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I am reading the book: "Theory of Functions of a Complex Variable" by A. I. Markushevich (Part 1) ...
I need some help with an aspect of the proof of Theorem 7.1 ...The statement of Theorem 7.1 reads as follows:
View attachment 9330At the start of the above proof by Markushevich we read the following:
"If $$f(z)$$ has a derivative $$f'_E(z_0)$$ at $$z_0$$, then by definition
$$\frac{ \Delta_E f(z) }{ \Delta z } = f'_E(z_0) + \epsilon ( z, z_0 )$$
where $$\epsilon ( z, z_0 ) \to 0$$ as $$\Delta z \to 0$$. ... ... "Now previously in Equation 7.1 at the start of Chapter 7, Markushevich has defined $$f'_E(z_0)$$ as follows:
$$f'_E(z_0) = \frac{ f(z) - f(z_0) }{ z - z_0 } = \frac{ \Delta_E f(z) }{ \Delta z }$$ ... ... ... (1)How exactly (formally and rigorously) is equation (1) exactly the same as $$\frac{ \Delta_E f(z) }{ \Delta z } = f'_E(z_0) + \epsilon ( z, z_0 )$$ ...... strictly speaking, shouldn't Markushevich be deriving $$\frac{ \Delta_E f(z) }{ \Delta z } = f'_E(z_0) + \epsilon ( z, z_0 )$$ ... from equation (1) ... Peter
I need some help with an aspect of the proof of Theorem 7.1 ...The statement of Theorem 7.1 reads as follows:
View attachment 9330At the start of the above proof by Markushevich we read the following:
"If $$f(z)$$ has a derivative $$f'_E(z_0)$$ at $$z_0$$, then by definition
$$\frac{ \Delta_E f(z) }{ \Delta z } = f'_E(z_0) + \epsilon ( z, z_0 )$$
where $$\epsilon ( z, z_0 ) \to 0$$ as $$\Delta z \to 0$$. ... ... "Now previously in Equation 7.1 at the start of Chapter 7, Markushevich has defined $$f'_E(z_0)$$ as follows:
$$f'_E(z_0) = \frac{ f(z) - f(z_0) }{ z - z_0 } = \frac{ \Delta_E f(z) }{ \Delta z }$$ ... ... ... (1)How exactly (formally and rigorously) is equation (1) exactly the same as $$\frac{ \Delta_E f(z) }{ \Delta z } = f'_E(z_0) + \epsilon ( z, z_0 )$$ ...... strictly speaking, shouldn't Markushevich be deriving $$\frac{ \Delta_E f(z) }{ \Delta z } = f'_E(z_0) + \epsilon ( z, z_0 )$$ ... from equation (1) ... Peter