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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.

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?

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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