Complex analysis - argument principle

[sweetvirgogirl]

(changes in arg h (z) as z traverses lambda)/(2pi) =
# of zeroes of h inside lambda +
# of holes of h inside lambda

now the doubt i have is what happens when the change i get in h (z) is say 9 pi/2 ... because then i would have a 2.5 on left side of the eqn ... so do i round it up and and say the function will have 2 zeros? (assuming it has no holes)

 

[shmoe]

Can you come up with an example where the change in arg h(z) around a closed curve is 9pi/2? Or anything that isn't an integer multiple of 2pi?

 

[mathwonk]

how can the change in argument, i.e. angle, be other than an integral multiple of 2pi, when you go around a closed loop?

 

[sweetvirgogirl]

okay ... so i believe i am doing something wrong then ...
f(z) = z^9 +5z^2 + 3
(i have to determine number of zeroes in the first quadrant)

i came up with the 9pi/2 ... i am sure i am doing something wrong ... i should be getting 8 pi/2, right?

 

[shmoe]

It would help if you show your work, but I have a guess where the problem is. You are probably looking at the contour from 0 to R, then along the circle of radius R centered at the origin to i*R, then back to zero, (for some R sufficiently large). I don't think you took into account what happens to the argument from i*R to 0. What is f doing along this segment?

 

[sweetvirgogirl]

It would help if you show your work, but I have a guess where the problem is. You are probably looking at the contour from 0 to R, then along the circle of radius R centered at the origin to i*R, then back to zero, (for some R sufficiently large). I don't think you took into account what happens to the argument from i*R to 0. What is f doing along this segment?

uhh ... sorry ...
now i am getting the answer quite fine ... just want to make sure what i am doing is right ...
so it travels 9pi/2 counting 0 to R and along the radius of R centered at origin... to iR ... but when it reaches iR, it travel clockwise to land on the positive real axis ... so i subtract that from 9 pi/2 ... which means in total... it has traveled 4 pi

 

[shmoe]

That's essentially it. I'd expect more detail on the changes over the pieces of the contour, I assume you have but haven't posted (which is fine if you're happy with that part!).

 

[mathwonk]

sweetness and light, are you actually thinking about these many questions before asking them? it just seems that is unlikely given how many questions you are asking per day on different topics.
just a suggestion.