complex analysis, evaluation using residual theorem on branch of logarithm given below help Given - log?/2 is a bratch of logarithm where log??/2 has nonzevo z and is only analytic on C {?iy:y?0} - with its derivative z1? por z?a

Question

complex analysis, evaluation using residual theorem on branch of logarithm given below help

Given -

lo g ? /2

is a bratch of logarithm where

lo g ? ? /2

has nonzevo

z

and is only analytic on

C \ {? i y : y ? 0}

- with its derivative

\frac{1}{z}

por

z ? a < ? ? i y : y ? 0}

. -

?_{R} (t) = R e^{i t}

por

t ? [0, ?]

where

0 < r < 1 < R

. - Let

?_{r, R}

a closed curve with

[r, R] ? ?_{2} ? [? R_{1} ? r] ? ? ?_{r}

. Hint:

? ?_{R}

is just traced in opposte direation of

?_{R}

waluate using the Residue Theorem and identipy the onder involued ou cull:

?_{? r, R} \frac{l o g ? 1/2}{( z ^{2} + 1 ) ^{2}} z d z

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