Periodic Function, Dirichlet's Condition, Fourier series, Even & Odd functions, Euler's Formula for Fourier Coefficients, Change of Interval, Fourier series in the intervals (0,2l), (-l,l) , (-pi, pi), (0, 2pi), Half Range Cosine & Sine series Root mean square, Complex Form of Fourier series, Parseval's Identity

2.  The knowledge of Fourier series is essential to understand some very useful concepts in Electrical
Engineering. Fourier Series is very useful for circuit analysis, electronics, signal processing etc. . The study
of Fourier series is the backbone of Harmonic analysis. The Fourier series is concerned with periodic
waves.
 Advanced noise cancellation and cell phone network technology uses Fourier series where digital filtering
is used to minimize noise and bandwidth demands respectively.
 Fourier series is broadly used in telecommunications system, for modulation and demodulation of voice
signals, also the input, output and calculation of pulse and their sine or cosine graph.
 One of many applications is compression. Everyone's favorite MP3 format uses this for audio compression.
You take a sound, expand its Fourier series. It'll most likely be an infinite series BUT it converges so fast
that taking the first few terms is enough to reproduce the original sound. The rest of the terms can be
ignored because they add so little that a human ear can likely tell no difference. So I just save the first few
terms and then use them to reproduce the sound whenever I want to listen to it and it takes much less
memory. Also JPEG for pictures is the same idea.
 The solution for Heat Transfer in a rod was first proposed by Joseph Fourier himself using Fourier series
 In signal processing, Image processing Fourier analysis is used in the design of many filters (LPFs, HPFs,
Band pass). [Variety of Filters are used in most of the high end sound systems]
 The frequency analysis of circuits that are used in so many appliances are done with the help of Fourier
Analysis.
 Systems whose responses are unknown can be analyzed using integral transforms like Fourier or Laplace
Transforms.

3. Periodic Function
• A function 𝑓 𝑥 is said to be periodic, if and
only if 𝑓 𝑥 + 𝑇 = 𝑓 𝑥 is true for some T and
every value of 𝑥.
• The smallest value of T for which 𝑓 𝑥 + 𝑇 =
𝑓 𝑥 is true for every 𝑥, will be called the
period of the function 𝑓(𝑥)

4. Examples:
1. 𝑠𝑖𝑛𝑥 = sin 𝑥 + 2𝜋 = sin(𝑥 + 4𝜋) = ⋯ . = 𝑠𝑖𝑛 (𝑥 + 2𝑛𝜋)
Here the smallest value is 2𝜋 such that 𝑠𝑖𝑛𝑥 = sin 𝑥 + 2𝜋
Hence 𝑠𝑖𝑛𝑥 is a periodic function with period 2𝜋
2. 𝐶𝑜𝑠 𝑥 is periodic with period 2𝜋
3. 𝑡𝑎𝑛 𝑥 is periodic with period 𝜋
4. 𝑠𝑖𝑛2𝑥 = sin 2𝑥 + 2𝜋 = 𝑠𝑖𝑛2 𝑥 + 𝜋 is periodic with period 𝜋
5. 𝐶𝑜𝑠 2𝑥 is periodic with period 𝜋
6. sin 𝑛𝑥 is periodic with period
2𝜋
𝑛
7. cos 𝑛𝑥 is periodic with period
2𝜋
𝑛
8. tan 𝑛𝑥 is periodic with period
𝜋
𝑛
9. Every constant function is periodic
10.Consider 𝑓: 𝑅 ⟶ 𝑅 defined by 𝑓 𝑥 =
0, 𝑖𝑓 𝑥 𝑖𝑠 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙
1, 𝑖𝑓 𝑥 𝑖𝑠 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙
Here 𝑓 is periodic 𝑝 where 𝑝 is any positive number.

