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Ted123
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[itex]S[/itex] is a star-shaped open subset of [itex]\mathbb{C}[/itex], [itex]f[/itex] is a holomorphic function from [itex]S[/itex] to [itex]\mathbb{C}[/itex], [itex]z_0[/itex] is an element of [itex]S[/itex].
I've just come out an exam and wondered whether the following 2 statements are true or false:
1 Let [itex]g[/itex] be a holomorphic function on [itex]S \subseteq \mathbb{C}[/itex], with the exception of a pole of order [itex]N[/itex] at [itex]z_0[/itex]. If the Laurent Series of [itex]g[/itex] around [itex]z_0[/itex] is
[itex]\displaystyle \sum_{n=-N}^{\infty} a_n ( z - z_0 )^n[/itex]
for [itex]z \in D'(z_0, R)[/itex] for some [itex]R>0[/itex] (and [itex]D(z_0 , R) \subseteq S[/itex]) and constants [itex]a_n \in \mathbb{C}[/itex], then the residue of [itex]g[/itex] at [itex]z_0[/itex] is given by [itex]a_{-1}[/itex].
2 Suppose [itex]S = D(z_0, R)[/itex] for some [itex]R>0[/itex] and
[itex]\displaystyle f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n[/itex]
for all [itex]z \in S[/itex] and some constants [itex]a_n \in \mathbb{C}[/itex]. Then necessarily [itex]a_0 = f(z_0)[/itex] and [itex]a_n = f^{(n)}(z_0)[/itex] for all [itex]n\geq 1[/itex].
I've just come out an exam and wondered whether the following 2 statements are true or false:
1 Let [itex]g[/itex] be a holomorphic function on [itex]S \subseteq \mathbb{C}[/itex], with the exception of a pole of order [itex]N[/itex] at [itex]z_0[/itex]. If the Laurent Series of [itex]g[/itex] around [itex]z_0[/itex] is
[itex]\displaystyle \sum_{n=-N}^{\infty} a_n ( z - z_0 )^n[/itex]
for [itex]z \in D'(z_0, R)[/itex] for some [itex]R>0[/itex] (and [itex]D(z_0 , R) \subseteq S[/itex]) and constants [itex]a_n \in \mathbb{C}[/itex], then the residue of [itex]g[/itex] at [itex]z_0[/itex] is given by [itex]a_{-1}[/itex].
2 Suppose [itex]S = D(z_0, R)[/itex] for some [itex]R>0[/itex] and
[itex]\displaystyle f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n[/itex]
for all [itex]z \in S[/itex] and some constants [itex]a_n \in \mathbb{C}[/itex]. Then necessarily [itex]a_0 = f(z_0)[/itex] and [itex]a_n = f^{(n)}(z_0)[/itex] for all [itex]n\geq 1[/itex].