f(2) = |2 +1|2.
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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Question
![HIN! ualculate tne CUElmcients UI 2, N 2 R = 1, III Che TayIor expansIoII UI J
around 0].
EXERCISE 38
Let f(z) = |2 + 1|2. Let y(t) = et,0 <t < 27 be the path that describes the
unit circle with centre 0 anticlockwise.
(i) Show that f is not holomorphic on any domain that contains y. [Hint: use
the Cauchy-Riemann equations.]
(ii) Find a function g that is holomorphic on some domain that contains y and
such that f(z) = g(z) at all points on the unit circle y. (It follows that
Sy f = S, 9.) [Hint: recall that if w e C then |w|2 = ww.]
(iii) Use Cauchy's Integral formula to show that
%3D
/ lz + 1|?dz
= 2ni](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcee8942c-e69a-487b-a05a-79aff5c61f9f%2F22fc6004-8adf-4af8-ab1a-edd86ee1bd68%2Ftylyflp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:HIN! ualculate tne CUElmcients UI 2, N 2 R = 1, III Che TayIor expansIoII UI J
around 0].
EXERCISE 38
Let f(z) = |2 + 1|2. Let y(t) = et,0 <t < 27 be the path that describes the
unit circle with centre 0 anticlockwise.
(i) Show that f is not holomorphic on any domain that contains y. [Hint: use
the Cauchy-Riemann equations.]
(ii) Find a function g that is holomorphic on some domain that contains y and
such that f(z) = g(z) at all points on the unit circle y. (It follows that
Sy f = S, 9.) [Hint: recall that if w e C then |w|2 = ww.]
(iii) Use Cauchy's Integral formula to show that
%3D
/ lz + 1|?dz
= 2ni
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