Solutions and/or Comments on Review Questions

1. The given number has polar form 8*exp(3*pi/4*i). So one cube root has polar form 2*exp((pi/4)*i). The others are 2*exp(11*(pi/12)*i) and 2*exp(19*(pi/12)*i)

2. This needs to be done by the definition. One parametric representation for the curve is Z(t) = t*(1 - i) + (1 - t)*i for t = 0 to 1. The value of the integral is 2/3 - 4/3 *i.

3. This integral can either be done by the definition or by the Fundamental Theorem of Calculus. Since the curve begins at 0 and ends at pi*i, the second approach yields exp(pi*i) - exp(0) = -2.

4. Since 2*i is outside the curve, the Cauchy Integral Theorem gives a value of 0 for the integral.

5. The integrand may be written as f(z)/(z - i) where f(z) = z/(z + i). Now we can apply the Cauchy Integral Formula to f at z = i. So the integral is 2*pi*i*f(i) = pi*i.

6. Using the FTC, the integral is -pi*cos(pi^2/pi) + pi*cos(0) = 2*pi.

7. This is 0 by Cauchy's Theorem.

8. pi*i*(-sin(0)) = 0, by the Generalized CIF

9. -pi*i/3 by the CIF. Here, f(z) = 1/(z - 3).

10. (a) This is false. A counterexample is exp(z).

    (b) This is true. Since |f(i)| = 2, by the Maximum Modulus Theorem there must be a point z on the enclosing circle where |f(z)| is greater than or equal to 2, and 2 is greater than 3/2.

11. (a) (The circle is centered at 0.) This is false by the Maximum Modulus principle -- |f(0)| is greater than values of |f(z)| on the unit circle.

    (b) This is true. The assumptions imply the function is entire and bounded in magnitude, so, by Liouville's Theorem, it must be constant. Since the bound approaches 0 as |z| approaches infinity, the constant must be 0.

12. If u is the real part of sin(pi/6 + ti) and v is the imaginary part of sin(pi/6 + ti), then we have 4u^2 + 1 = (4/3) v^2, which describes a hyperbola in the uv-plane.

Last modified: February 19, 1999