![FORMULA
The formula for a Fourier series on an interval [𝑐, 𝑐 + 2𝑙] is :
𝐹 𝑥 =
𝑎0
2
+
𝑛=1
∞
( 𝑎 𝑛 𝑐𝑜𝑠
𝑛𝜋𝑥
𝑙
+ 𝑏 𝑛 𝑠𝑖𝑛
𝑛𝜋𝑥
𝑙
)
Where,
𝑎 𝑛 =
1
𝑙 𝑐
𝑐+2𝑙
𝐹 𝑥 𝑐𝑜𝑠
𝑛𝜋𝑥
𝑙
𝑑𝑥
𝑏 𝑛 =
1
𝑙 𝑐
𝑐+2𝑙
𝐹 𝑥 𝑠𝑖𝑛
𝑛𝜋𝑥
𝑙
𝑑𝑥
𝑎0 =
1
𝑙 𝑐
𝑐+2𝑙
𝐹 𝑥 𝑑𝑥
And “𝑙” defines period, if period is specified then, period = 2𝑙
and if it is not then, the maximum limit will be the value of “𝑙” .](https://image.slidesharecdn.com/presentation-170731162516/85/Fourier-Series-Engineering-Mathematics-7-320.jpg)
Fourier Series - Engineering Mathematics
- 2. Welcome to our Presentation Topic : “FOURIER SERIES”
- 3. INDEX • Definition. • Conditions. • Formula. • Example. • Conclusion.
- 4. DEFINITION • FOURIER SERIES : Fourier Series is an infinite series representation of periodic function in terms of the trigonometric sine and cosine functions. • Most of the single valued functions which occur in applied mathematics can be expressed in the form of Fourier series, which is in terms of sines and cosines.
- 5. DEFINITION • Fourier series is to be expressed in terms of periodic functions- sines and cosines. Fourier series is a very powerful method to solve ordinary and partial differential equations, particularly with periodic functions appearing as non-homogeneous terms.
- 6. CONDITONS Let F(x) satisfy the following conditions : 1. F(x) is defined in the interval, c < x < c+2l. 2. F(x) and F’(x) sectionally continuous in c < x < c+2l. 3. F(x+2l) = F(x) i.e. F(x) is periodic with period 2l. If these 3 conditions remains, then we can say F(x) is Fourier series.
- 7. FORMULA The formula for a Fourier series on an interval [𝑐, 𝑐 + 2𝑙] is : 𝐹 𝑥 = 𝑎0 2 + 𝑛=1 ∞ ( 𝑎 𝑛 𝑐𝑜𝑠 𝑛𝜋𝑥 𝑙 + 𝑏 𝑛 𝑠𝑖𝑛 𝑛𝜋𝑥 𝑙 ) Where, 𝑎 𝑛 = 1 𝑙 𝑐 𝑐+2𝑙 𝐹 𝑥 𝑐𝑜𝑠 𝑛𝜋𝑥 𝑙 𝑑𝑥 𝑏 𝑛 = 1 𝑙 𝑐 𝑐+2𝑙 𝐹 𝑥 𝑠𝑖𝑛 𝑛𝜋𝑥 𝑙 𝑑𝑥 𝑎0 = 1 𝑙 𝑐 𝑐+2𝑙 𝐹 𝑥 𝑑𝑥 And “𝑙” defines period, if period is specified then, period = 2𝑙 and if it is not then, the maximum limit will be the value of “𝑙” .
