Complex conjugate:

Complex conjugate:
The complex conjugate ,or simply the conjugate of acomplex number z=x+iy is defined as the complex number x-iy and is defined by z ? that is
z ?=x-iy……………….(1)
=(x,-y)
Note that:
z ?=z and ?z ??=?z?
If z1=x1+iy1 & z2=x2+iy2 ,then
z1+z2=(x1+x2)-i(y1+y2)=(x1-iy1)+(x2-iy2)= z ?_1+z ?_2
i.e.
z1+z2= z ?_1+z ?_2…………………(2)
and
z1-z2= z ?_1-z ?_2…………………(3)
z1z2= z ?_1 z ?_2…………………(4)
(z1/z2)= z ?_1/z ?_2…(z2?0)..(5)
z+z ?=2Rez?Rez=(z+z ?)/2……(6)
z-z ?=2iImz?Imz=(z-z ?)/2i……(7)
zz ?=?z?2……………………………..(8)
with the aid of identity (8) one can easily obtain various other properties
?z1z2?=?z1??z2?………………….(9)
?z1/z2?=?z1?/?z2?…(z2?0)...(10)
For properties (9)
?z1z2?2=(z1z2)(z1z2) =(z1z2)(z ?_1 z ?_2)=(z1z ?_1) (z2z ?_2)=?z1?2?z2?2
Then
?z1z2?=?z1??z2?
4-Triangle inequality :
Properties of moduli and conjugates in sec.3 enable us to give an algebraic derivative of the triangle inequality
1. ?z1 +z2???z1?+?z2?
Proof:
?z1 +z2?2=(z1 +z2)(z1 +z2) =(z1 +z2)( z ?_1+z ?_2)=(z1z ?_1)+(z1z ?_2)+(z ?_1z2)+(z2z ?_2)
=(z1z ?_1)+(z1z ?_2)+ (z1z ?_2)+(z2z ?_2)
But
(z1z ?_2)+ (z1z ?_2)=2Re(z1z ?_2)?2?z1z ?_2?=2?z1??z ?_2?=2?z1??z2?
And so
?z1 +z2?2??z1?2+2?z1??z2?+?z2?2=(?z1?+?z2?)2
Then
?z1 +z2???z1?+?z2?
2. ?z1 +z2????z1?-?z2??
Proof:
? z1?= ?z1- z2+ z2???z1+z2?+?(-z2)?=?z1+z2?+?z2?
Then
?z1?-?z2???z1+z2?……………………(*)
Interchange z1 & z2 ,we obtain
? z2?-?z1???z2+z1?
Then
-(?z1?-?z2?)??z1+z2?………………..(**)
From (*) and (**) we deduce that
?z1 +z2????z1?-?z2??
Useful alternative forms of inequalities (1),(2) when z2 replaced by - z2
3. ?z1-z2???z1?+?z2?
4. ?z1-z2????z1?-?z2??
Home work:
1.show that
?z1-z2???z1-z3?+?z3-z2?
2.use properties of conjugates and moduli to show that
a. z ?+3i=z-3i
b. iz =-iz ?
c.?(2z ?+5)(?2-i)?=?3?2z ?+5?
3.show that
?z1 +z2?2+?z1 -z2?2=2{?z1?2+?z2?2}
4.If iz2-z ?=0 , find values of ?z?