A) Write CRE for both functions.
B) See here.
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:
.
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
For part (c), the Laurent series for is kind of messy. Could certainly use long division to get the first few terms. Here's one way to get the rest of the terms: First look at Mathworld under Bernoulli numbers. There they show how to derive:
Now, if:
then split up the series for into two sums, and figure out how the series for could be modified to represent those two series. Here's a start: what happens when I substitute in the series for ?
Probably an easier way to get the terms though. Also, the radius of convergence extends to the nearest singularity which is
As shawsend has said, R = π. To find the principal part of the Laurent expansion, you might guess that since sin(z) is close to z when z is small, the principal part of 1/sin(z) ought to be 1/z. You can then justify that guess as follows.
Let . Since z-sin(z) has degree 3 and z*sin(z) has degree 2, it follows that g(z) has a removable singularity at the origin and is therefore holomorphic in a neighbourhood of the origin. Therefore has the same principal part as 1/z (namely 1/z).
I just realized I over-killed it: You only wanted the singular part. Do what Opalg said or just use long division:
and the first term when you do that is then all the rest are positive powers of . But hey Mathfied, thanks a bunch. I hadn't known about all that stuff I posted above until I researched it and worked it out myself today. The Bernoulli stuff is interesting I think.