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# Laurent's Theorem watch

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

Part (A)
----------------------------
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:
.

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

- Well I know that:
.
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
2. (A) Look to show u_x = u_y = 0
(B) Apply an estimation theorem to that expression for c_n
(C) Do you know the sinz series?
3. hmm so for part (A)
u_x = v_y = -u_x AND
u_y = -v_x = v_x

so u and v are constant because u_x = -u_x and -v_x = v_x

is that correct?
4. for part (C):
yes the sin z series is:

what would be the next step please?
5. Personally, I'd look at sin z / z.

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