# Laurent's Theorem

by mathfied
Tags: laurent, theorem
 P: 16 Hi just a bit of help needed here as I don;t know where to start: Part (A) ---------------------------- Suppose $f(z) = u(x,y) + iv(x,y)\;and\;g(z) = v(x,y) + iu(x,y)$ 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) ---------------------------- Let f be a holomorphic function on the punctured disk $D'(0,R) = \left\{ {z \in C:0 < |z| < R} \right\}$ where R>0 is fixed. What is the formulae for c_n in the Laurent expansion: $f(z) = \sum\limits_{n = - \infty }^\infty {c_n z_n }$. Using these formulae, prove that if f is bounded on D'(0,R), it has a removable singularity at 0. - Well I know that: $c_n = \frac{1} {{2\pi i}}\int\limits_{\gamma _r }^{} {\frac{{f(s)}} {{(s - z_0 )^{n + 1} }}} ds = \frac{{f^{(n)} (z_0 )}} {{n!}}$. Any suggestions from here? PART (C) ------------------- Find the maximal radius R>0 for which the function $f(z) = (\sin z)^{ - 1}$ 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 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution