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Some pecularities of derivative of complex function

To transit in (24) from partial derivative with respect to to those with respect to x and y, we have to make a substitution:

25)

And it follows from (20) that

Substituting sequentially (26) into (25) and (24), we yield

(27)

To the point, when Caushy - Riemann conditions

are true, the intermediate expression (27) becomes independent on the angle  . This last is one of the proofs that in their essence Caushy - Riemann conditions only define the class of functions of a complex variable having a central symmetry.

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To write the form of the second total derivative for the function of a complex variable, use the principle of the double sequential mapping

Note that the differentiation directions of the first and second derivatives for the function of a complex variable can be generally not the same. So

To present the second derivative in co-ordinate form, substitute to (27) the value of dw/dz taken from (27). After transformation we yield

For functions of complex variable satisfying Caushy - Riemann conditions, we can find the second derivative, noting that according to (25), the following equality is a complex analogue of Caushy - Riemann equations:

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