(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.
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![(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.
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