The Cauchy-Riemann equations

In order for a complex function of a single complex variable to be differentiable it must be differentiable both parallell to the imaginary axis δy →0 and parallell to the real axis δx →0.

This condition leads to the CAuchy –Riemann equations-

The Cauchy–Riemann equations on a pair of real-valued functions of two real variables u(x,y) and v(x,y) are the two equations:

(1a) $\dfrac{ \partial u }{ \partial x } = \dfrac{ \partial v }{ \partial y } \,$

and

(1b) $\dfrac{ \partial u }{ \partial y } = -\dfrac{ \partial v }{ \partial x } \,$

Proof:

Suppose that

$f(z) = u(z) + i \cdot v(z)$

is a function of a complex number z. Then the complex derivative of ƒ at a point z₀ is defined by

$\lim_{\underset{h\in\mathbb{C}}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = f'(z_0)$

provided this limit exists.

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis; in either case it should give the same result. Approaching along the real axis, one finds

$\lim_{\underset{h\in\mathbb{R}}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = \frac{\partial f}{\partial x}(z_0).$

On the other hand, approaching along the imaginary axis,

$\lim_{\underset{h\in \mathbb{R}}{h\to 0}} \frac{f(z_0+ih)-f(z_0)}{ih} =\frac{1}{i}\frac{\partial f}{\partial y}(z_0).$

The equality of the derivative of ƒ taken along the two axes is

$i\frac{\partial f}{\partial x}(z_0)=\frac{\partial f}{\partial y}(z_0),$

Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of $\mathbb{C}$ (this is called a domain in $\mathbb{C}$ ).

function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function. This is not true for real differentiable functions.