Defining Analyticity at Infinity: How Do You Define and Calculate It?

  • Thread starter Thread starter Palindrom
  • Start date Start date
  • Tags Tags
    Infinity
Click For Summary
Analyticity at infinity for a complex function involves substituting z with w=1/z, effectively analyzing the function as w approaches 0. The derivative at infinity can be calculated by relating it to the derivative at zero through the transformation, specifically using f'(\infty)=g'(0) where g(z)=f(1/z). This method is common but often not explicitly taught, leading to confusion among students. Clarifying these definitions and calculations is crucial for understanding complex analysis. Mastery of these concepts enhances comprehension of function behavior at infinity.
Palindrom
Messages
263
Reaction score
0
How does one define, given a complex function, the following:

  • The function is analytic at infinity.
  • The derivative of the function at infinity.

It turns out that it's supposed to be quite common to define these terms, however I have never been shown either of them. I have a few guesses, but along with some lecture notes I could put my hands on, it's all become a big mess for me.
 
Physics news on Phys.org
Presumably one replaces z with w=1/z and works w=0.
 
O.K., but then if g(z)=f(1/z) do I take f'(\infty)=g'(0)? Just like this?
 

Similar threads

Undergrad Determining the Rate at Which Functions approach Infinity
  • · Replies 3 ·
Replies
3
Views
9K
Undergrad Analytic continuation of a dilogarithm
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Summing over continuum and uncountable numerocities
  • · Replies 1 ·
Replies
1
Views
3K
High School How do I invert this exponential function?
  • · Replies 3 ·
Replies
3
Views
4K
Graduate Infinities in QFT (and physics in general)
  • · Replies 113 ·
4
Replies
113
Views
10K
Definition of Analytic Functions in Complex Analysis
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad How do I use the four axioms of a neighborhood to define an open set?
  • · Replies 2 ·
Replies
2
Views
2K
Analytical definitions vs intuitive (or perhaps first year ) definitions
  • · Replies 1 ·
Replies
1
Views
3K
Analytic and continuouse functions
  • · Replies 5 ·
Replies
5
Views
2K