2. Fourier Series
• As we know that the fourier series of function f(x) in any interval (-l, l) is given by:
• 𝑓 𝑥 = 𝑎0 + 𝑛=1
∞
𝑎 𝑛 cos
𝑛𝜋𝑥
𝐿
+ 𝑏 𝑛 sin
𝑛𝜋𝑥
𝐿
• Where:-
• 𝑎0 =
1
2𝑙 −𝑙
𝑙
𝑓 𝑡 𝑑𝑡
• 𝑎 𝑛=
1
𝑙 −𝑙
𝑙
𝑓 𝑡 𝑐𝑜𝑠
𝑛𝜋𝑡
𝑙
𝑑𝑡
• 𝑏 𝑛=
1
𝑙 −𝑙
𝑙
𝑓 𝑡 𝑠𝑖𝑛
𝑛𝜋𝑡
𝑙
𝑑𝑡
3. Fourier Integral
• Let f(x) be a function which is piecewise continuous in every finite interval in
(−∞, ∞) and absolute integral in (−∞, ∞).
• Then 𝑓 𝑥 =
1
𝜋 0
∞
( −∞
∞
𝑓 𝑡 𝑐𝑜𝑠𝜔 𝑡 − 𝑥 𝑑𝑡)𝑑𝜔
• Where :
• 𝜔 =
𝑛𝜋
𝑙
• 𝑙 → ∞
7. Fourier cosine integrals
• When 𝑓(𝑥) is an even function:
• 𝐴 𝜔 =
2
𝜋 0
∞
𝑓 𝑡 𝑐𝑜𝑠𝜔𝑡𝑑𝑡 and B 𝜔 = 0
• So the fourier integrals of an even function is given by:
• 𝑓(𝑥) = 0
∞
𝐴 𝜔 𝑐𝑜𝑠𝜔𝑥𝑑𝜔
8. Fourier sin integral
• When 𝑓(𝑥) is an odd function:
• 𝐴 𝜔 = 0 and B 𝜔 =
2
𝜋 0
∞
𝑓 𝑡 𝑠𝑖𝑛𝜔𝑡𝑑𝑡
• So the fourier integral of odd function is given by:
• 𝑓(𝑥) = 0
∞
𝐵 𝜔 𝑠𝑖𝑛𝜔𝑥𝑑𝜔
9. Fourier cosine sum
• Find the fourier cosine integral of 𝒇 𝒙 = 𝒆−𝒌𝒙, where 𝒙 > 𝟎, 𝒌 > 𝟎 hence show that
𝟎
∞ 𝒄𝒐𝒔𝝎𝒙
𝒂 𝟐+𝝎 𝟐 𝒅𝝎 =
𝝅
𝟐𝒂
𝒆−𝒂𝒙
The fourier cosine integral of 𝑓 𝑥 is given by:
𝑓 𝑥 =
0
∞
𝐴 𝜔 𝑐𝑜𝑠𝜔𝑥𝑑𝜔
𝐴 𝜔 =
2
𝜋 0
∞
𝑓 𝑡 𝑐𝑜𝑠𝜔𝑡𝑑𝑡
=
2
𝜋 0
∞
𝑒−𝑘𝑡 𝑐𝑜𝑠𝜔𝑡𝑑𝑡
=
2
𝜋
𝑒−𝑘𝑡
𝑘2 + 𝜔2
(−𝑘𝑐𝑜𝑠𝜔𝑡 + 𝜔𝑠𝑖𝑛𝜔𝑡 (𝑓𝑟𝑜𝑚 0 𝑡𝑜∞)
=
2
𝜋
(
𝑎
𝑎2 + 𝜔2)
11. Fourier sine integral sum
• Find the sine integral of 𝑓 𝑥 = 𝑒−𝑏𝑥
, hence show that
𝜋
2
𝑒−𝑏𝑥
=
0
∞ 𝜔𝑠𝑖𝑛𝜔𝑥
𝑏2+𝜔2 𝑑𝜔
The fourier sine integral of 𝑓 𝑥 is given by:
𝑓(𝑥) =
0
∞
𝐵 𝜔 𝑠𝑖𝑛𝜔𝑥𝑑𝜔