Cauchy-Riemann Equation

Cauchy-Riemann Equation
Theoreom 1(Cauchy-Riemann equation): Suppose that
f(z) = f(x + iy) = u(x, y) + iv(x, y)
is differentiable at the point z 0 = x 0 + iy 0 . Then the partial derivative of u and v exist
at the point (x 0 , y 0 ), and is given by
and also
f ′ (z 0 ) = u x (x 0 , y 0 ) + iv x (x 0 , y 0 ) (1)
f ′ (z 0 ) = v y (x 0 , y 0 ) − iu y (x 0 , y 0 ). (2)
Equating the real and imaginary parts of equations (1) and equations (2) gives
u x (x 0 , y 0 ) = v y (x 0 , y 0 ), v x (x 0 , y 0 ) = −u y (x 0 , y 0 ). (3)
Remarks:
1. If f is differentiable at z 0 , then the Cauchy-Riemann equation (3) will be satisfied
at z 0 , and we can use either (1) or (2) to compute f ′ (z 0 ).
2. Taking the contrapositive, if equations (3) are not satisfied at z 0 , then we know
that f is not differentiable at z 0
3. Even if equations (3) are satisfied at z 0 , we can not necessarily conclude that
f is differentiable at z 0 . That is the mere salification of the Cauchy-Riemann
equation is not sufficient to guarantee the differentiability of a function. The
following theorem, however, gives conditions that guarantee the differentiability
of f at z 0 .
Theoreom 2(Cauchy-Riemann conditions for differentiability): Let
f(z) = f(x + iy) = u(x, y) + iv(x, y)
be a continuous function that is defined in some neighborhood of the point z 0 = x 0 +iy 0 .
If all the partial derivative u x , u y , v x and v y are continuous at the point (x 0 , y 0 and
if the Cauchy-Riemann equations u x (x 0 , y 0 ) = v y (x 0 , y 0 ) and u y (x 0 , y 0 ) = −v x (x 0 , y 0 )
hold at (x 0 , y 0 ), then f is differentiable at z 0 , and the derivative f ′ (z 0 ) can be computed
with either equation (1) or (2).
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Theoreom 3(fundamental theorem characterizing complex functions): Let
Ω ∈ C be an open set. The following properties are equivalent:
(a) The function f(z) has a continuous complex derivative f ′ (z) for all z ∈ Ω;
(b) The real and imaginary parts of f(z) have continuous partial derivative and
satisfy the Cauchy-Riemann equations (3) in Ω;
(c) The function f(z) is analytic for all z ∈ Ω, and so is the differentiable and has
a convergent power series expansion as each point z 0 ∈ Ω. The radius of convergence
ρ is at least as large as the distance from z 0 to the boundary ∂Ω.
Definition of Analytic. We say that the complex function f is analytic at the point
z 0 if there is some ε > 0 such that f ′ (z) exists for all z ∈ D ε (z 0 ). In other words, f
must be differentiable not only at z 0 , but also at all points in some ε neighborhood of
z 0 .
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