M417
Let f(z) have an isolated singularity at z = 0.
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1.
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Suppose that |f(z)| is bounded away from 0 for z near 0; i\. e\. , there are R > 0, δ > 0, such that for 0 < |z| < R, f(z) is analytic and |f(z)| > δ.
Then f does not have an essential singularity at z=0.
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2.
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If f has an essential singularity at z = 0, and w0 is
any complex number, then there is a sequence {zn} such
that
limn → ∞zn = 0 and limn→∞f(zn)=w0.
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3.
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If limz → 0|f(z)| = ∞, then
f has a pole at z=0, but f(z) does not have an essential
singularity at z = 0.
One can now show the following result:
Theorem Suppose that g(z) is an entire function such that limz → ∞g(z) = ∞. Then g(z) is a polynomial.
The function f(z) = g([1/z])
satisfies the hypotheses of the third exercise. For z near 0, f(z) = ∑n=0Nbn z−n and
g(z) = ∑n=0Nbn zn.
Why are there no other
nonzero terms?
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On 06 Feb 2017, 13:23.