1550251515-Classical_Complex_Analysis__Gonzalez_

40 Chapter^1

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Fig. 1.11

EXERCISES 1.4

1. Do alf operations geometrically.

(a) (-3,4)+(-2,-5) (b) (1+2i)+(3-4i)

(c) (4+i)+(1+3i)+(-1+2i) (d) (5+2i)-(1+4i)

(e) (-2 + i) - (3 - 4i) (f) (1+4i) + (3 - 2i) - (2 + i)

(g) (2 + 2i)(l + J3i) (h) 3i(2 + i)

(i) 2ei7r/6 x 3ei7r/4 (j) 1 ~ i

(k) 5i (1)^9 + 7i

1+2i 2 + 3i

(m) 4e2i7r/3 + 2ei7r/5 (n) 6e5i7r/3 + 3e-i7r/4

2. A vertex of an equilateral triangle of side 1 lies on the negative imag-
   inary axis, and the center of the triangle is at the origin. Find the
   complex numbers corresponding to the three vertices.


3. A regular hexagon lies on the upper half of the complex plane with one
   vertex at (0, 0) and a consecutive vertex at (1, 0). Find the complex
   numbers corresponding to the other vertices.


4. Let a and f3 be the vertices of a square. Find the complex numbers
   corresponding to the other two vertices. Examine all possible cases.

5. (a) If the points z 1 , z 2 , z 3 lie on the unit circle (center at the origin)

and satisfy z1 + z 2 + z 3 = O, show that they are the vertices of an

equilateral triangle.

(b) If z1, z2, za are the vertices of an isosceles triangle, with a right

angle at the vertex z 2 , show that

z1^2 + 2z2^2 + za^2 = 2z2(z1 + za)

( c) Prove that three given points z 1 , z 2 , z 2 are vertices of an equilateral

triangle iff