5. Continuous function
• A function 𝑓(𝑥) is said to be continuous at 𝑥 = 𝑎, if lim
𝑥⟶𝑎−
𝑓(𝑥) = lim
𝑥⟶𝑎+
𝑓 𝑥 = 𝑓(𝑎)
• A function 𝑓(𝑥) is continuous in [𝑎 𝑏], if 𝑓(𝑥) is continuous at each and every point in [𝑎 𝑏].
Example:
1. Consider the function 𝑓: 𝑅 ⟶ 𝑅 defined by 𝑓 𝑥 = 𝑥 , for all 𝑥
Here for any = 𝑎 , lim
𝑥⟶𝑎−
𝑓(𝑥) = lim
𝑥⟶𝑎+
𝑓 𝑥 = 𝑓 𝑎
𝑖. 𝑒 𝑤ℎ𝑒𝑛 𝑥 = 1, lim
𝑥⟶1−
𝑓(𝑥) = 1& lim
𝑥⟶1+
𝑓 𝑥 = 1
Hence 𝑓(𝑥) is continuous
2. Consider the function 𝑓: 𝑅 ⟶ 𝑅 defined by 𝑓 𝑥 =
𝑥, 𝑖𝑓 𝑥 < 1
𝑥
2
, 𝑖𝑓 𝑥 ≥ 1
When 𝑥 = 1 , lim
𝑥⟶1−
𝑓(𝑥) = 1 , (∵ 𝑓 𝑥 = 𝑥 , 𝑖𝑓 𝑥 < 1)
lim
𝑥⟶1+
𝑓 𝑥 =
1
2
, (∵ 𝑓 𝑥 =
𝑥
2
, 𝑖𝑓 𝑥 ≥ 1)
Here lim
𝑥⟶1−
𝑓(𝑥) ≠ lim
𝑥⟶1+
𝑓 𝑥
Hence 𝑓(𝑥) is not continuous

6. Fourier Series
A series of sines and cosines of an angle and its multiples
of the form
𝑎0
2
+ 𝑎1 cos 𝑥 + 𝑎2 𝑐𝑜𝑠2𝑥 + 𝑎3 𝑐𝑜𝑠3𝑥 + ⋯ + 𝑎 𝑛 𝑐𝑜𝑠𝑛𝑥 +
⋯ . +𝑏1 𝑠𝑖𝑛𝑥 + 𝑏2 𝑠𝑖𝑛2𝑥 + 𝑏3 𝑠𝑖𝑛3𝑥 + ⋯ + 𝑏 𝑛 𝑠𝑖𝑛𝑛𝑥 + ⋯
=
𝑎0
2
+ 𝑛=1
∞
𝑎 𝑛 𝑐𝑜𝑠𝑛𝑥 + 𝑛=1
∞
𝑏 𝑛 𝑠𝑖𝑛𝑛𝑥
is called Fourier series,
where 𝑎0, 𝑎1, … 𝑎 𝑛, . . 𝑏1, 𝑏2, … 𝑏 𝑛 … are called Fourier
constants (or) Euler’s constants (or) Fourier Coefficients.

7. Note :
• 𝑓 𝑥 is defined in (0, 2𝜋) means 0 ≤ 𝑥 ≤ 2𝜋
• 𝑓 𝑥 is defined in (−𝜋, 𝜋) means −𝜋 ≤ 𝑥 ≤
𝜋
• 𝑓 𝑥 is defined in (0, 2𝑙) means 0 ≤ 𝑥 ≤ 2𝑙
• 𝑓 𝑥 is defined in (−𝑙, 𝑙) means −𝑙 ≤ 𝑥 ≤ 𝑙

8. Euler’s formula for the Fourier
Coefficients
If 𝑓(𝑥) is defined in (0, 2𝜋) can be expressed as
𝑓(𝑥) =
𝑎0
2
+ 𝑛=1
∞
𝑎 𝑛 𝑐𝑜𝑠𝑛𝑥 + 𝑛=1
∞
𝑏 𝑛 𝑠𝑖𝑛𝑛𝑥
Where 𝑎0 =
1
𝜋 0
2𝜋
𝑓 𝑥 𝑑𝑥 ------- (1)
𝑎 𝑛 =
1
𝜋 0
2𝜋
𝑓 𝑥 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥 , 𝑛 ≥ 0 -------- (2)
𝑏 𝑛 =
1
𝜋 0
2𝜋
𝑓 𝑥 𝑠𝑖𝑛𝑛𝑥 𝑑𝑥 , 𝑛 ≥ 1 -------- (3)
The formulae (1),(2) & (3) are called Euler’s
formula

