Page 41 - Aeronautical Engineer Data Book

28 Aeronautical Engineer’s Data Book
If x + iy = a + ib then x = a and y = b
(a + ib) + (c + id) = (a + c) = i(b + d)
(a + ib) – (c + id) = (a – c) = i(b + d)
(a + ib)(c + id) = (ac – bd) + i(ad + bc)
a + ib ac + bd bc –ad
33 = 33 + i 33
2
2
c+ id c + d 2 c + d 2
Every complex number may be written in polar
form. Thus
x + iy = r(cos + i sin ) = r
r is called the modulus of z and this may be
written r = |z|
r = � �� 2
2
x
y
+
is called the argument and this may be written
= arg z
y
tan = 33
x
If z = r (cos + i sin ) and z = r (cos + i
1
1
2
2
2
1
sin )
2
z z = r r [cos( + ) + i sin( + )]
1
1 2
1
2
2
1 2
r ( + )
= r 1 2 1 2
[cos( – ) + i sin( + )]
r 1 1 2 1 2
z \z = 3333
1
2
r 2
r 1
= 33 ( – )
2
1
r 2
2.8.6 Standard series
Binomial series n(n – 1)
n
n
(a + x) = a + na n–1 x + 3 n–2 x 2
3 a
2!
n(n –1)(n –2)
n
a
+ 33 –3 x 3
3!
2
2
+ ... (x < a )
The number of terms becomes inifinite when n
is negative or fractional.