Page 880 - Advanced_Engineering_Mathematics o'neil

860 Answers to Selected Problems

25. Label the vertices z,w,u in counterclockwise order. The sides are vectors represented by the complex numbers
w − z,u − w, z − u. The triangle is equilateral if and only if
|w − z|=|u − w|= z − u|,

and each side can be rotated 2π/3 radians to align with the next side. Therefore,
u − w = (w − z)e −2πi/3 and z − u = (u − w)e −2πi/3 .
2
2
2
Check that this gives z + w + u = zw + zu + wu.
27. If |z|= 1, then zz = 1. Divide by zz and manipulate the denominator to show that

z − w
z − w z − w
= = .
1 − zw zz − zzzzw z − w

Similarly, if |w|= 1, divide (z − w)(1 − zw) by ww and take the absolute value.
29. M is an open half-plane consisting of all z = x + iy with y < 7. Boundary points are points x + 7i, a horizontal line.
31. U is the vertical strip of points z = x + iy with 1 < x ≤ 3. U is neither open nor closed. Boundary points are points
1 + iy (not in U) and points 3 + iy (in U). U is not bounded.
2
2
33. z = x + iy is in W if and only if x > y . This is the open region "inside" the parabola x = y . Boundary points are
2
points y + iy on the parabola. W is not closed, because it does not contain all of its boundary points. W is open,
because all points of W are interior points.
Section 19.2 Complex Functions
1. u(x, y) = x,v(x, y) = y − 1. The Cauchy-Riemann equations hold everywhere, and f is differentiable for all z.

2
2
3. u(x, y) = x + y ,v(x, y) = 0. The Cauchy-Riemann equations hold nowhere, and f is differentiable at no z.
2
2
5. u(x, y) = 0,v(x, y) = x + y . The Cauchy-Riemann equations hold only at z = 0. From the limit definition of the
derivative, we obtain f (0) = 0.

7. u(x, y = 1,v(x, y) = y/x, The Cauchy-Riemann equations hold nowhere, and f is not differentiable at any point at
which f (z) is defined.
2
2
9. u(x, y) = x − y ,v(x, y) =−2xy. The Cauchy-Riemann equations hold only at z = 0. From the limit definition of the

derivative, f (0) = 0.
11.
x y
u(x, y) =−4x + ,v(x, y) =−4y + .
x + y 2 x + y 2
2
2
The Cauchy-Riemann equations hold for all z
= 0. f is differentiable for all nonzero z (the partial derivatives are
continuous for z
= 0).
Section 19.3 The Exponential and Trigonometric Functions
1. cos(1) + i sin(1) 3. cos(3)cosh(2) − i sin(3)sinh(2)
5 1 i
5. e [cos(2) + i sin(2)] 7. [(1 − cos(2)cosh(2)]+ sin(2)sinh(2)
2 2
9. i
11. u(x, y)= e x 2 −y 2 cos(2xy),v(x, y)e x 2 −y 2 sin(2xy),
∂u x 2 −y 2 ∂v
= e (2x cos(2xy) − 2y sin(2xy) = ,
∂x ∂y
∂u x 2 −y 2 ∂v
= e (−2y cos(2xy) − 2x sin(2xy) =−
∂y ∂x
13.
x
x
u(x, y) = xe cos(y) − ye sin(y),
x
x
v(x, y) = ye cos(y) + xe sin(y),
The Cauchy-Riemann equations hold at all (x, y).
π
15. ln(2) + i(4k + 1) , k any integer 17. ln(2) + (2k + 1)π
2
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October 14, 2010 17:50 THM/NEIL Page-860 27410_25_Ans_p801-866