Complex variable analysis
Please do these problems:
#3,4, and 5 from oldtest2.pdf
#28 from extraprobs.pdf
total 4 problems
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MATH 132/3, WINTER 2012, EXTRA PROBLEMS #1. Let C = {(x, y) | x, y ? R} be the set of all vectors in the plane, which we add, subtract and multiply with real numbers as usual, i.e., (u, v) + (x, y) = (u + x, v + y), t(x, y) = (tx, ty) (u, v, x, y, t ? R). Clearly (x, y) = x(1, 0) + y(0, 1) so that if we abbreviate (1, 0) = 1 for the unit vector on the x-axis and we let i = (0, 1) for the unit vector on the y-axis, we can rewrite this equation as (x, y) = x1 + yi = x + yi. Suppose we define multiplication zw on C so that the following three conditions hold: (1) All the usual rules of multiplication and addition hold, in particular, (uv)w = u(vw), uv = vu, w(u + v) = wu + wv, and for every w 6= 0 there is a number z such that zw = 1. (2) For any real number t, (t, 0)(x, y) = (tx, ty), so that tz is exactly as before when t is on the x-axis (a real number). (3) For all z,w, |zw| = |z||w|. Technically (1) means that C is a field; and you can find a definition of a field in any algebra book or in the Field (mathematics) entry of Wikipedia. Prove that i2 = -1 and hence (u + iv)(x + iy) = (ux – yv) + i(xv + uy). Hint: Prove that if i2 = A + iB, then A = -1 and B = 0. #2. Let z = v3 + i. Write in the form x + iy the following numbers: (a) z + i (b) z(2i – 1) (c) z – 1 z + 1 (d) 2z (e) Re(z2) (f) Im(z2) (g) |z2(i + 1)| (h) i101 Math. 132/3, Winter 2012, Yiannis N. Moschovakis, extra problems 1
2 MATH 132/3, WINTER 2012, EXTRA PROBLEMS #3. Find the argument of each of the following complex numbers and write in polar form rcis(): (a) v3 (b) – v3 + i (c) 1 v3 + i #4. Write (1 + i)103 in the form a + bi. #5. Find the argument and write in polar form the following complex numbers: (a) 13 (b) – 3 i (c) (1 + v3i)4 #6. Find all the values of the following: (a) (1 + v3i) 13 (b) i 1 + i16 #7. Describe the projections on the Riemann sphere of the following sets in the complex plane: (a) The lower half planes {z | Im(z)
MATH 132/3, WINTER 2012, EXTRA PROBLEMS #1. Let C = {(x, y) | x, y ? R} be the set of all vectors in the plane, which we add, subtract and multiply with real numbers as usual, i.e., (u, v) + (x, y) = (u + x, v + y), t(x, y) = (tx, ty) (u, v, x, y, t ? R). Clearly (x, y) = x(1, 0) + y(0, 1) so that if we abbreviate (1, 0) = 1 for the unit vector on the x-axis and we let i = (0, 1) for the unit vector on the y-axis, we can rewrite this equation as (x, y) = x1 + yi = x + yi. Suppose we define multiplication zw on C so that the following three conditions hold: (1) All the usual rules of multiplication and addition hold, in particular, (uv)w = u(vw), uv = vu, w(u + v) = wu + wv, and for every w 6= 0 there is a number z such that zw = 1. (2) For any real number t, (t, 0)(x, y) = (tx, ty), so that tz is exactly as before when t is on the x-axis (a real number). (3) For all z,w, |zw| = |z||w|. Technically (1) means that C is a field; and you can find a definition of a field in any algebra book or in the Field (mathematics) entry of Wikipedia. Prove that i2 = -1 and hence (u + iv)(x + iy) = (ux – yv) + i(xv + uy). Hint: Prove that if i2 = A + iB, then A = -1 and B = 0. #2. Let z = v3 + i. Write in the form x + iy the following numbers: (a) z + i (b) z(2i – 1) (c) z – 1 z + 1 (d) 2z (e) Re(z2) (f) Im(z2) (g) |z2(i + 1)| (h) i101 Math. 132/3, Winter 2012, Yiannis N. Moschovakis, extra problems 1
2 MATH 132/3, WINTER 2012, EXTRA PROBLEMS #3. Find the argument of each of the following complex numbers and write in polar form rcis(): (a) v3 (b) – v3 + i (c) 1 v3 + i #4. Write (1 + i)103 in the form a + bi. #5. Find the argument and write in polar form the following complex numbers: (a) 13 (b) – 3 i (c) (1 + v3i)4 #6. Find all the values of the following: (a) (1 + v3i) 13 (b) i 1 + i16 #7. Describe the projections on the Riemann sphere of the following sets in the complex plane: (a) The lower half planes {z | Im(z)
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