Proving f(z)=e^(g(z)) on a Convex Set Omega

In summary, the conversation discusses proving that a given analytic function can be represented as e^(g(z)) for some analytic function g defined on a convex set. The conversation also mentions using the Cauchy-Goursat theorem and manipulating the function f'/f to find a solution.
  • #1
michael.wes
Gold Member
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Homework Statement



Suppose that f is analytic on a convex set omega and that f never vanishes on omega. Prove that f(z)=e^(g(z)) for some analytic function g defined on omega.
Hint: does f'/f have a primitive on omega?

Homework Equations



[tex]f(z)=\sum_{k=0}^\infty a_k(z-p)^k[/tex]

The Attempt at a Solution



I was able to prove that f'/f has a primitive on omega by the Cauchy-Goursat theorem, but I'm not sure where to go from here. Any help is appreciated!
 
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  • #2


Let F be a primitive of f'/f. Now, consider the function [tex]G(z)=e^{F(z)}/f(z)[/tex]. What is it's derivative? What can you conclude from that?
 
  • #3


I got that [tex]e^{g(z)}=cf(z)[/tex], for some complex constant c and some analytic function g. It's usually easy in these problems to show that the constant is 1, but this is not a concrete function, so I'm not sure how to do that.
 
  • #4


Well, the constant is not necessairly 1, so you'll have to find something else. You'll have to modify your function g in some way such that the equation is right...
 

FAQ: Proving f(z)=e^(g(z)) on a Convex Set Omega

What does it mean for a function to be "convex"?

A convex function is one in which the line connecting any two points on the graph of the function lies above or on the graph itself. In other words, the function is always "curving upwards" and does not have any "dips" or "valleys".

How do you prove that a function is convex on a set?

To prove that a function is convex on a set, you must show that the line connecting any two points on the graph of the function lies above or on the graph itself. This can be done by using the definition of a convex function and applying it to the specific set in question. This may involve showing that the function's second derivative is positive on the set, or using other techniques such as the convexity criterion or the convexity-preserving operations theorem.

What is the significance of proving that f(z)=e^(g(z)) on a convex set Omega?

The significance of proving this statement is that it allows us to confidently use properties and operations of convex functions on this particular function, as we know it will also be convex. This can be useful in various applications in mathematics and science.

Can a function be convex on one set and not on another?

Yes, a function can be convex on one set and not on another. A function's convexity is dependent on the set it is being evaluated on. The same function can be convex on one set and concave on another. It is important to specify the set when discussing convexity of a function.

Are there any alternative ways to prove f(z)=e^(g(z)) on a convex set Omega?

Yes, there are alternative ways to prove this statement. One method is to use the Taylor series expansion of f(z) and g(z) and then compare the two series to show they are equivalent. Another method is to use the definition of convexity and properties of exponential and logarithmic functions to show that the function is convex on the given set.

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