9) The functions ln(z) and z^α

Question 1

What does the notation X±i0 mean

Answer 1

Question 2

What are the two common choices of branches for ln(z) and how do they differ

Answer 2

Question 3

What are branch cuts and branch points in the complex plane

Answer 3

Question 4

How do branch cuts for the function f(z)=ln(z+1)−ln(z−1) arise, and how are they determined

Answer 4

Question 5

What are two important things to consider when dealing with multivalued functions and their expansions

Answer 5

• When a function consists of multiple terms with potential multivaluedness (e.g., logarithms or square roots), apply the branch definition separately to each term before combining them.
  This ensures the function is well-defined and avoids inconsistencies caused by branch cuts
• Use Laurent expansions to understand the behavior of functions at large distances in the complex plane

Question 6

How is z^α defined as a single-valued function

Answer 6