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)
and
- (1b)
- Proof:
- Suppose that
is a function of a complex number z. Then the complex derivative of ƒ at a point z0 is defined by
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
On the other hand, approaching along the imaginary axis,
The equality of the derivative of ƒ taken along the two axes is
- Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of
(this is called a domain in ). - 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.
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