Essential formulae 663
Complex Numbers
z = a + j b =r(cos θ + j sin θ)= r∠θ =re j θ where
j 2 =−1
Modulus r =|z| =
(a 2 + b 2 )
Argument θ = arg z =tan −1 b
a
Addition: (a + j b) +(c + j d) =(a + c) + j (b +d)
Subtraction: (a + j b) −(c + j d) =(a − c) + j (b −d)
Complex equations: If m + j n = p + j q then m = p
and n = q
Multiplication: z 1 z 2 =r 1 r 2 ∠(θ 1 + θ 2 )
Division:
z 1
z 2
=
r 1
r 2
∠(θ 1 − θ 2 )
De Moivre’s theorem:
[r∠θ] n =r n ∠nθ =r n (cos nθ + j sin nθ)=re j θ
Matrices and Determinants
Matrices:
If A =
a b
c d
and B =
e f
g h
then
A + B =
a + e b + f
c + g d + h
A − B =
a − e b − f
c − g d − h
A × B =
ae + bg a f + bh
ce + dg c f + dh
A
−1
=
1
ad − bc
d −b
−c a
If A =
⎛
⎜
⎝
a 1 b 1 c 1
a 2 b 2 c 2
a 3 b 3 c 3
⎞
⎟
⎠ then A −1 =
B T
|A|
where
B T = transpose of cofactors of matrix A
Determinants:
a b
c d
= ad − bc
a 1 b 1 c 1
a 2 b 2 c 2
a 3 b 3 c 3
= a 1
b 2 c 2
b 3 c 3
− b 1
a 2 c 2
a 3 c 3
+ c 1
a 2 b 2
a 3 b 3
Differential Calculus
Standard derivatives:
y or f (x)
dy
dx
or f
(x)
ax n
anx n−1
sin ax
a cos ax
cos ax
−a sin ax
tan ax
a sec 2 ax
sec ax
a sec ax tan ax
cosec ax
−a cosec ax cot ax
cot ax
−a cosec
2 ax
e ax
ae ax
ln ax
1
x
sinh ax
a cosh ax
cosh ax
a sinh ax
tanh ax
a sech 2 ax
sech ax
−a sech ax tanh ax
cosech ax −a cosech ax coth ax
coth ax
−a cosech
2 ax
sin −1 x
a
1
√
a 2 − x 2
sin −1 f (x)
f (x)
1 − [ f (x)] 2
cos −1 x
a
−1
√
a 2 − x 2