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

Part (A)

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Suppose $\displaystyle f(z) = u(x,y) + iv(x,y)\;and\;g(z) = v(x,y) + iu(x,y)$ 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 $\displaystyle D'(0,R) = \left\{ {z \in C:0 < |z| < R} \right\}$ where R>0 is fixed. What is the formulae for c_n in the Laurent expansion:

$\displaystyle

f(z) = \sum\limits_{n = - \infty }^\infty {c_n z_n }$.

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

- Well I know that:

$\displaystyle c_n = \frac{1}

{{2\pi i}}\int\limits_{\gamma _r }^{} {\frac{{f(s)}}

{{(s - z_0 )^{n + 1} }}} ds = \frac{{f^{(n)} (z_0 )}}

{{n!}}$.

Any suggestions from here?

PART (C)

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Find the maximal radius R>0 for which the function $\displaystyle

f(z) = (\sin z)^{ - 1}$ 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