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b. Let f(z) = z² on the set S which is the union of the semi-discs S₁ = {z : |z] ≤ 2; Re z ≤ 0} and S₂ = {z |z| ≤ 1; Rez ≥ 0}. Show that there are points z on the boundary of S for which f(z) is an interior point of f(S).

b. Let f(z) = z² on the set S which is the union of the semi-discs S₁ = {z : |z] ≤ 2; Re z ≤ 0} and S₂ = {z |z| ≤ 1; Rez ≥ 0}. Show that there are points z on the boundary of S for which f(z) is an interior point of f(S).

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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EXPLAIN THE PART B) SOLVE IN DIGITAL FORMAT

b) f maps S₁ onto D(0; 4). If z € S₁, we can write z = reit with 0 ≤ r ≤ 2 and π/2 ≤ t ≤ 3π/2 (and all of these values of r and t occur), so f(z) = z² = r²e²it, and 0 ≤ r² ≤ 4 and π ≤ t ≤ 3, so we cover all of D(0; 4). Also f maps S₂ onto the closed unit disc (with the same reasoning). Hence ƒ(S₁ U S₂) = D(0; 4).
Transcribed Image Text:b) f maps S₁ onto D(0; 4). If z € S₁, we can write z = reit with 0 ≤ r ≤ 2 and π/2 ≤ t ≤ 3π/2 (and all of these values of r and t occur), so f(z) = z² = r²e²it, and 0 ≤ r² ≤ 4 and π ≤ t ≤ 3, so we cover all of D(0; 4). Also f maps S₂ onto the closed unit disc (with the same reasoning). Hence ƒ(S₁ U S₂) = D(0; 4).
3. a. Suppose f is nonconstant and analytic on S and f (S) = T. Show that if f(z) is a boundary point of T, z is a boundary point of S. b. Let f(z) = z² on the set S which is the union of the semi-discs S₁ = {z : [z] ≤ 2; Re z ≤ 0} and S₂ = {z |z| ≤ 1; Rez ≥ 0}. Show that there are points z on the boundary of S for which f(z) is an interior point of f (S).
Transcribed Image Text:3. a. Suppose f is nonconstant and analytic on S and f (S) = T. Show that if f(z) is a boundary point of T, z is a boundary point of S. b. Let f(z) = z² on the set S which is the union of the semi-discs S₁ = {z : [z] ≤ 2; Re z ≤ 0} and S₂ = {z |z| ≤ 1; Rez ≥ 0}. Show that there are points z on the boundary of S for which f(z) is an interior point of f (S).
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