Analytic function

Analytic function?-
suppose f is defined on a domain D,suppose z is a point of D ,then f is said
to be analytic at z if f is differentiable not only at z but also differentiable
at each point of some nbhd of z .
further ,f is said to be analytic on D if f is analytic at each point of D.
entire function:-
A function f is said to be an entire function if f is analytic at each point
of complex plane ? .
note:-since the derivative of the polynomail exists
every where it follows that every polynomial is an
entire function.
and we proof that lemma:-
let f be analytic on a domain D,then the function f reduces to a constant if one
of the function is true.
1.f is real valued.
2.f(z ? ) is analytic on D.
3.|f(z)| is constant on D.
lemma:-
let f be analytic on a domain D,then the function f reduces to a constant if one
of the function is true.and each analytic function f on a domain D determines two families curves which
are orthogonal to one-another .

1.f is real valued.
2.f(z ? ) is analytic on D.
3.|f(z)| is constant on D.