ATAR Notes: Forum
Uni Stuff => Science => Faculties => Mathematics => Topic started by: Gloamglozer on August 27, 2011, 03:46:35 pm
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Please note that the below two questions are assignment questions. Please feel free to reveal as much or as little as you desire. Thank you.
Let =u(x,y) + iv(x,y))
be a complex function. Consider a general change of variables )
and )
producing the function:
 = f(x(\eta,\phi),y(\eta,\phi)))
1. Consider the change of variables 
and 
and show that
 + \frac{i}{2} (\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}))
 + \frac{i}{2} (\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}))
2. Explain why this shows that an analytic function is independent of 
.
EDIT: Post edited to put tildes in.
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that looks pretty confusing because
was used to define two different things, I am assuming that we are using the "new" in the partial derivatives, that's why I advise you to firstly write  = f(x(\eta,\phi),y(\eta,\phi)))
. Then you should be able to use the chain rule to find 
etc.
Who's your lecturer?
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Ahh ooops. There were supposed to be tildes on the partial derivatives of
etc.
Iwan Jensen is the lecturer this semester. Last semester they had Alex Ghitza/Paul Norbury. I think it was supposed to be Paul Pearce because we're using his notes as a backbone but for some reason Iwan was drafted in at short notice.
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Suppose
)
is an entire function such that | \leq |z|)
. Show that  = a + bz^2)
with 
.
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It suffices to show that if is an entire function with
| \leq z)
for all 
, then  = bz)
for some . The key is that being entire means that
 = \sum^{\infty}_{n=0}{a_n z^n})
with
}{\frac{f(z)}{z^n} \, dz})
for all 
. As you can choose 
as big or as small as you like, you can estimate simply and show that 
unless 
, in which case 
.