- 8. FORMULA To do this math we need a shortcut formula, because we have trigonometric term in this formula. And we know that trigonometric term never ends.so we have to use this shortcut formula- 𝐹 𝑥 = 𝑢0 𝑣0 𝑑𝑥 ⇒ 𝑢0 𝑣0 − 𝐷𝑢0 𝑣1 + 𝐷𝑢1 𝑣2 − ⋯ 𝑢𝑛𝑡𝑖𝑙 0
- 9. EXAMPLE • Expand F(x) = 𝑥2; 0<x<2𝜋 and period = 2𝜋 𝑆𝑜𝑙 𝑛 : Here, period = 2𝜋 or, 2𝑙 = 2𝜋 or, 𝑙 = 𝜋 Now, 𝑎 𝑛 = 1 𝑙 𝑐 𝑐+2𝑙 𝐹 𝑥 𝑐𝑜𝑠 𝑛𝜋𝑥 𝑙 𝑑𝑥 ⇒ 1 𝜋 0 2𝜋 𝑥2 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥 ⇒ 1 𝜋 𝑥2 sin 𝑛𝑥 𝑛 − 2𝑥 𝑛 − cos 𝑛𝑥 𝑛 + 2 𝑛2 − sin 𝑛𝑥 𝑛 0 2𝜋 ⇒ 1 𝜋 𝑥2 𝑛 sin 𝑛𝑥 + 2𝑥 𝑛2 cos 𝑛𝑥 − 2 𝑛3 sin 𝑛𝑥 0 2𝜋 ⇒ 1 𝜋 4𝜋2 𝑛 sin 2𝑛𝜋 + 4𝜋 𝑛2 cos 2𝑛𝜋 − 2 𝑛3 sin 2𝑛𝜋 − 0 + 0 − 0 ⇒ 1 𝜋 0 + 4𝜋 𝑛2 − 0 ⇒ 4 𝑛2
- 10. EXAMPLE 𝑏 𝑛 = 1 𝑙 𝑐 𝑐+2𝑙 𝐹 𝑥 𝑠𝑖𝑛 𝑛𝜋𝑥 𝑙 𝑑𝑥 ⇒ 1 𝜋 0 2𝜋 𝑥2 . sin 𝑛𝑥 𝑑𝑥 ⇒ 1 𝜋 𝑥2 − cos 𝑛𝑥 𝑛 − 2𝑥 𝑛 − 𝑠𝑖𝑛 𝑛𝑥 𝑛 + 2 𝑛2 cos 𝑛𝑥 𝑛 0 2𝜋 ⇒ 1 𝜋 − 𝑥2 𝑛 𝑐𝑜𝑠 𝑛𝑥 + 2𝑥 𝑛2 sin 𝑛𝑥 + 2 𝑛3 cos 𝑛𝑥 0 2𝜋 ⇒ 1 𝜋 − 4𝜋2 𝑛 cos 2𝑛𝜋 + 4𝜋 𝑛2 sin 2𝑛𝜋 + 2 𝑛3 cos 2𝑛𝜋 − 0 + 0 + 2 𝑛3 ⇒ 1 𝜋 − 4𝜋 𝑛 + 0 + 2 𝑛3 − 2 𝑛3 ⇒ − 4𝜋 𝑛
- 11. EXAMPLE 𝑎0 = 1 𝑙 𝑐 𝑐+2𝑙 𝐹 𝑥 𝑑𝑥 ; ⇒ 1 𝜋 0 2𝜋 𝑥2 𝑑𝑥 ⇒ 1 𝜋 𝑥3 3 0 2𝜋 ⇒ 1 𝜋 8𝜋3 3 − 0 ⇒ 8𝜋2 3 So. The Fourier series is : 𝐹 𝑥 = 8𝜋2 3 2 + 𝑛=1 ∞ ( 4 𝑛2 𝑐𝑜𝑠 𝑛𝜋𝑥 𝑙 + − 4𝜋 𝑛 𝑠𝑖𝑛 𝑛𝜋𝑥 𝑙 ) ⇒ 4𝜋2 3 + 𝑛=1 ∞ ( 4 𝑛2 𝑐𝑜𝑠 𝑛𝜋𝑥 𝑙 − 4𝜋 𝑛 𝑠𝑖𝑛 𝑛𝜋𝑥 𝑙 ) (Answer :)
- 12. Conclusion Conclusions To continue researching Fourier Series there are a few areas and specific problems that we would address. Fourier is a lengthy math, So we have to be careful about the formula while doing this math.
- 13. Thank you all for having patience.