Hi just a bit of help needed here as I don;t know where to start:(adsbygoogle = window.adsbygoogle || []).push({});

Part (A)

----------------------------

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

[itex]

f(z) = \sum\limits_{n = - \infty }^\infty {c_n z_n }[/itex].

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

- Well I know that:

[itex]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!}}[/itex].

Any suggestions from here?

PART (C)

-------------------

Find the maximal radius R>0 for which the function [itex]

f(z) = (\sin z)^{ - 1}[/itex] 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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Laurent's Theorem

**Physics Forums | Science Articles, Homework Help, Discussion**