Complex Logarithm: question seems simple, must be missing something

  • Thread starter Thread starter Mathmos6
  • Start date Start date
  • Tags Tags
    Complex Logarithm
Click For Summary
SUMMARY

The discussion centers on the analytic function h: ℂ → ℂ\{0}, which is defined to have no zeros. The key conclusion is that there exists an analytic function H: ℂ → ℂ such that h(z) = exp(H(z)). The participant questions the validity of using the principal branch of the logarithm, log(h(z)), to define H(z) and seeks clarification on potential issues with analyticity. The suggestion to consider the integral of h'/h indicates a deeper exploration of the logarithmic properties in complex analysis.

PREREQUISITES
  • Understanding of complex analysis, specifically analytic functions
  • Familiarity with the properties of the complex logarithm
  • Knowledge of the principal branch of logarithmic functions
  • Concept of complex integration, particularly involving h'/h
NEXT STEPS
  • Study the properties of the complex logarithm and its branches
  • Learn about the implications of analyticity in complex functions
  • Explore the integral of h'/h in the context of complex analysis
  • Investigate the conditions under which logarithmic functions remain analytic
USEFUL FOR

This discussion is beneficial for students and professionals in mathematics, particularly those studying complex analysis, as well as educators seeking to clarify concepts related to analytic functions and logarithmic properties.

Mathmos6
Messages
76
Reaction score
0

Homework Statement


Hi all, I'm having some trouble seeing why this question isn't trivial, maybe someone can help explain what I actually need to show - shouldn't take you long! :)

Suppose h:\mathbb{C} \to \mathbb{C}-\{0\} is analytic with no zeros. Show there is an analytic function H:\mathbb{C} \to \mathbb{C} such that h(z)=exp(H(z)) for all z.

Now surely H(z) is just log(h(z)), defined on (say) the principal branch? Is there some reason why the principal branch won't necessarily work, perhaps? The logarithm should be analytic on a domain with no zeros in too, right? In which case the composition with h will be analytic too. I must be missing something!

Thanks very much in advance :)
 
Physics news on Phys.org
You could consider the integral of h'/h.
 

Similar threads

[Complex analysis] Contradiction in the definition of a branch
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Question about branch of logarithms
  • · Replies 14 ·
Replies
14
Views
4K
Show that the real part of a certain complex function is harmonic
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Question about branch of logarithm
  • · Replies 2 ·
Replies
2
Views
3K
Can Discrete Logarithms Be Added in Modular Arithmetic?
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad Apparent counterexample to Cauchy-Goursat theorem (Complex Analysis)
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
3K
Order of Zero of Complex Logarithm
  • · Replies 5 ·
Replies
5
Views
3K
Possible branch cuts for arcsin derivative
  • · Replies 6 ·
Replies
6
Views
3K
Graduate Existence of a limit implies that a function can be harmonic extended
  • · Replies 1 ·
Replies
1
Views
3K