function

function:-
-suppose s is a set of complex number a function on s is a rule that arranges one
value to every member of s ,we write it as w=f(z),z?s.
-if u&v are the real & imaginary parts of w and x & y are real and imaginary
part of z then
f(x+iy)=u+iv
u=u(x,y) , v=v(x,y)
in the polar form we can write
f(re^i? )=u+iv ,u=u(r,?) ,v=v(r,?)
in the case v=0 ,then a real valued function on the domains.
example: suppose f(z)=(z ? )^2+2z
if z=x+iy ,then
f(z)=(x-iy)^2+2x+2iy
=x^2-y^2-i2xy+2x+i2y
=(x^2-y^2+2x)+i(2y-2xy)
example?-consider f(z)=|z|^2
if z=x+iy then f(z)=|z|^2=x^2+y^2=x^2+y^2+i0
so f is real value



Chapter two concept of limit point
suppose f is defined at all points in some nbhd of a point z_0 ,by the statment
that
lim?(z?z_0 )??f(z)=w_0 …(*)?
we mean that w=f(z) can be made arbitrary close to w_0 provided z
is sufficiently close to z_0 and distinct from it.
mathematically:-
this means for each ?>0 arbitarly small there exist ?>0 ?
|f(z)-w_0 |i.e. ???(?,zo).

uniqueness of the limit?-
suppose lim?(z?z_0 )??f(z)=w_0 ,? lim?(z?z_0 )??f(z)=w_1 ,?
for ?>0 arbitarely small there exist
?_1??_1 (?,z_0 )>0 such that
|f(z)-w_0 |simillarly ,there exist ?_2??_2 (?,z_0 )
0<|z-z_0 |choose ?=min??{?_1,?_2 }>0?
then
0<|z-z_0 |but then
|w_0-w_1 |=|w_0-f(z)+f(z)-w_1 |?|w_0-f(z)|+|f(z)-w_1 |=|f(z)-w_0 |+|f(z)-w_1 |since w_0-w_1 is afixed complex number & ?>0 is arbitarly small we get
|w_0-w_1 |=0
?w_0-w_1=0
?w_0=w_1