Lecture 8 Complex Analysis

1.8 DE Moivrés Theorem:-
The product of the two numbers Z_1=r_1 (cos???_1 ?+i sin?sin???_1 ? ), Z_2=r_2 (cos???_2+i sin???_2 ? ? ) is
Z_1 Z_2=r_1 r_2 (cos???_1+i sin???_1 ? ? )(cos???_2+i sin???_2 ? ? )
=r_1 r_2 [cos???_1 cos???_2+i cos???_1 sin???_2+i sin???_1 cos???_2- ? ? ? ? ? ?
sin???_1 sin???_2 ? ?]
=r_1 r_2 [(cos???_1 cos???_2-sin???_1 sin???_2 ? ? ? ? )+i ( sin???_1 cos???_2 ? ?
+cos???_2 sin???_2 ? ?)]
Thus Z_1 Z_2=r_1 r_2 [cos??(?_1+?_2 )+i sin??(?_1+?_2 )]? ? *
We get arg(Z_1 Z_2 )=?_1+?_2=arg(Z_1 )+arg(Z_2 ), that is the angle of the vector Z_1 Z_2 is the sum of the angles ?_1 and ?_2
as fig.
A generalization of equation *, if we let
Z_1=r_1 (cos???_1 ?+i sin???_1 ? ),
Z_2=r_2 (cos???_2+i sin???_2 ? ? ) ,…,
Z_n=r_n (cos???_n+i sin???_n ? ? ), then
Z_1 Z_2…Z_n=r_1 r_2…r_n [cos?(?_1+?_2+?+?_n )+i sin??(?_1+?_2+?+?_n )]?
If Z_1=Z_2=?=Z_n=Z=r(cos??+i sin?? )
Z^n=r^n (cos??+i sin?? )^n=r^n (cos?n?+i sin?n? )
When r=1 we get (cos??+i sin?? )^n=(cos?n?+i sin?n? ) which is called DE Moivrés Theorem .
Remark:- We can find multiplication inverse of complex numbers Z?0 by polar form .
Z^(-1)=1/Z=1/r(cos??+i sin?? ) =1/r(cos??+i sin?? ) ?((cos??-i sin?? ))/((cos??-i sin?? ) )
= ( cos??-i sin??)/(r[ ?cos?^2???+?sin?^2?? ?])=1/r[cos??(-?)+i sin??(-?)]? ?
and we get
Z_1/Z_2 =r_1 (cos???_1 ?+i sin???_1 ? ) 1/r_2 [cos??(?-??_2 )+i sin??(?-??_2 )]? ?
=r_1/r_2 (cos???_1 ?+i sin???_1 ? )[cos??(?-??_2 )+i sin??(?-??_2 )]? ?
=r_1/r_2 [cos??(?_1 ?-??_2 )+i sin??(?_1 ?-??_2 )]? ?
1.9 Euler s Formula:-
The equation e^i?=cos??+i sin?? is called Euler s formula, that is we define the complex number Z by Euler s formula as follow
Z=r(cos??+i sin?? )=re^i?
Thus
Z^(-1)=1/Z=1/?re?^i? =1/r e^(-i?) and Z ?=re^(-i?)
and the product
Z_1 Z_2=r_1 e^(?i??_1 ) r_2 e^(?i??_2 )=r_1 r_2 e^i(?_1+?_2 )
Z_1/Z_2 =r_1/r_2 e^i(?_1-?_2 )
The Euler s formula
e^iy=?_(n=0)^??(iy)^n/n!=1+iy-y^2/2!-i y^3/3!+y^4/4!+i y^5/5!-y^6/6!-y^7/7!+?
=(1-y^2/2!+y^4/4!-y^6/6!+?)+i(y-y^3/3!+y^5/5!-y^7/7!+?)
= cos??+i sin??
H.W:-Write the Complex number in polar form by two method ?2+i?2