9. Sufficient Condition for Fourier series
expansion (Dirichlet’s Condition)
• Any function 𝑓(𝑥) defined on (𝑎, 𝑏) can be
expressed as Fourier series
𝑎0
2
+ 𝑛=1
∞
𝑎 𝑛 𝑐𝑜𝑠𝑛𝑥 +
𝑛=1
∞
𝑏 𝑛 𝑠𝑖𝑛𝑛𝑥 where 𝑎0, 𝑎 𝑛, 𝑏 𝑛 are constants,
provided
• 𝑓(𝑥) is well defined, single valued and periodic.
• 𝑓(𝑥) is piecewise continuous with finite number
of discontinuity in (𝑎, 𝑏)
• 𝑓 𝑥 has only finite number of maximum and
minimum in 𝑎, 𝑏

10. Note:
• Dirichlet’s conditions are not necessary but
sufficient for the existence of Fourier series.

11. 𝑡𝑎𝑛 𝑥 can’t be expressed as Fourier series
because it has infinite number of discontinuity.

12. 𝐶𝑜𝑠𝑒𝑐 𝑥 can’t be expressed as Fourier series

13. Convergence of Fourier Series
If 𝑓(𝑥) is continuous at 𝑥 = 𝑎, then lim
𝑥→𝑎
𝑓 𝑥 =
𝑓(𝑎), the series converges to 𝑓(𝑎)
(𝑖. 𝑒) the sum of the Fourier series= 𝑓(𝑎)
If 𝑓(𝑥) is discontinuous at 𝑥 = 𝑎, then the
Fourier series converges to the average value of
right & left hand limits at 𝑥 = 𝑎
(i.e) the sum of the Fourier series =
1
2
[𝑓 𝑎− + 𝑓 𝑎+ ]

14. Even function
A function 𝑓(𝑥) is said to be even if 𝑓 −𝑥 =
𝑓 𝑥 , for all 𝑥
Examples
1.𝑓 𝑥 = 𝑥2
2.𝑓 𝑥 = 𝑥
3.𝑓 𝑥 = 𝑐𝑜𝑠𝑥
4.𝑓 𝑥 = 𝑥𝑠𝑖𝑛𝑥

15. Note:
Even functions are symmetric about y- axis

16. Odd function
A function 𝑓(𝑥) is said to be odd if 𝑓(−𝑥) =
− 𝑓(𝑥)
Examples
1. 𝑓 𝑥 = 𝑥
2. 𝑓 𝑥 = 𝑠𝑖𝑛𝑥
3. 𝑓 𝑥 = 𝑥 cos 𝑥
4. 𝑓 𝑥 = 𝑥3

17. Note:
Odd functions are symmetric about the origin
(𝑖. 𝑒) Odd functions are symmetric in opposite
quadrants

18. Note
Even x Even = Even
Odd x Odd = Even
Odd x Even = Odd
Examples
1. 𝑓 𝑥 = 𝑥2 is even & 𝑔 𝑥 = 𝑐𝑜𝑠𝑥 is even
𝑓 𝑥 𝑔 𝑥 = 𝑥2
𝑐𝑜𝑠𝑥 is even
2. 𝑓 𝑥 = 𝑥 is odd & 𝑔 𝑥 = 𝑠𝑖𝑛𝑥 is odd
𝑓 𝑥 𝑔 𝑥 = 𝑥𝑠𝑖𝑛𝑥 is even
3. 𝑓 𝑥 = 𝑥2 is even & 𝑔 𝑥 = 𝑥 is odd
𝑓 𝑥 𝑔 𝑥 = 𝑥3 is odd

