functions

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.
definition?
by an ? -neighbourhood of apoint z_0 denoted by |z-z_0 |by adeleted nbhd of z_0 we mean 0<|z-z_0 |a point z_0 is said to be an interior point of a set s if there exist an nbhd of z_0,
which contains only a point of s.(the nbhd is contained in s)
a point z_0 is said to be an exterior point of s if there exist an nbhd of z_0 ,which
does not contain any point of s.(i.e contaning no point of s)
if a point z_0 is neither a exterior point nor an interior point then this point is called
a boundary point of s the totally of such points is called aboundary of s,
this means if we consider any nbhd of boundary point z_0 ,then this nbhd will
have points from the set s as well as points outside s.
a set s is said to be open if s which does not have any boundary point.
equivalently ,s is open iff each of its point is interior point .
A set is said to be closed if it contains all the boundary points .for instance
{z:|z|<2} is open while
{z:|z|?1} is closed
the collection of all boundary point of s is called boundary of s.
notice that |z|=1 is boundary of {z: |z|<1}
as well as {z:|z|?1}
observse that for a set s not to be open that must be a boundary point of s which
will be inside s while a set s not to be closed
there must be aboundary point of s which is outside of s.
for instant the desk
0<|z|?1
is neither open nor closed . on the otherhand complex plane ? is both
open and closed.a set s is said to be connected .if any two points in s can be joined
by polygonal lines consisting of line segments joined end to end and lying inside
the set .for instance {z:|z|<3}is connected {z:|lnz|?1}is unconnected since contain the haph plane lnz?1 and lnz?-1 and
an open set which is connected is called a domain .
every nbhd of the point is a domain .
definition??: a region it is every nbhd.?