Reference no: EM132610650
ASSIGNMENT - COMPLEX VARIABLES
(1) By writing i = (0, 1) and y = (y, 0), show that -(iy) = (-i)y = i(-y).
(2) Solve the equation z2 + z + 1 = 0 for z = (x, y) by writing
(x, y)(x, y) + (x, y) + (1, 0) = (0, 0)
and then solving a pair of simultaneous equations in x and y.
(3) Reduce each of these quantities to a real number:
(1 + 2i)/(3 - 4i) + (2 - i)/5i; (1 - i)4.
(4) For any two complex numbers z1, z2, verify the following inequality
||z1| - |z2|| ≤ |z1 + z2| ≤ |z1| + |z2| .
(5) In each case, sketch the set of points determined by the given condition:
(a) |z - 1 + i| = 1; (b) |z + i| ≤ 3; (c) |z - 4i| ≥ 4.
(6) Prove that z is real if and only if z¯ = z.
(7) Show that the equation |z -z0| = R of a circle, centered at z0 with radius R, can be written
|z|2 - 2Re(zz¯0) + |z0|2 = R2.
(8) Find the principle argument Arg z when
z = i/(-2 - 2i)
(9) Find the square roots of 2i and 1-√3i and express them in rectangular coordinates.
(10) Find the four zeros of the polynomial z4 + 4.
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