19. Note:
1.If 𝑓(𝑥) is even, then −𝑎
𝑎
𝑓 𝑥 𝑑𝑥 =
2 0
𝑎
𝑓 𝑥 𝑑𝑥
2.If 𝑓(𝑥) is odd, then −𝑎
𝑎
𝑓 𝑥 𝑑𝑥 = 0

20. Neither Even nor Odd function
A function which not either even or odd is called
neither even nor odd function
Examples
1.𝑥 + 𝑥2
2.−𝑥 + 𝑥𝑠𝑖𝑛𝑥

21. Formulae
1. Bernoulli’s formula 𝑢𝑣 𝑑𝑥 = 𝑢𝑣1 − 𝑢′ 𝑣2 + 𝑢′′ 𝑣3 −
⋯ where 𝑢 & 𝑣 are functions of 𝑥
𝑢′
=
𝑑𝑢
𝑑𝑥
, 𝑢′′
=
𝑑2 𝑢
𝑑𝑥2 … 𝑣1 = 𝑣 𝑑𝑥 , 𝑣2 = 𝑣1 𝑑𝑥 … .
2. 𝑒 𝑎𝑥
𝑠𝑖𝑛𝑏𝑥 𝑑𝑥 =
𝑒 𝑎𝑥
𝑎2+𝑏2 [𝑎𝑠𝑖𝑛𝑏𝑥 − 𝑏𝑐𝑜𝑠𝑏𝑥]
3. 𝑒 𝑎𝑥 𝑐𝑜𝑠𝑏𝑥 𝑑𝑥 =
𝑒 𝑎𝑥
𝑎2+𝑏2 [𝑎𝑐𝑜𝑠𝑏𝑥 + 𝑏𝑠𝑖𝑛𝑏𝑥]
4.𝑐𝑜𝑠𝑛𝜋 = −1 𝑛
5.𝑐𝑜𝑠2𝑛𝜋 = 1
6. 𝑆𝑖𝑛𝑛𝜋 = 0
7. −1 𝑛+1
= −1 𝑛−1

22. Fourier series of an even function
𝒇(𝒙) in (−𝝅, 𝝅)
𝑓 𝑥 =
𝑎0
2
+ 𝑛=1
∞
𝑎 𝑛 𝑐𝑜𝑠𝑛𝑥
Where 𝑎0 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑑𝑥 =
2
𝜋 0
𝜋
𝑓 𝑥 𝑑𝑥
𝑎 𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥 =
2
𝜋 0
𝜋
𝑓 𝑥 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥
Note:
here there is no 𝑏 𝑛 term, because
𝑏 𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑠𝑖𝑛𝑛𝑥 𝑑𝑥 = 0 (∵
𝑓 𝑥 𝑖𝑠 𝑒𝑣𝑒𝑛, 𝑠𝑖𝑛𝑛𝑥 𝑖𝑠 𝑜𝑑𝑑 , ℎ𝑒𝑛𝑐𝑒 𝑓 𝑥 𝑠𝑖𝑛𝑛𝑥 𝑖𝑠 𝑜𝑑𝑑)

23. Fourier series of an odd function 𝒇(𝒙)
in (−𝝅, 𝝅 )
𝑓 𝑥 = 𝑛=1
∞
𝑏 𝑛 𝑠𝑖𝑛𝑛𝑥
Where 𝑏 𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑠𝑖𝑛𝑛𝑥 𝑑𝑥 =
2
𝜋 0
𝜋
𝑓 𝑥 𝑠𝑖𝑛𝑛𝑥 𝑑𝑥
Note: here there are no 𝑎0 & 𝑎 𝑛terms, because 𝑎0 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑑𝑥 = 0
𝑎 𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥 = 0
(∵ 𝑓 𝑥 𝑖𝑠 𝑜𝑑𝑑, 𝑐𝑜𝑠𝑛𝑥 𝑖𝑠 𝑒𝑣𝑒𝑛 , ℎ𝑒𝑛𝑐𝑒 𝑓 𝑥 𝑐𝑜𝑠𝑛𝑥 𝑖𝑠 𝑜𝑑𝑑)

