(a) Let R be a simply connected region and let f : R → C be a holomorphic functio Fix a point z in R and define F: R → C via F(z) = √( f (6) ds, where C(z) is any contour in R that starts at z and ends at z. i. Show that = √(20.20+41 5 (5) ds, [20,zo+h] F (²0 + h) − F (zo) = √ - 20 for all zo ER and h E C, where h is small enough so that the line segme [zo, zo+h] is contained in R. You may assume without proof that integra of holomorphic functions satisfy contour independence on simply connect regions.
(a) Let R be a simply connected region and let f : R → C be a holomorphic functio Fix a point z in R and define F: R → C via F(z) = √( f (6) ds, where C(z) is any contour in R that starts at z and ends at z. i. Show that = √(20.20+41 5 (5) ds, [20,zo+h] F (²0 + h) − F (zo) = √ - 20 for all zo ER and h E C, where h is small enough so that the line segme [zo, zo+h] is contained in R. You may assume without proof that integra of holomorphic functions satisfy contour independence on simply connect regions.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(a) Let R be a simply connected region and let ƒ : R → C be a holomorphic function.
Fix a point z in R and define F: R→ C via
F(z) = √ƒ(6) ds.
where C(z) is any contour in R that starts at z and ends at z.
i. Show that
F (zo + h) — F'(zo) = √____. f (C) dç,
[20,20+h]
for all zo E R and h = C, where h is small enough so that the line segment
[zo, zo+h] is contained in R. You may assume without proof that integrals
of holomorphic functions satisfy contour independence on simply connected
regions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb0d2dbea-0987-4c02-a00c-0b87f219c8f9%2F07b49cbe-1a99-4fce-83e5-73aa6c03aa0e%2Fa0bgoz6_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Let R be a simply connected region and let ƒ : R → C be a holomorphic function.
Fix a point z in R and define F: R→ C via
F(z) = √ƒ(6) ds.
where C(z) is any contour in R that starts at z and ends at z.
i. Show that
F (zo + h) — F'(zo) = √____. f (C) dç,
[20,20+h]
for all zo E R and h = C, where h is small enough so that the line segment
[zo, zo+h] is contained in R. You may assume without proof that integrals
of holomorphic functions satisfy contour independence on simply connected
regions.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
