Cauchy-Riemann conditions in Cartesian coordinates
The Cauchy-Riemann equations offer a test for analyticity, connecting the real and imaginary components of f(z) in Cartesian coordinates.
A complex function f(z) can be written as f(z) = u(x, y) + i v(x, y), where u(x, y) and v(x, y) are real-valued functions, and z = x + i y. The function is analytic at a point if its derivative with respect to z exists and is unique. Explicitly, this derivative is:
f^\prime(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z}
This limit must not depend on the direction along which \Delta z approaches zero. Expressing f(z) in terms of u and v, and requiring this uniqueness, yields the Cauchy-Riemann conditions:
\begin{aligned} \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} &= -\frac{\partial v}{\partial x} \end{aligned}
If these equations hold at a point, and the partial derivatives of u and v are continuous there, f(z) is analytic at that point.
Example: Analyticity of f(z) = z^2
Take f(z) = z^2. Expanding in Cartesian form,
f(z) = (x + i y)^2 = x^2 - y^2 + i 2 x y
u(x, y) = x^2 - y^2 and v(x, y) = 2 x y. The partial derivatives are:
\begin{aligned} & \frac{\partial u}{\partial x} = 2x \\ & \frac{\partial u}{\partial y} = -2y \\ & \frac{\partial v}{\partial x} = 2y \\ & \frac{\partial v}{\partial y} = 2x \end{aligned}
Substituting into the Cauchy-Riemann equations, both are satisfied for all (x, y). All partial derivatives are continuous everywhere, so f(z) = z^2 is analytic throughout the complex plane.
The existence of these partial derivatives and the satisfaction of the Cauchy-Riemann conditions provides an efficient criterion to assess analyticity. This approach simplifies the task of checking for complex differentiability for various functions.
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