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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
EXPLAIN THE PART B) SOLVE IN DIGITAL FORMAT
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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc95a7fa8-f21a-4709-81cf-64d281fbc8e2%2F877185f2-8e44-470c-96f7-669285abb23b%2Fixr5qs_processed.jpeg&w=3840&q=75)
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).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 5 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