24. Fourier series of neither even nor odd
function 𝒇(𝒙) in (−𝝅, 𝝅)
𝑓(𝑥) =
𝑎0
2
+ 𝑛=1
∞
𝑎 𝑛 𝑐𝑜𝑠𝑛𝑥 + 𝑛=1
∞
𝑏 𝑛 𝑠𝑖𝑛𝑛𝑥
Where 𝑎0 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑑𝑥
𝑎 𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥 , 𝑛 ≥ 0
𝑏 𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑠𝑖𝑛𝑛𝑥 𝑑𝑥 , 𝑛 ≥ 1

25. Change of interval
Fourier series in the interval (𝟎, 𝟐𝒍)
In many problems, we need to find Fourier series expansion
over an interval of length ‘2𝑙’ instead of ′2𝜋′
In this case the Fourier series expansion of 𝑓(𝑥) is given by
𝑓(𝑥) =
𝑎0
2
+
𝑛=1
∞
𝑎 𝑛 cos
𝑛𝜋𝑥
𝑙
+
𝑛=1
∞
𝑏 𝑛 sin
𝑛𝜋𝑥
𝑙
Where 𝑎0 =
1
𝑙 0
2𝑙
𝑓 𝑥 𝑑𝑥
𝑎 𝑛 =
1
𝑙 0
2𝑙
𝑓 𝑥 cos
𝑛𝜋𝑥
𝑙
𝑑𝑥
𝑏 𝑛 =
1
𝑙 0
2𝑙
𝑓 𝑥 sin
𝑛𝜋𝑥
𝑙
𝑑𝑥

26. Fourier series of a function 𝒇(𝒙) in
(−𝒍, 𝒍)
Case (i) If 𝒇(𝒙) is even in (−𝒍, 𝒍 )
𝑓 𝑥 =
𝑎0
2
+ 𝑛=1
∞
𝑎 𝑛 cos
𝑛𝜋𝑥
𝑙
Where 𝑎0 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 𝑑𝑥 =
2
𝑙 0
𝑙
𝑓 𝑥 𝑑𝑥
𝑎 𝑛 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 cos
𝑛𝜋𝑥
𝑙
𝑑𝑥 =
2
𝑙 0
𝑙
𝑓 𝑥 cos
𝑛𝜋𝑥
𝑙
𝑑𝑥

27. Case (ii) If 𝒇(𝒙) is odd in (−𝒍, 𝒍)
𝑓 𝑥 = 𝑛=1
∞
𝑏 𝑛 𝑠𝑖𝑛𝑛𝑥
Where 𝑏 𝑛 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 sin
𝑛𝜋𝑥
𝑙
𝑑𝑥 =
2
𝑙 0
𝑙
𝑓 𝑥 sin
𝑛𝜋𝑥
𝑙
𝑑𝑥

28. Case (iii) If 𝒇(𝒙) is neither even nor
odd function in (−𝒍, 𝒍 )
𝑓(𝑥) =
𝑎0
2
+ 𝑛=1
∞
𝑎 𝑛 cos
𝑛𝜋𝑥
𝑙
+ 𝑛=1
∞
𝑏 𝑛 sin
𝑛𝜋𝑥
𝑙
Where 𝑎0 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 𝑑𝑥
𝑎 𝑛 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 cos
𝑛𝜋𝑥
𝑙
𝑑𝑥
𝑏 𝑛 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 sin
𝑛𝜋𝑥
𝑙
𝑑𝑥

30. Half Range Cosine series of 𝒇(𝒙) in
(𝟎, 𝝅)
𝑓 𝑥 =
𝑎0
2
+ 𝑛=1
∞
𝑎 𝑛 𝑐𝑜𝑠𝑛𝑥
Where 𝑎0 =
2
𝜋 0
𝜋
𝑓 𝑥 𝑑𝑥
𝑎 𝑛 =
2
𝜋 0
𝜋
𝑓 𝑥 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥

