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Show that if f(2) is analytic for |z| <1+e, then for any z = rei® with r < 1, 5. eit z dt = 0 1– žeit 1 (a) 2n Jo eit -dt = f(z) eit – z (b) f(e“)- 27

Show that if f(2) is analytic for |z| <1+e, then for any z = rei® with r < 1, 5. eit z dt = 0 1– žeit 1 (a) 2n Jo eit -dt = f(z) eit – z (b) f(e“)- 27

Advanced Engineering Mathematics
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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[Complex Variables] How do you solve this? The second picture is a hint

5. Show that if f(2) is analytic for |2| < 1+ €, then for any z = reio with r < 1, (a) eit z dt = 0 1– žeit 27 eit -dt = f(2) (b) f(e“)- eit
Transcribed Image Text:5. Show that if f(2) is analytic for |2| < 1+ €, then for any z = reio with r < 1, (a) eit z dt = 0 1– žeit 27 eit -dt = f(2) (b) f(e“)- eit
Step1. Write the integral as a curve integral , F(w)dw with w = e", and let z be a parameter, which is a fixed conatant. Now what is F(w)? Step2. Use Cauchy's theorem. (hint of the hint: if z is inside the unit circle, where should - be?)
Transcribed Image Text:Step1. Write the integral as a curve integral , F(w)dw with w = e", and let z be a parameter, which is a fixed conatant. Now what is F(w)? Step2. Use Cauchy's theorem. (hint of the hint: if z is inside the unit circle, where should - be?)
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