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Some features of derivative of complex function

As it follows from the above analysis, in (42) only is the variable, all other variables - , ₁ and ₂ - become the parameters. Their numerical values are not given in the statement of problem, which we can use in finding the solution of (40).

We would like to recall one more detail. After we wrote the total derivative in the form (23), (29), the yielded derivatives with respect to become the derivatives in a trivial meaning, so the substitution

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is true and (42) will take the form

(43)

Let us seek the solution of (43) in the form

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

Let us take

With it we yield

If in (47) the term

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this equation turns into the known Helmholtz differential equation having the standard solution

The possibility of (48) turning into zero is caused by the free choice (under the statement of problem) of the parameters , ₁  and   ₂.