When all in
are differentiable, the derivative
is a function of
in
. This is called a derivative of
and denoted by
.
(1) If are differentiable in
, then the followings hold.
(i)
(ii)
(iii)
(2) If is differentiable in
,
is differentiable in
, then the composite function
is differentiable in
and
.
(3) If is differentiable, then
is continuous.
A function
defined in
is differentiable at
in
. Then keeping
constant and bringing
colser to 0, we have
When is differentiable at any point in the domain
,
is said to be analytic in
.
An analytic function in entire plane
is called entire function.
Note1. When a function is analytic at , it means that it is analytic including not only
but also its neighborhood.
Note2. If is analytic, it is continuous (becouse it is differentiable).
Note We assumed that and
have continuous second-order partial derivatives, but We don't need this assumption because the holomorphic function is separately proved to be differentiable many times.
Note This theorem is a strong enough condition and effective for determining anayticity.
(2)
implies
,
,
,
. Then,
. Thus,
is a harmonic function. Next,
is analytic. Then by the Cauchy-Riemann's equation, we have
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2. Differentiate the following functions.
3. When
, check the analyticity of the following functions and if it is analytic, find the derivative.