31. Half Range Sine series of 𝒇(𝒙) in
(𝟎, 𝝅)
𝑓 𝑥 = 𝑛=1
∞
𝑏 𝑛 𝑠𝑖𝑛𝑛𝑥
Where 𝑏 𝑛 =
2
𝜋 0
𝜋
𝑓 𝑥 𝑠𝑖𝑛𝑛𝑥 𝑑𝑥

32. Half Range Cosine series of 𝒇(𝒙) in
(𝟎, 𝒍)
𝑓 𝑥 =
𝑎0
2
+ 𝑛=1
∞
𝑎 𝑛 cos
𝑛𝜋𝑥
𝑙
Where 𝑎0 =
2
𝑙 0
𝑙
𝑓 𝑥 𝑑𝑥
𝑎 𝑛 =
2
𝑙 0
𝑙
𝑓 𝑥 cos
𝑛𝜋𝑥
𝑙
𝑑𝑥

33. Half Range Sine series of 𝒇(𝒙) in (𝟎, 𝒍)
𝑓 𝑥 = 𝑛=1
∞
𝑏 𝑛 sin
𝑛𝜋𝑥
𝑙
Where 𝑏 𝑛 =
2
𝑙 0
𝑙
𝑓 𝑥 sin
𝑛𝜋𝑥
𝑙
𝑑𝑥

34. Complex form of Fourier series
Let 𝑓(𝑥) be a periodic function of period 2𝜋 in 0,2𝜋 .The complex
form of Fourier series is given by
𝑓 𝑥 = 𝑐0 + 𝑛=1
∞
(𝑐 𝑛 𝑒 𝑖𝑛𝑥 + 𝑐−𝑛 𝑒−𝑖𝑛𝑥)
Where 𝑐0 =
𝑎0
2
, 𝑐 𝑛 =
𝑎 𝑛−𝑖𝑏 𝑛
2
, 𝑐−𝑛 =
𝑎 𝑛+𝑖𝑏 𝑛
2
𝑖. 𝑒 𝑐0 =
𝑎0
2
=
1
2𝜋 0
2𝜋
𝑓 𝑥 𝑑𝑥
𝑐 𝑛 =
1
2𝜋 0
2𝜋
𝑓 𝑥 𝑒−𝑖𝑛𝑥
𝑑𝑥
𝑐−𝑛 =
1
2𝜋 0
2𝜋
𝑓 𝑥 𝑒 𝑖𝑛𝑥
𝑑𝑥
𝑖. 𝑒 The complex form can be written as 𝑓 𝑥 = 𝑛=−∞
∞
𝑐 𝑛 𝑒 𝑖𝑛𝑥
Where 𝑐 𝑛 =
1
2𝜋 0
2𝜋
𝑓 𝑥 𝑒−𝑖𝑛𝑥
𝑑𝑥 for all 𝑛 = 0, ±1, ±2, …

35. Complex form of Fourier series in
various intervals
 In (0,2𝜋), 𝑓 𝑥 = 𝑛=−∞
∞
𝑐 𝑛 𝑒 𝑖𝑛𝑥
Where 𝑐 𝑛 =
1
2𝜋 0
2𝜋
𝑓 𝑥 𝑒−𝑖𝑛𝑥
𝑑𝑥 for all 𝑛 = 0, ±1, ±2, …
 In (−𝜋, 𝜋), 𝑓 𝑥 = 𝑛=−∞
∞
𝑐 𝑛 𝑒 𝑖𝑛𝑥
Where 𝑐 𝑛 =
1
2𝜋 −𝜋
𝜋
𝑓 𝑥 𝑒−𝑖𝑛𝑥 𝑑𝑥 for all 𝑛 = 0, ±1, ±2, …
 In (0,2𝑙), 𝑓 𝑥 = 𝑛=−∞
∞
𝑐 𝑛 𝑒
𝑖𝑛𝜋𝑥
𝑙
Where 𝑐 𝑛 =
1
2𝑙 0
2𝑙
𝑓 𝑥 𝑒−
𝑖𝑛𝜋𝑥
𝑙 𝑑𝑥 for all 𝑛 = 0, ±1, ±2, …
 In (−𝑙, 𝑙), 𝑓 𝑥 = 𝑛=−∞
∞
𝑐 𝑛 𝑒
𝑖𝑛𝜋𝑥
𝑙
Where 𝑐 𝑛 =
1
2𝑙 −𝑙
𝑙
𝑓 𝑥 𝑒−
𝑖𝑛𝜋𝑥
𝑙 𝑑𝑥 for all 𝑛 = 0, ±1, ±2, …

