# Integral qWatch

#1
see q 5.11 in attachment.

in the solutions, why are we going in a clockwise direction on the inner boundary?
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9 years ago
#2
(Original post by latentcorpse)
???
latex - there is some kind of encoding for that attachment
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9 years ago
#3
Open office doesn't like it. Could you post it up on TSR please?
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#4
ok. suppose that f(z) is holomorphic on the annulus . Prove that if , then

If R is the closed annulus then R is entirely contained in U. Apply Cauchy's Integral Formula for to get the result and remember that the inner boundary gets a clockwise orientation (which explains the negative in the 2nd term)

my question is why is the inner boundary orientated clockwise???
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9 years ago
#5
Did you mean ?

my question is why is the inner boundary orientated clockwise???
This is impossible to explain without a diagram.

Have a look at Figure 1.19 (page 33) of http://media.wiley.com/product_data/...3527406379.pdf. The idea is to use one closed contour to go around both your original contours, by using 'contour walls' (the bits A and B in the diagram) to connect the two contours. Because A, B are the same path, only in opposite directions, they cancel out. But if you trace the entire contour, you'll see you end up going clockwise around the inner contour.

Incidentally, I find it very hard to believe this hasn't been covered in your lecture notes.
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#6
it has. i found it just after i posted this lol - although the diagram in those notes is much clearer than the one i copied down in lectures.

a simple question that i am unsure about.

Is a satisfactory definition of holomorphic:

"If U is an open subset of and be a class function. f is holomorphic iff it's complex differentiable on U iff it satisfies the Cauchy Riemann equations on U."

The reason I ask is that my notes seem to be making use of the fact that holomorphic functions are continuous and this isn't mentioned in the definition we were given

cheers
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9 years ago
#7
I don't see that's a satisfactory definition. (Simply put, if you have more than one set of "iff" in your statement, it probably isn't a definition).

I'd go for something like:

"If U is an open subset of , then is holomorphic on U iff f is complex differentiable on U.

Note that complex differentiable trivially implies continuity.
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