Higher Engineering Mathematics, Sixth Edition

674 Higher Engineering Mathematics

The Laplace transforms of derivatives:

First derivative

L

{

dy

dx

}

=sL{y}−y(0)

wherey( 0 )is the value ofyatx=0.

Second derivative

L

{

dy

dx

}

=s^2 L{y}−sy( 0 )−y′(0)

wherey′( 0 )is the value of

dy

dx

atx=0.

Fourier Series

If f(x)is a periodic function of period 2πthen its

Fourier series is given by:

f(x)=a 0 +

∑∞

n= 1

(ancosnx+bnsinnx)

where, for the range−πto+π:

a 0 =

1

2 π

∫π

−π

f(x)dx

an=

1

π

∫π

−π

f(x)cosnxdx (n= 1 , 2 , 3 ,...)

bn=

1

π

∫π

−π

f(x)sinnxdx (n= 1 , 2 , 3 ,...)

Iff(x)isa periodicfunctionofperiodLthen itsFourier

series is given by:

f(x)=a 0 +

∑∞

n= 1

{

ancos

(

2 πnx

L

)

+bnsin

( 2 πnx

L

)}

where for the range−

L

2

to+

L

2

:

a 0 =

1

L

∫L/ 2

−L/ 2

f(x)dx

an=L^2

∫ L/ 2

−L/ 2

f(x)cos

( 2 πnx

L

)

dx(n= 1 , 2 , 3 ,...)

bn=L^2

∫ L/ 2

−L/ 2

f(x)sin

( 2 πnx

L

)

dx(n= 1 , 2 , 3 ,...)

Complex or exponential Fourier series:

f(x)=

∑∞

n=−∞

cnej

2 πnx

L

where cn=

1

L

∫ L 2

−L 2

f(x)e−j

2 πnx

L dx

For even symmetry,

cn=

2

L

∫ L 2

0

f(x)cos

( 2 πnx

L

)

dx

For odd symmetry,

cn=−j

2

L

∫ L

2

0

f(x)sin

( 2 πnx

L

)

dx