it is further assumed that ƒ(x) is such that ∫ l 0 ƒ(x)sin (nπx /l) dx has a definite meaning for every value of n.

Before we proceed to examine the justification for the assumptions made, it is desirable to examine the result obtained, and to deduce other series from it. In order to obtain a series of the form

B0 + B1 cos πx+ B2 cos 2πx+ ... + Bn cos nπx+ ...
l ll

for the representation of ƒ(x) in the interval o to l, let us apply the series (1) to represent the function ƒ(x) sin (πx / l); we thus find

2Σ∞ 1 sin nπx∫ l 0 ƒ(x) sin πxsin nπxdx,
l ll l

or

1Σ∞ 1 sin nπx∫ l 0 ƒ(x) { cos (n − 1) πx− cos (n + 1) πx} dx.
l ll l

On rearrangement of the terms this becomes

1sin πx∫ l 0 ƒ(x) dx + 2Σ sin πxcos nπx∫ l 0 ƒ(x) cos nπxdx.
l ll ll l