http://www.math.ualberta.ca/~runde/files/ass411-1.pdf
I can't do #3 or #4
I have a feeling I can do 3 but I'm not quite sure where to go from
and for 4 I know that to show it's not injective show that exp(0,0) = exp(0,2pi)
http://www.math.ualberta.ca/~runde/files/ass411-1.pdf
I can't do #3 or #4
I have a feeling I can do 3 but I'm not quite sure where to go from
and for 4 I know that to show it's not injective show that exp(0,0) = exp(0,2pi)
3)The problem here is that how is really defined what means? I assume it means that we think of as (since the complex numbers are simply points in formally).
Now define and .
This means .
However, and by Cauchy-Riemann equations.
4)Note thus it is not injective.
May I offer a different perspective?
If and then:
.....
Well there you go: If is analytic then the partials satisfy the Cauchy-Riemann equations so and the only way for
is for the partials to satisfy the Cauchy-Riemann equations, that is, must be analytic.
thank you but i got that one already
also for #4
where it says show that exp(C) = C/{0}
what does that mean? the notation is unfamiliar to me and hasn't been explained in course lecture or notes
also there is no textbook for the class which makes this particularly frustrating
Hey jb. It's a function at a domain. Suppose I define the unit circle as the domain
Then I could say, what is the function at that domain? or what is meaning how does the function map the domain . Same with . How does the exponential function map all of ? It does so by the expression:
meaning it maps all of to the deleted neighborhood: which is minus the origin.
It may be late (since it's past 3pm...), but I don't care
I'm also learning how to do this stuff !
For #5 c), use Cauchy-Hadamard theorem : Cauchy-Hadamard theorem - Wikipedia, the free encyclopedia
My teacher says the best way above all to solve this problem is to use Abel's theorem :
(R is the radius of convergence)
Here, Hadamard's theorem is more straightforward.
#6 is above my capacities...