Complex Analysis: Two Questions About Non-Constant Analytic Functions

In summary, if a function is not zero in a neighborhood around zero, then there exists a logarithmic derivative.
  • #1
arthurhenry
43
0
Two questions:
1)Quote comes from a textbook:

Each non-constant function analythic function with f(0)=0 is,in a small nbhd of 0, the composition of a conformal map with the nth-power map...The proof is given and I think I am comfortable with it..

My question is a lot simpler (I think): Can we say the same for a function such that f(a)=b?
I am kindly asking someone to explain why one loses no generality if we
assume a=b=f(a)=0.

2) If an analytic function is not zero in a nbhd, what can I say about the derivative there? i.e. what restriction does this impose on f' (f prime)

Around a circle that lies in the nbhd above we should have
# Zeros=Integral (f'/f)=0

Should this not say f'=0 on that circle?
Thank you
 
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  • #2
1) If f(a)=b, then f(z) looks like b+(z-a)^n near a.

2) Could you explain why you think f'=0 on the circle?
 
  • #3
I was thinking: if f' were to be never zero, then how can Integral[f'/f] around a circle be zero.
The integral should be zero as f has no zeros inside the circle...Is what I am thinking.

Than you for your time again
 
  • #4
The contour integral of a nonzero function can very well be zero. Take for instance f(z)=z-2 and integrate f'/f=1/(z-2) on the unit circle.
 
  • #5
Somewhat embarrassing...

Thank you very much for the help
 
  • #6
Actually, if f is analytic and 1-1, then f' is not zero; if f'(z) is 0 , then factor:

f(z)-zo=z^k(g(z)) (i.e., use Taylor series, where k is the index of the zero of f ),

and g(z) is analytic and non-zero in a 'hood U of zo (the zeros of an analytic non-constant

function are isolated) . The idea is that these conditions on f allow you to define a local k-th root function

in the ball, and k-th roots are k-to-1, and so in particular not 1-1.Under these conditions , take a

small ball B(z,r) around f(z). This allows you to define a log locally, so that you can

then define a k-th root of g(z) (technically, the ball is simply-connected, and does not

wind around 0). Then you can define a k-th root using the log, and k-th roots are not

1-1; they are actually 1-1. Maybe needs work, but it is late.
 

1. What is a non-constant analytic function in complex analysis?

In complex analysis, a non-constant analytic function is a function that is differentiable at every point in its domain. This means that the function has a well-defined derivative at every point, and the derivative itself is also analytic. Non-constant analytic functions are important because they allow for a greater understanding of the behavior of complex functions.

2. How is a non-constant analytic function different from a constant analytic function?

A constant analytic function is a function that is equal to a single complex number at every point in its domain. This means that the function has a derivative of 0 at every point, and the function does not vary in any direction. On the other hand, a non-constant analytic function has a non-zero derivative at every point, and the function varies in different directions. Essentially, a constant analytic function is "flat" while a non-constant analytic function has some curvature.

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