Criterion for differentiability
⑴ real valued functions
and
are continuous.
⑵ have continuous 1st order partial derivative in the neighborhood of point
⑶
and
satisfy C-R equations at point
즉 복소함수 미분가능성은
- 실함수 u, v가 연속
- z 근방에서 연속인 1st order 편미분을 가짐
- u, v는 z에서 코시리만방정식을 만족
Criterion for analyticity
⑴ real valued functions
and
are continuous.
⑵ have continuous 1st order partial derivatives in domain D
⑶
and
satisfy C-R equations at all points of domain D
즉 복소함수 해석가능성은 위 미분가능성에서, 점(point) 대신 domain으로만 바꾸면 됨
To check for differentiability, we use C-R equations.
Is there an easy way to check if
is analytic in a given domain?
(Ans: Yes)
Theorem
If
is analytic in domain D
then
and
are harmonic functions
in the same domain.
Harmonic functions
A real valued function such as
and
- has continuous 2nd order partial derivatives in domain D
- satisfies Laplace equations
Example
(a) verify
is harmonic in the entire complex plane.
Soln.
으로 라플라스 방정식을 만족.
(b) Find the harmonic conjugate function of
Soln. Since both
-
we have this
and
-
we are trying to find this
must satisfy CR eqns
......① and
......②
we use partial integration of eqn ① with respect to
- we need to find
// y^3앞에 부호 -인듯
To find
and substitute into eqn ②
gives
Therefore
and the analytic fn is
tmp; 내 생각은
각각
로 편적분하면
이렇게 하는게 아니라 저렇게 원래 식에 넣어야..
TBW
Exponential, logarithmic, trigonometric and hyperbolic functions
Exponential functions
Example
Evaluate
Soln.
Analyticity
is analytic for all
because
1.
(이 두 함수가) continuous and have continuous 1st order partial derivatives at every point of complex plane
2. also satisfy C-R eqns at all points of the complex plane
Properties of
Periodicity