11) Analytic Continuation

Question 1

What is the definition of analytic continuation

Answer 1

Analytic continuation extends a function f(z) from one domain to another while preserving regularity
Case A -
(i) f(z) is regular in a domain D ⊂ C;
(ii) g(z) is regular in a domain E ⊂ C;
(iii) f(z) = g(z) in D ∩ E
Case B -
(i) f(z) is regular in a domain D ⊂ C;
(ii) g(z) is regular in a domain E ⊂ C;
(iii) f(z) = g(z) on some line L ∈ D ∩ E.

Question 1

What happens if a function f(z) is regular in a domain D and has a zero at a∈D

Answer 1

If f(z) is regular in a domain D and f(a)=0:
Either a is an isolated zero (no other zeros nearby) Or f(z)≡0 (identically zero) throughout D

Question 2

What is the uniqueness property of analytic continuation

Answer 2

Analytic continuations are unique.

If two regular functions f(z) and g(z) agree on a connected region where both are defined, then f(z)=g(z) everywhere in their shared domain of regularity

Question 3

How do you extend a function or identity to a larger domain using analytic continuation

Answer 3

• Prove it works in a smaller domain where the function or identity is clearly valid
• Check the function is analytic (smooth and differentiable) in the desired larger domain
• Extend step by step using overlapping regions where the function is already defined
• Confirm consistency by ensuring the extended function matches the original in the overlap

Question 4

What is the “mirror-image” property for regular functions

Answer 4

Question 5

What is the weak form of Schwarz’s Reflection Principle

Answer 5

Question 6

What is contact continuation in analytic continuation

Answer 6

Suppose
(i) regions D1 and D2 are in contact along the line γ
(ii) functions f1(z) and f2(z) are regular in regions D1 and D2 respectively
(iii) and are continuous in regions D1 ∪ γ and D2 ∪ γ respectively; also
(iv) f1(z) = f2(z) on γ
Then each function f1, f2 is the analytic continuation of the other

Question 7

What is the strong form of Schwarz’s Reflection Principle

Answer 7