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Some pecularities of derivative of complex function

It is obvious from the above analysis that in (42) only is a variable, all other variables -  , ₁ and ₂  - become the parameters. And their quantities are not fixed by the statement of problem; this can be used when finding the solution of (40).

Note also that after writing the total derivative in the forms (23) and (29), the obtained derivatives with respect to become the derivatives in a common sense. In this connection the substitution

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is true, and (42) takes the following form:

(43)

We will seek the solution of (43) in the form

where  is some parameter independent of . Substituting (44) into (43), we yield

Take

If in (47)

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then this equation becomes a known Helmholtz differential equation having the standard solution

The equation (48) can be zero because of free choosing the parameters  , ₁   and  ₂   given by the statement of problem.

To determine the conditions at which (48) is zero, equalise to zero the real and imaginary parts of the left-hand part of the equality: