I'm not very clear of the problems below,so I may make some mistakes,if you point out them and explain to me,I'm reallly grateful.(adsbygoogle = window.adsbygoogle || []).push({});

1.If f(z) is an analytic function,why can we derivate it as a real function to get it's derivation?

I mean f'(z) should be [tex]f^' (z) = \frac{{\partial u}}{{\partial x}} + i\frac{{\partial v}}{{\partial x}}[/tex],we can get the derivation by this formula,but why can we just derivate it as a real function?For example,if [tex]f(z) = \log (z - a)[/tex],then it's derivation is

[tex]f'(z) = \frac{1}{{z - a}}[/tex]?

2.What on earth is principle-valued branch?

Why (z)^(1/2) is multiple-valued?Why we may choose for [tex]\Omega [/tex] the complement of the negative real axis z<=0 then it is a single-valued function?

I'm really confused of it.And why once the continuity is established the analyticity follows by derivation of the inverse function?

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# Some questions in Complex Analysis

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