50.1.1 Cauchy Riemann Equations
Next consider the very important Cauchy Riemann equations which give conditions
under which complex valued functions of a complex variable are analytic.
Theorem 50.1.5 Let Ω be an open subset of ℂ and let f : Ω → ℂ be a function, such
that for z = x + iy ∈ Ω,
Then f is analytic if and only if u,v are C1
Proof: Suppose f is analytic first. Then letting t ∈ ℝ,
This verifies the Cauchy Riemann equations. We are assuming that z → f′
continuous. Therefore, the partial derivatives of
are also continuous. To see this,
note that from the formulas for f′
given above, and letting
is continuous since
if and only if
z1 → z.
The other cases are similar.
Now suppose the Cauchy Riemann equations hold and the functions, u and v are
Then letting h
We know u and v are both differentiable and so
Dividing by h and using the Cauchy Riemann equations,
Taking the limit as h → 0,
It follows from this formula and the assumption that u,v are C1
It is routine to verify that all the usual rules of derivatives hold for analytic functions.
In particular, the product rule, the chain rule, and quotient rule.