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ди Эт = c for some real constant ди ди θυ = = 0 for all ду მ Əy 3. Suppose f(z) = u(x, y) + iv(x, y) is analytic on |2| ≤ 1 and |f(z)| c for all || ≤ 1. Use Cauchy-Riemann equations to show that u |2|≤ 1. Hence, f(2) is a constant function on |2|≤ 1. 2 Hint: Consider f(2)|2 and take partial derivatives. 4. Suppose f(z) is analytic on |2| ≤ 1. Suppose |f(2)| is constant on the boundary |2| = 1. Show that either f has a zero in || < 1 or f is a constant function in |2| ≤ 1. Hint: Apply maximum and minimum modulus principle and question 3.

ди Эт = c for some real constant ди ди θυ = = 0 for all ду მ Əy 3. Suppose f(z) = u(x, y) + iv(x, y) is analytic on |2| ≤ 1 and |f(z)| c for all || ≤ 1. Use Cauchy-Riemann equations to show that u |2|≤ 1. Hence, f(2) is a constant function on |2|≤ 1. 2 Hint: Consider f(2)|2 and take partial derivatives. 4. Suppose f(z) is analytic on |2| ≤ 1. Suppose |f(2)| is constant on the boundary |2| = 1. Show that either f has a zero in || < 1 or f is a constant function in |2| ≤ 1. Hint: Apply maximum and minimum modulus principle and question 3.

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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ди Эт = c for some real constant ди ди θυ = = 0 for all ду მ Əy 3. Suppose f(z) = u(x, y) + iv(x, y) is analytic on |2| ≤ 1 and |f(z)| c for all || ≤ 1. Use Cauchy-Riemann equations to show that u |2|≤ 1. Hence, f(2) is a constant function on |2|≤ 1. 2 Hint: Consider f(2)|2 and take partial derivatives. 4. Suppose f(z) is analytic on |2| ≤ 1. Suppose |f(2)| is constant on the boundary |2| = 1. Show that either f has a zero in || < 1 or f is a constant function in |2| ≤ 1. Hint: Apply maximum and minimum modulus principle and question 3.
Transcribed Image Text:ди Эт = c for some real constant ди ди θυ = = 0 for all ду მ Əy 3. Suppose f(z) = u(x, y) + iv(x, y) is analytic on |2| ≤ 1 and |f(z)| c for all || ≤ 1. Use Cauchy-Riemann equations to show that u |2|≤ 1. Hence, f(2) is a constant function on |2|≤ 1. 2 Hint: Consider f(2)|2 and take partial derivatives. 4. Suppose f(z) is analytic on |2| ≤ 1. Suppose |f(2)| is constant on the boundary |2| = 1. Show that either f has a zero in || < 1 or f is a constant function in |2| ≤ 1. Hint: Apply maximum and minimum modulus principle and question 3.
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