88 - 90

Some features of derivative of complex function

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

25)

Noting that it follows from (20) that

and substituting sequentially these expressions to (24), we yield

(27)

To the point, the intermediate expression (25) in case if the Caushy - Riemann conditions

are true becomes independent of the angle . This last is one of proofs that the Caushy - Riemann conditions only define the class of centrally symmetrical functions of complex variable.

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To write the form of second total derivative of the function of complex variable, it is sufficient to use the principle of double sequential mapping:

Given the direction in which we take the first and second derivatives can generally be not same,

To yield the second derivative with respect to z in coordinate form, substitute to (29) the expression for dw/dz from (27). After transformation yield

For functions of complex variable satisfying the Caushy – Riemann conditions, we can find the second derivative, noting that in accordance with (25) and (26), the equality

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is the complex analogue of Caushy - Riemann equations.