Page 347 - Schaum's Outline of Theory and Problems of Advanced Calculus

338 FOURIER SERIES [CHAP. 13

(3) f ðxÞ and f ðxÞ are piecewise continuous in ð L; LÞ.
0
Then the series (1) with Fourier coefficients converges to
f ðxÞ if x is a point of continuity
ðaÞ
if x is a point of discontinuity
f ðx þ 0Þþ f ðx 0Þ
2
ðbÞ
Here f ðx þ 0Þ and f ðx 0Þ are the right- and left-hand limits of f ðxÞ at x and represent lim f ðx þ Þ and
lim f ðx Þ, respectively. For a proof see Problems 13.18 through 13.23. !0þ
!0þ
The conditions (1), (2), and (3) imposed on f ðxÞ are sufficient but not necessary, and are generally
satisfied in practice. There are at present no known necessary and sufficient conditions for convergence
of Fourier series. It is of interest that continuity of f ðxÞ does not alone ensure convergence of a Fourier
series.

ODD AND EVEN FUNCTIONS
5
3
3
A function f ðxÞ is called odd if f ð xÞ¼ f ðxÞ. Thus, x ; x 3x þ 2x; sin x; tan 3x are odd
functions.
6
4
x
2
A function f ðxÞ is called even if f ð xÞ¼ f ðxÞ. Thus, x ; 2x 4x þ 5; cos x; e þ e x are even
functions.
The functions portrayed graphically in Figures 13-1(a)and 13-1ðbÞ are odd and even respectively,
but that of Fig. 13-1(c)is neither odd nor even.
In the Fourier series corresponding to an odd function, only sine terms can be present. In the
Fourier series corresponding to an even function, only cosine terms (and possibly a constant which we
shall consider a cosine term) can be present.
HALF RANGE FOURIER SINE OR COSINE SERIES
A half range Fourier sine or cosine series is a series in which only sine terms or only cosine terms are
present, respectively. When a half range series corresponding to a given function is desired, the function
is generally defined in the interval ð0; LÞ [which is half of the interval ð L; LÞ, thus accounting for the
name half range] and then the function is specified as odd or even, so that it is clearly defined in the other
half of the interval, namely, ð L; 0Þ.In such case, we have
8 ð L
2 n x
>
> a n ¼ 0; f ðxÞ sin dx for half range sine series
> b n ¼
< L
L 0
2 ð L n x ð4Þ
>
>
> f ðxÞ cos dx for half range cosine series
: b n ¼ 0; a n ¼
L 0 L
PARSEVAL’S IDENTITY
If a n and b n are the Fourier coefficients corresponding to f ðxÞ and if f ðxÞ satisfies the Dirichlet
conditions.

1 ð L a 0 2 X
1
2
2
2
Then f f ðxÞg dx ¼ þ ða n þ b n Þ (5)
L L 2
n¼1
(See Problem 13.13.)