a0 2

n 1

Fourier series of f(x) in 0,2

f (x )

a0 n x n x a n cos b n sin 2 n 1

1 a0 an

1

1 bn

a

2

n

cos

n x nx b n sin

Fourier series of f(x) in ,

a0 2 n 1

f (x)

f ( x ) dx

1 a0

nx 0 f ( x ) cos dx

1 an

nx 0 f ( x ) sin dx

1 bn

0

2

2

Even Function a nx f (x) 0 a n cos 2 n 1

n x nx bn sin a n cos

f ( x ) dx

f ( x ) cos

nx dx

f ( x ) sin

nx dx

Odd Function

b

f (x)

n 1

n

sin

nx

a0

2 f ( x ) dx 0 2 n x f ( x ) cos dx 0

a0 0

an 0

an

2 nx f (x) sin dx 0

bn

bn 0 Convergence of Fourier Series: At a continuous point x = a, Fourier series converges to f(a)

At end point c or c+2l in (c, c+2l), Fourier series converges to At a discontinuous point x = a, Fourier series converges to

f ( c ) f ( c 2 ) 2

f (a ) f (a ) 2

Fourier series in the Interval of length 2 f (x)

a0 2

a n 1

n

cos nx b n sin nx

Fourier Series of f(x) in (- , )

Fourier Series of f(x) in (0,2 )

f (x )

a0 a n cos nx b n sin nx 2 n 1

1 a0 an

1

1 bn

2

f ( x ) dx 0

2

f ( x ) cos

nxdx

0

2

f ( x ) sin

nxdx

0

a0

a0 a n cos nx 2 n 1

2

an

f ( x ) dx 0

2 f ( x ) cos nx dx 0

bn 0

a0 a n cos nx b n sin nx 2 n 1

1 a0

Even Function

f (x)

f (x )

an

1

1 bn

f ( x ) dx

f ( x ) cos nxdx

f ( x ) sin

nxdx

Odd Function

f (x )

b n 1

n

sin nx

a0 0

an 0

2 b n f (x) sin nx dx 0

Half Range Fourier series

Fourier Cosine Series

f (x)

Fourier Sine Series

a0 nx an cos 2 n1

2 a0

nx f (x) bn sin n 1

f ( x ) dx

2 n x bn f ( x ) sin dx

0

0

2 nx a n f ( x ) cos dx 0

Convergence of Fourier Cosine series: At a continuous point x = a, Fourier cosine series converges to f(a). At end point 0 in(0,l), Fourier cosine series converges to f(0+) At end point l in(0,l), Fourier cosine series converges to f(l-)

Convergence of Fourier Sine series: At a continuous point x = a, Fourier Sine series converges to f(a). At both end points Fourier Sine series converges to 0.

Harmonic Analysis: a

0

2

y , N

an

nx y cos 2 N

,

b

n

2

nx y sin N

Parseval’s Theorem: If f (x)

a0 nx nx a n cos bn sin 2 n 1

is the Fourier series of f(x) in (c, c+2l),

c 2 2 2 a 2 1 Then y 0 (a n 2 b n 2 ) (or) 1 [ f ( x )] 2 dx a 0 1 ( a n 2 b n 2 ) 4 2 n 1 2 c 4 2 n 1

Root Mean Square Value:

y 2 is the effective value (or) Root Mean square (RMS) value of the function y = f(x), which is given by c 2

[ f ( x )]

y

2

dx

c

2

Some Important Results: 1.

Sin n =0 for all integer values of n

2. Cos n= (–1)n for all integer values of n 3. Cos2n=1 for all integer values of n 4. Sin2n = 0 for all integer values of n 5. If f( –x ) = f( x ) then f(x) is even and If f( –x ) = – f( x ) then f( x ) is odd. 1 ( x ) ( ,0 ) is even if either 1 ( x ) 2 ( x ) or 2 ( x ) 1 ( x ) 6. f (x ) ( x ) ( 0 , ) 2

1 ( x ) 7. f ( x ) 2 ( x )

( ,0) ( 0, )

is odd if either 1 (x) 2 (x) or 2 ( x ) 1 ( x )

x, x 0 8. x x, x 0 9.

ax e cos bxdx

10 .

e

ax

e ax a cos bx b sin bx a2 b2

e ax sin bxdx 2 a sin bx b cos bx a b2

11 . udv uv 1 u v 2 u v 3 ......... Where u

du d2u , u , ........ v 1 dx dx 2

dv ,

v2

v

1

dx , v 3

v

2

dx .......... ..