True or False? (Complex Analysis) Watch

TheEd
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#1
Report Thread starter 7 years ago
#1
S is a star-shaped open subset of \mathbb{C}, f is a holomorphic function from S to \mathbb{C}, z_0 is an element of S.

I've just come out an exam and wondered whether the following 2 statements are true or false:

1 Let g be a holomorphic function on S \subseteq \mathbb{C}, with the exception of a pole of order N at z_0. If the Laurent Series of g around z_0 is

\displaystyle \sum_{n=-N}^{\infty} a_n ( z - z_0 )^n

for z \in D'(z_0, R) for some R>0 (and D(z_0 , R) \subseteq S) and constants a_n \in \mathbb{C}, then the residue of g at z_0 is given by a_{-1}.

2 Suppose S = D(z_0, R) for some R>0 and

\displaystyle f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n

for all z \in S and some constants a_n \in \mathbb{C}. Then necessarily a_0 = f(z_0) and a_n = f^{(n)}(z_0) for all n\geq 1.
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nuodai
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#2
Report 7 years ago
#2
At a glance, #1 is true and #2 is partly true but not quite (you're missing out a n! somewhere).
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TheEd
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#3
Report Thread starter 7 years ago
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(Original post by nuodai)
At a glance, #1 is true and #2 is partly true but not quite (you're missing out a n! somewhere).
OK yes that's what I thought. I'm confident about the ones below but could you just quickly verify:

(i) if f is bounded on S then f is a constant function FALSE

(ii) if g is another function which is holomorphic on S then the product fg is holomorphic on S TRUE

(iii) the series \sum_{n=0}^{\infty} z^n / n^2 converges for |z|=1 TRUE

(iv) since S is star-shaped, if \gamma is a contour in S then \int_{\gamma} f(z)\;dz = 2\pi i FALSE

(v) if |f(z)| \leq 1 on S and \gamma has length 2\pi then \left | \int_{\gamma} f(z) \;dz \right | \leq 2\pi TRUE
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