36. Root Mean Square Value (RMS value)
(or) Effective Value
The root mean square of a function 𝑓(𝑥)
defined in the interval (𝑎, 𝑏) is given by
𝑎
𝑏 𝑓 𝑥 2 𝑑𝑥
𝑏−𝑎
, it is denoted by 𝑌
∴ 𝑌2
= 𝑎
𝑏
𝑓 𝑥 2 𝑑𝑥
𝑏−𝑎
=
1
𝑏−𝑎 𝑎
𝑏
𝑓 𝑥 2
𝑑𝑥

37. Parseval’s Identity
If 𝑓(𝑥) is periodic with period 2𝜋 defined in (−𝜋, 𝜋).
Then
1
2𝜋 −𝜋
𝜋
𝑓 𝑥 2 𝑑𝑥 =
𝑎0
2
4
+
1
2 𝑛=1
∞
(𝑎 𝑛
2 + 𝑏 𝑛
2)
𝑖. 𝑒
1
𝜋 −𝜋
𝜋
𝑓 𝑥 2 𝑑𝑥 =
𝑎0
2
2
+ 𝑛=1
∞
(𝑎 𝑛
2 + 𝑏 𝑛
2)
Where 𝑎0, 𝑎 𝑛, 𝑏 𝑛 are called Fourier coefficients of 𝑓(𝑥) is
called Parseval’s Identity
𝑎0 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑑𝑥
𝑎 𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥
𝑏 𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑠𝑖𝑛𝑛𝑥 𝑑𝑥

38. Parseval’s Identity in the interval
(−𝒍, 𝒍)
It is given by
1
𝑙 −𝑙
𝑙
𝑓 𝑥 2
𝑑𝑥 =
𝑎0
2
2
+ 𝑛=1
∞
(𝑎 𝑛
2
+ 𝑏 𝑛
2
)
Where 𝑎0 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 𝑑𝑥 𝑎 𝑛 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 cos
𝑛𝜋𝑥
𝑙
𝑑𝑥
𝑏 𝑛 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 sin
𝑛𝜋𝑥
𝑙
𝑑𝑥

39. Half Range Sine and Cosine
Parseval’s Identity in (𝟎, 𝒍)

40. Half Range Cosine Parseval’s Identity
It is given by
2
𝑙 0
𝑙
𝑓 𝑥 2
𝑑𝑥 =
𝑎0
2
2
+ 𝑛=1
∞
𝑎 𝑛
2
Where 𝑎0 =
2
𝑙 0
𝑙
𝑓 𝑥 𝑑𝑥
𝑎 𝑛 =
2
𝑙 0
𝑙
𝑓 𝑥 cos
𝑛𝜋𝑥
𝑙
𝑑𝑥

41. Half Range Sine Parseval’s Identity
It is given by
2
𝑙 0
𝑙
𝑓 𝑥 2
𝑑𝑥 = 𝑛=1
∞
𝑏 𝑛
2
Where 𝑏 𝑛 =
2
𝑙 0
𝑙
𝑓 𝑥 sin
𝑛𝜋𝑥
𝑙
𝑑𝑥

42. Note:
In 0,2𝜋 & −𝜋, 𝜋 , 𝑢𝑠𝑒 ′𝑛𝑥′
In 0,2𝑙 & −𝑙, 𝑙 , 𝑢𝑠𝑒 ′
𝑛𝜋𝑥
𝑙
′