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Question:

For the function $f(z) = 1 + e^{(1/2)\log z}$

a) sketch range

b) is it open, bounded closed or connected

c) find $\dfrac{d}{dz}$

for the range I was thinking to expand it to its complex log form as $e^{\log|z| + I\arg z} = e^{\log z} e^{\arg z}$ but am not sure from there.

I can't answer $b$ unless I solve $a$, but will be able to with the range in mind.

then for $c$ I think the derivative would be the same as if it were a real function correct? which would be $e^{(1/2)\log z} . 1/( 1/2\ln z)$ .

any help is appreciated thanks !

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    Please use MathJax to format math-text. Here is a useful link to help with the formatting: http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – 2017-02-16
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    What is the domain? – 2017-02-16

1 Answers 1

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Remember that we define a branch of a power as $e^{a\log(z)}$, so that by definition $e^{\frac{1}{2}\log(z)} = \sqrt{z}$. Hence the range of $f$ is just the half plane perpendicular to the choice of ray for the branch and then shifted by 1.

As for the derivative you are right except you did not take the derivative of $\log(z)$ or $e^{\frac{1}{2}z}$ correctly when you did the chain rule it looks like, so if you change that you should be fine. Alternatively you can just take the derivative of $1+\sqrt{z}$.