Harmonic functions.
Suppose that
is analytic in a neighborhood
of . Moreover suppose that the partial derivatives of
and
are differentiable and that the second partial
derivatives are continuous functions on
From Cauchy-Riemann equations follows:
As the second partial derivatives are continuous on
we have
and
and
vxy  v yx . It follows that:
Definition
be a function of two real variables
Let
defined over a domain
and
. Suppose that
has second order partial derivatives on
. The function
is called an harmonic function if it verifies the equation:
Example
Let
. The function is an entire function, as we proved
previously. With
and
On the one hand, we have:
On the other hand, we have:
The functions
and
are both harmonic.
, we have:
Example
Let
This function has partial derivatives of any order over
Let us compute the needed ones:
Thus,
i.e.
.
.
is harmonic over
Definition
Let be a function of two real variables, harmonic over a domain
be a function of two real variables, defined over
. Let
and such that
is analytic over
. Then
is called an harmonic conjugate of
Example
Let
. This is a polynomial function,
thus it has partial derivatives of any order.
The function
If there exists
then
and
.
is harmonic:
such that
is analytic over
verify the Cauchy-Riemann equations:
respectively.
are independent of and
where
and
As these two formulas define the same function,
and
must be constant, i.e
where
Finally, we have:
Example
.
Let
be the function defined by
This function has partial derivatives of any order over
We have:
Let us look for points where
therefore,
is harmonic:
is harmonic
.