674 Higher Engineering Mathematics
The Laplace transforms of derivatives:
First derivative
L
dy
dx
= sL{y} − y(0)
where y(0) is the value of y at x = 0.
Second derivative
L
dy
dx
= s
2
L{y} − sy(0) − y
(0)
where y (0) is the value of
d y
dx
at x = 0.
Fourier Series
If f (x) is a periodic function of period 2π then its
Fourier series is given by:
f (x) = a 0 +
∞
n=1
(a n cos nx + b n sin nx)
where, for the range −π to +π:
a 0 =
1
2π
π
−π
f (x) dx
a n =
1
π
π
−π
f (x) cos nx dx (n = 1, 2, 3, ...)
b n =
1
π
π
−π
f (x) sin nx dx (n = 1, 2, 3, ...)
If f (x) is a periodic function of period L then its Fourier
series is given by:
f (x) = a 0 +
∞
n=1
a n cos
2π nx
L
+ b n sin
2πnx
L
where for the range −
L
2
to +
L
2
:
a 0 =
1
L
L/2
−L/2
f (x) dx
a n =
2
L
L/2
−L/2
f (x) cos
2πnx
L
dx (n = 1, 2, 3, ...)
b n =
2
L
L/2
−L/2
f (x) sin
2πnx
L
dx (n = 1, 2, 3, ...)
Complex or exponential Fourier series:
f (x) =
∞
n=−∞
c n e
j
2πnx
L
where c n =
1
L
L
2
−
L
2
f (x)e
− j
2πnx
L dx
For even symmetry,
c n =
2
L
L
2
0
f (x) cos
2πnx
L
dx
For odd symmetry,
c n = − j
2
L
L
2
0
f (x) sin
2πnx
L
dx