Property of the index of a function

In summary, the conversation discusses the relationship between a complex analytic function and the index of the function on a simply closed curve. It is stated that for the function to have a unique g(z) on the curve for a specific branch of the logarithm, the index of the function must be zero. The question is raised as to why this is necessary and the response is that it is a result of the math. The conversation also considers the converse, when the index is not zero, and the idea is proposed that understanding this relationship can be aided by considering the opposite scenario.
  • #1
Jim Kata
197
6
Hi, I'm working through a paper and I am quite stupid so some things that maybe obvious are not obvious to me. Say you have some have some complex analytic function that is defined on some simply closed curve, and the index of this function defined on this curve is zero,

[tex] \int_C \frac{f'(z)}{f(z)}dz =\Delta Arg(f(z)) = 0[/tex] Then there exists for a specific branch of the logarithm a g(z) such that exp(g(z))=f(z). My question is why do you need the index of f(z) to be zero for this curve for g(z) to be unique on the curve for a specific branch.
 
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  • #2
Um. Because that's how the math works out.
It sounds a bit like asking if x-a=0 then why does x have to be equal to a to make the relation true?

... what happens when the index is not zero?
 
  • #3
I'll rephrase the question. Let [tex] f(z)=R(z)exp(i\theta(z))[/tex] . Why is it when

[tex] ind_C(f(z))=0[/tex] then [tex] \alpha < \theta (z) < \alpha +2\pi [/tex] for [tex] \alpha \in \mathbb{R}[/tex]. When [tex] z \in C [/tex]
 
  • #4
Same answer. That's how the math works out.
Can you think of some way that it shouldn't work out that way?

It can help to understand these things by considering the converse:
What does it mean when indC(f(z)) ≠ 0 ?

What's the paper you are working through?
 

What is the property of the index of a function?

The property of the index of a function refers to the power or exponent that is applied to the independent variable in a mathematical equation. It determines the relationship between the input and output values of a function.

Why is the index of a function important?

The index of a function is important because it helps us understand the behavior of the function. It can tell us if the function is increasing or decreasing, and how quickly it is changing. It also helps us solve equations and make predictions about the function's values.

How do you find the index of a function?

The index of a function can be found by looking at the power or exponent that is applied to the independent variable in the function's equation. It can also be determined by analyzing the graph of the function and identifying the slope or curvature at different points.

Can the index of a function be negative?

Yes, the index of a function can be negative. This indicates that the function has a negative slope or is decreasing. It is also possible for the index to be a fraction or decimal, which would result in a curved or non-linear function.

How does the index of a function affect its graph?

The index of a function can greatly impact its graph. A higher index (such as 2 or 3) will result in a steeper curve, while a lower index (such as 0.5 or 0.25) will result in a flatter curve. A negative index will cause the graph to decrease rather than increase. Changes in the index can also shift the graph horizontally or vertically.

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