(a) Consider the function f(z) = u(x, y)+iv(x, y) = x − 2x² +2y² +i(y− 4xy). This function defines two families of level curves u(x, y) = const. and v(x, y) = const. Sketch or plot these two families of level curves in the same plane. Verify, analytically, that the families are orthogonal. (b) Consider u(x, y) = xy + x + 2y. Verify that u is harmonic in an appropriate domain S. Find its harmonic conjugate v and find the analytic function f(z) = u + iv that satisfies f(2i) = -1 +5i. (c) Find the image of the line z = 2+iy, y ≤ R under the map w = ez. Describe the shape and features. (d) Express the value of sin(+ i) in the form a + ib. Hence determine sin((2k + 1)π + i) for every integer k (justify your answer).
(a) Consider the function f(z) = u(x, y)+iv(x, y) = x − 2x² +2y² +i(y− 4xy). This function defines two families of level curves u(x, y) = const. and v(x, y) = const. Sketch or plot these two families of level curves in the same plane. Verify, analytically, that the families are orthogonal. (b) Consider u(x, y) = xy + x + 2y. Verify that u is harmonic in an appropriate domain S. Find its harmonic conjugate v and find the analytic function f(z) = u + iv that satisfies f(2i) = -1 +5i. (c) Find the image of the line z = 2+iy, y ≤ R under the map w = ez. Describe the shape and features. (d) Express the value of sin(+ i) in the form a + ib. Hence determine sin((2k + 1)π + i) for every integer k (justify your answer).
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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Transcribed Image Text:Problem 3 (Complex differentiation)
(a) Consider the function ƒ(z) = u(x, y)+iv(x, y) = x − 2x²+2y² +i(y — 4xy). This function
defines two families of level curves u(x, y) = const. and v(x, y) = const. Sketch or plot
these two families of level curves in the same plane. Verify, analytically, that the families
are orthogonal.
(b) Consider u(x, y) = xy + x + 2y. Verify that u is harmonic in an appropriate domain S.
Find its harmonic conjugate v and find the analytic function f(z) = u + iv that satisfies
f(2i) = −1+5i.
(c) Find the image of the line z = 2+iy, y € R under the map w = e. Describe the shape
and features.
(d) Express the value of sin(+ i) in the form a + ib. Hence determine sin((2k + 1)π + i) for
every integer k (justify your answer).
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