proof of the Cauchy-Riemann equations
Existence of complex derivative implies the Cauchy-Riemann equations.
Suppose that the complex derivative
(1) |
exists for some . This means that for all , there exists a , such that for all complex with , we have
Henceforth, set
If is real, then the above limit reduces to a partial derivative in , i.e.
Taking the limit with an imaginary we deduce that
Therefore
and breaking this relation up into its real and imaginary parts gives the Cauchy-Riemann equations.
The Cauchy-Riemann equations imply the existence of a complex derivative.
Suppose that the Cauchy-Riemann equations
hold for a fixed , and that all the partial derivatives are continuous at as well. The continuity implies that all directional derivatives exist as well. In other words, for and we have
with a similar relation holding for . Combining the two scalar relations into a vector relation we obtain
Note that the Cauchy-Riemann equations imply that the matrix-vector product above is equivalent to the product of two complex numbers, namely
Setting
we can therefore rewrite the above limit relation as
which is the complex limit definition of shown in (1).
Title | proof of the Cauchy-Riemann equations |
---|---|
Canonical name | ProofOfTheCauchyRiemannEquations |
Date of creation | 2013-03-22 12:55:39 |
Last modified on | 2013-03-22 12:55:39 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 6 |
Author | rmilson (146) |
Entry type | Proof |
Classification | msc 30E99 |
Defines | complex derivative |