Let R be a simply connected region and let f : R → C be a holomorphic function. Fix a point z in R and define F: R→ C via F(z) = √( f(s) ds, where C(z) is any contour in R that starts at z* and ends at z. F(²0 + h) − F (20) = √2+1 ƒ (C) ds, (zo [zo,zo+h] √ 2020+ (F(S) — ƒ(20)) dz|
Let R be a simply connected region and let f : R → C be a holomorphic function. Fix a point z in R and define F: R→ C via F(z) = √( f(s) ds, where C(z) is any contour in R that starts at z* and ends at z. F(²0 + h) − F (20) = √2+1 ƒ (C) ds, (zo [zo,zo+h] √ 2020+ (F(S) — ƒ(20)) dz|
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let R be a simply connected region and let f: R→ C be a holomorphic function.
Fix a point z in R and define F : R → C via
F (2) = √( 5 (C) ds.
where C(z) is any contour in R that starts at z* and ends at z.
F(zo + h) − F (zo) :
=
[zo,zo+h]
√20.2
[zo,zo+h]
f(s) ds,
(ƒ ($) — ƒ (zo)) dz <ɛ |h|
-
Deduce that F is an antiderivative for f on R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb0d2dbea-0987-4c02-a00c-0b87f219c8f9%2Fa2d6d21e-577c-4936-a1a1-ec47477f7889%2Fw2sm20d_processed.png&w=3840&q=75)
Transcribed Image Text:Let R be a simply connected region and let f: R→ C be a holomorphic function.
Fix a point z in R and define F : R → C via
F (2) = √( 5 (C) ds.
where C(z) is any contour in R that starts at z* and ends at z.
F(zo + h) − F (zo) :
=
[zo,zo+h]
√20.2
[zo,zo+h]
f(s) ds,
(ƒ ($) — ƒ (zo)) dz <ɛ |h|
-
Deduce that F is an antiderivative for f on R.
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