Mathematical preliminaries - Continuous and Discrete Time Signals and Systems - page 794

= 1 sin t cos θ

2 [sin( t + θ ) + sin( t − θ )]
a cos t + b sin t = √

− 1 ( − b / a )

= C cos( t + θ ) , C a 2 + b 2 ,θ = tan

√

b
a

− 1

a cos( mt ) + b sin( mt ) = a 2 + b 2 cos( mt − θ ) ,θ = tan

a cos( mt ) + b sin( mt ) = √

a
b

− 1

a 2 + b 2 sin( mt + φ ) ,φ = tan

A.2 Power series

ln(1 + t ) = t − t 2
2

+ t 3
3

− t 4
4

+

= 1 + t + t 2
2!

+ t 3
3!

+ t 4
4!

e t

+

= t − t 3
3!

+ t 5
5!

− t 7
7!

sin t +

= 1 − t 2
2!

+ t 4
4!

− t 6
6!

cos t +

= t + t 3
3

+ 2 t 5
15

+ 17 t 7
315

tan t +

t 3
3

t 5
5

= t + 1
2

+ 1 . 3
2 . 4

− 1 t

sin +

A.3 Series summat ion

Arithmetic series

n

[ a + ( n − 1) d ] = N

2 [2 a + ( N

− 1) d ]

n = 1

n

= N ( N

+ 1)

n = 1 + 2 ++ N

2

n = 1

Geometric series

N

= a (1 − r N + 1 )
1 − r

ar n

n = 0

N − 1

j 2 π kn
N

01 ≤ k ≤ ( N − 1)
Nk = 0 ,

exp =

n = 0

∞

1
1 − r , r < 1

r n =

n = 0

∞

r
(1 − r ) 2 , r < 1

nr n

=

n = 0