303 ME Advanced Engineering Mathematics Fourier Cosine and Sine Series
M.M. Yovanovich
Fourier series. The Fourier series of a periodic function f x with period 2L
is de ned as the trigonometric series
1 X 1 nx + X B sin nx f x = A0 + An cos L n L n=1 n=1
Many physical problems in engineering and science are solved by the use of 1
where the Fourier coe cients A0; An; Bn are given by ZL A0 = 21 L ,L f x dx 1 Z L f x cos nx dx An = L L ,L and 1 Z L f x sin nx dx Bn = L L ,L The kth term of the Fourier cosine and sine series are ! ! kx and B sin kx Ak cos L k L The kth partial sum of the Fourier series is
k nx + X B sin nx f x A0 + An cos L n L n=1 n=1 k X
2 3 4 5
6
Othogonality of Cosine and Sine Functions
n
o1 The family of cosine functions cos nx n=0 L relation 8 ZL mx cos nx dx = 0; L; cos L 2 L 0 : L;
satis es the orthogonality
m 6= n m = n 6= 0 m=n=0
7
n
o1 The family of sine functions sin nx n=1 satis es the orthogonality reL lation ZL mx sin nx dx = 0; m 6= n sin L 8 L; m = n L 0 2 The following properties of cosine and sine functions when n = 0; 1; 2; 3 : : : can be used to simplify the Fourier coe cients:
sinn = 0 and sin,n = 0 and cosn = ,1n and cos,n = 0 n 6= 0
9 10
Even and Odd Functions
The Fourier series expansion can be obtained with less e ort if we take advantage of the even or odd property of the function. An even function requires that f ,x = f x 11 which means that the graph of the function is symmetric with respect to vertical the axis. For an odd function,
f ,x = ,f x
12
The cosine function is an even function, whereas the sine function is an odd function. If f x is an even function,
ZL
L
f x dx = 2
ZL
ZL
0
f x dx
13
If f x is an odd function,
,L
f x dx = 0
14
The product of even and odd functions is and odd function, whereas the products of two even functions or two odd functions produce even functions. 2
An even function f x has the Fourier expansion 1 X f x = A0 + An cos nx ; 0 x L 15 L n=1 where 2 Z L f x dx; 2 Z L f x cos nx dx; A0 = L 0 An = L 0 n = 1; 2; 3; L 16 For an odd function f x, the Fourier series has the form 1 X nx ; 0 x L f x = Bn sin L 17 n=1 where 2ZL Bn = L f x sin nx dx; n = 1; 2; 3; 18 L
The rst example is the periodic sawtooth wave de ned as f x = x; ,L x L with the periodic condition f x + 2L = f x This function, as de ned, is an odd function. The Fo...