On Analytic Continuations of complex-valued functions

  • Context: Graduate 
  • Thread starter Thread starter logicalmoron
  • Start date Start date
  • Tags Tags
    Functions
Click For Summary
SUMMARY

The discussion centers on constructing analytic continuations of complex-valued functions, specifically the logarithm function, log(z), with a branch cut from (-∞, 0). It is established that if two functions have Laurent expansions with overlapping radii of convergence, they must be equal in that domain. The user seeks guidance on how to create an analytic continuation of log(z) that excludes the pole at z = 0, emphasizing the function's analyticity at zero when represented by a Taylor series around -1.

PREREQUISITES
  • Understanding of complex analysis concepts, particularly analytic functions.
  • Familiarity with Laurent series and their properties.
  • Knowledge of branch cuts in complex functions.
  • Experience with Taylor series expansions and their applications.
NEXT STEPS
  • Study the properties of analytic functions in complex analysis.
  • Learn about constructing analytic continuations for various functions.
  • Explore the implications of branch cuts on function behavior.
  • Investigate the use of Taylor series in determining function behavior near singularities.
USEFUL FOR

Mathematicians, students of complex analysis, and anyone interested in advanced topics related to analytic functions and their continuations.

logicalmoron
Messages
1
Reaction score
0
Hey Folks, first post here,

I'm having significant difficulty understanding how to create an analytic continuation of a function. The topic seems straightforward (please stop me if I am wrong): if you have two functions whose laurent expansions have a radius of convergence > 0, then the functions must be equal on the domain where those radii of convergence overlap (this is likely a gross oversimplification as the topic was just introduced yesterday.)

My question is as to how you actually construct an analytic continuation of a function — say the log(z) function, with a branch cut taken from (-∞,0), to find an analytic continuation of the function on that branch cut minus the pole at z = 0.

Again this is still probably a gross oversimplification so I would really appreciate any advice/suggestions you guys have.
 
Physics news on Phys.org
Log is analytic at zero when it is represented by a Taylor series about -1.
 

Similar threads

Undergrad Expansion of a complex function around branch point
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
4K
Graduate Existence of a limit implies that a function can be harmonic extended
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Analytic continuation of Airy function
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Integrating a function of which poles appear on the branch cut
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Is the following sum a part of any known generalized function?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Extending radius of convergence by analytic continuation
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Analytic continuation of Dilogarithm
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Understanding the Order of Poles in Complex Functions
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Finding the derivative of a characteristic function
  • · Replies 5 ·
Replies
5
Views
3K