Higher Engineering Mathematics

Ess-For-H8152.tex 19/7/2006 18: 2 Page 720

720 ESSENTIAL FORMULAE

Laplace Transforms

Function Laplace transforms

f(t) L{f(t)}=

∫∞

0 e

−stf(t)dt

(^11) s
k ks
eat s−^1 a
sinat s (^2) +aa 2
cosat s (^2) +sa 2
t s^12
tn(n=positve integer) snn+! 1
coshat s (^2) −sa 2
sinhat s (^2) −aa 2
e−attn (s+na!)n+ 1
e−atsinωt (s+aω) (^2) +ω 2
e−atcosωt (s+as+) 2 a+ω 2
e−atcoshωt (s+as+) 2 a−ω 2
e−atsinhωt (s+aω) (^2) −ω 2
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
Iff(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) is a periodic function of periodLthen its
Fourier 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=^2 L
∫L/ 2
−L/ 2
f(x) cos
( 2 πnx
L
)
dx(n=1, 2, 3,...)
bn=^2 L
∫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