Hi just a bit of help needed here as I don;t know where to start:

Part (A)

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Suppose are analytic in some domain D. Show that both u and v are constant functions..?

I guess we have to use the CRE here but not really sure how to approach this..?

Part (B)

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Let f be a holomorphic function on the punctured disk where R>0 is fixed. What is the formulae for c_n in the Laurent expansion:

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Using these formulae, prove that if f is bounded on D'(0,R), it has a removable singularity at 0.

- Well I know that:

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Any suggestions from here?

PART (C)

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Find the maximal radius R>0 for which the function is holomorphic in D'(0,R) and find the principal part of its Laurent expansion about z_0=0

??

Any help would be greatly appreciated.

Thanks a lot