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Copy of Complex Numbers -- Homework #5

# Copy of Complex Numbers -- Homework #5 - MA 459 Homework 5...

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Unformatted text preview: MA 459 - Homework # 5 1. Write each of the following functions in the form u(x, y) + iv(x, y). 2+1” 1 222+3 C)‘ 61) 22+1 2 12—1] a)322+52+i+1 b) Find the domain of deﬁnition of each of these functions. 2. Describe the range of each of the following functions: a) f (z) = z + 5 for Re(z) > 0 b) g(z) = z2 for z in the first quadrant Re(z) 2 O, Im(z) Z 0. c) p(z) i —Zz3 for z in the quarter — disk |z| < 1, 0 < Arg(z) < 71/2 3. Let P(z) =‘z —— 3, G(z) = —iz, H(z) = 2. Sketch the image of the circle Izl = 1 under each of the following composite mappings: l1) C(P(Z)) l7)G(H(Z)) C) H(P(Z)) d)F(G(H(Z))) 4. Decide whether each of the following sequences converges, and if so,-find its limit. 1' z' 1——ir" a) zn = it b) z,.l = Arg(—1 + 12-) c) Zn 2 (T) 5. Find each of the following limits 2 2 . _ .2 . 2 +9 . z +1 ‘0 Ens-(Z .51) ”1133.24.- 01.13.1244 ...
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