harmonic function

singular point: if a function f fails to be analytic at a point z_0 but is analytic at some point
in every neighborhood of z_0 ,then z_0 is called a singular point ,or singularity.
definition:-(isolated sigular point )
if z_0 is singular point and there is anbhd to z_0 such that f is analytic in every
point in this nbhd except z_0 then z_0 is called isolated singular point .
- a function f has an isolated singularity at z=a if there is
an R>0 ,such that f is defined and analytic in B(a,R)-{a}
but not in a .
removable singularity:the point a is called a removable singularity if
there is an analytic function
g:B(a,R)?? such that g(z)=f(z) for 0<|z-a|example?-
the function sinz/z,1/z,and exp??1/z,all have ? isolated singularities at z=0
however ,only sinz/z has aremovable singularity .and if f has an isolated singularity at a then the point z=a is
a removable singularity iff.
lim?(z?a)?(z-a)f(z)=0,a harmonic function real valued function h of two variables x and y is said to be harmonic
on a domain D ,if
1- h is continuous partial derivatives of first and second order.
(i.e hx ,hy ,hxx ,hyy ,hxy are continuous on D)
2- h satisfies laplace^ s equation .
h_xx+h_yy=0
i.e, (?^2 h)/(?x^2 )+(?^2 h)/(?y^2 )=0