# True or False? (Complex Analysis)Watch

#1
is a star-shaped open subset of , is a holomorphic function from to , is an element of .

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

1 Let be a holomorphic function on , with the exception of a pole of order at . If the Laurent Series of around is

for for some (and ) and constants , then the residue of at is given by .

2 Suppose for some and

for all and some constants . Then necessarily and for all .
0
7 years ago
#2
At a glance, #1 is true and #2 is partly true but not quite (you're missing out a somewhere).
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#3
(Original post by nuodai)
At a glance, #1 is true and #2 is partly true but not quite (you're missing out a 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 converges for TRUE

(iv) since S is star-shaped, if is a contour in S then FALSE

(v) if on S and has length then TRUE
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