Cauchy-Riemann equations

March 31, 2011

Definition 1. Let f(z) be a complex valued function defined in a neighborhood of a point z_{1} = x_{1} + iy_{1}. Then f is complex differentiable at z_{1} if

f(z) = f(z_{1}) + p(z)(z − z_{1}),  (1)

where p is continuous at z_{1}. We define the complex derivative of f at z_{1} to be the value of p at z_{1}.

f′(z_{1}) = p(z_{1}).  (2)

Theorem 1. Let f(z) = u(x, y) + iv(x, y). Then f is complex differentiable at z_{1} if and only if u and v are real differentiable at (x_{1}, y_{1}) and

u_{x}(x_{1}, y_{1}) = v_{y}(x_{1}, y_{1}), u_{y}(x_{1}, y_{1}) = −v_{x}(x_{1}, y_{1}).  (3)

These are called the Cauchy-Riemann equations.

Proof. Let us write equation (1) in matrix form:

[\begin{matrix}u(x, y) \\ v(x, y \end{matrix}] = [\begin{matrix}u(x_{1}, y_{1}) \\ v(x_{1}, y_{1}) \end{matrix}] + [\begin{matrix}a(x, y) & −b(x, y) \\ b(x, y) & a(x, y) \end{matrix}] [\begin{matrix}x − x_{1} \\ y − y_{1} \end{matrix}]  (4)

We can write this line by line

u(x, y) =u(x_{1}, y_{1}) + a(x, y)(x − x_{1}) − b(x, y)(y − y_{1})  (5)
v(x, y) =v(x_{1}, y_{1}) + b(x, y)(x − x_{1}) + a(x, y)(y − y_{1}).  (6)

In this form we see that u and v are real differentiable at (x_{1}, y_{1}) and u_{x} = v_{y}, u_{y} = −v_{x} at (x_{1}, y_{1}).

Next suppose u and v are real differentiable at (x_{1}, y_{1}) and (3) is true. Let a_{1} = u_{x}(x_{1}, y_{1}) = v_{y}(x_{1}, y_{1}), b_{1} = v_{x}(x_{1}, y_{1}) = −u_{y}(x_{1}, y_{1}). Let’s write the definition of real differentiability as follows

u(x, y) =u(x_{1}, y_{1}) + a_{1}(x − x_{1}) − b_{1}(y − y_{1}) + e(x, y)  (7)
v(x, y) =v(x_{1}, y_{1}) + b_{1}(x − x_{1}) + a_{1}(y − y_{1}) + d(x, y),  (8)

where e(x, y)/|z − z_{1}| → 0, d(x, y)/|z − z_{1}| → 0 as z → z_{1}. So

f(z) = f(z_{1}) + (a_{1} + ib_{1})(z − z_{1}) + ϵ(z)(z − z_{1}), where ϵ(z) = \frac{e(z) + id(z)}{z − z_{1}}.  (9)

and ϵ(z) → 0 as z → z_{1}. This can be rewritten as

f(z) = f(z_{1}) + p(z)(z − z_{1}),  (10)

where p(z) = a_{1} + ib_{1} + ϵ(z) and p(z) → a_{1} + ib_{1} as z → z_{1} so f is complex differentiable at z_{1}.