Complex analysis - argument principle

In summary, the conversation discusses the calculation of the change in argument of a function h(z) as z traverses a closed curve. The equation states that this change is equal to the number of zeroes of h inside the curve plus the number of holes of h inside the curve. There is a question about what happens when the change is not an integer multiple of 2pi, and a request for an example. The conversation also touches on the calculation of the number of zeroes of a function in a specific quadrant, with some discussion about the contour used and its impact on the change in argument. Finally, there is a suggestion to consider the amount of thought put into asking multiple questions on different topics in a short period of time.
  • #1
sweetvirgogirl
116
0
(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)
 
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  • #2
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?
 
  • #3
how can the change in argument, i.e. angle, be other than an integral multiple of 2pi, when you go around a closed loop?
 
  • #4
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?
 
  • #5
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?
 
  • #6
shmoe said:
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
 
  • #7
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!).
 
  • #8
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.
 

1. What is the argument principle in complex analysis?

The argument principle is a fundamental theorem in complex analysis that relates the number of zeros and poles of a function to the number of times the function's argument winds around a point in the complex plane. It is also known as the winding number theorem.

2. How is the argument principle used in solving complex integration problems?

The argument principle is used to evaluate integrals that cannot be solved using traditional methods. It involves finding the number of zeros and poles of a function inside a closed contour, and using this information to calculate the integral using the Cauchy residue theorem.

3. Can the argument principle be applied to functions with branch cuts?

Yes, the argument principle can be applied to functions with branch cuts. However, the contour used in the calculation must be chosen carefully to avoid crossing the branch cuts, as this can lead to incorrect results.

4. How does the argument principle relate to the concept of analytic continuation?

The argument principle is closely related to the concept of analytic continuation, as both involve the behavior of complex functions around singularities. The argument principle helps to determine the behavior of a function near a singularity, which can then be used to analytically continue the function to a larger domain.

5. Are there any practical applications of the argument principle?

Yes, the argument principle has many practical applications in mathematics, physics, and engineering. It is used in the study of differential equations, signal processing, and fluid mechanics, to name a few. It also has applications in number theory, where it is used to study the distribution of prime numbers.

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