Answer Problems 1–12 without referring back to the text. Fill in the blank or answer true/false.

1. A function f is analytic at a point z₀ if f can be expanded in a convergent power series centered at z₀. _____
2. A power series represents a continuous function at every point within and on its circle of convergence. _____
3. For f(z) = 1/(z − 3), the Laurent series valid for |z| > 3 is z⁻¹ + 3z⁻² + 9z⁻³ + … . Since there are an infinite number of negative powers of z = z − 0, z = 0 is an essential singularity. _____
4. The only possible singularities of a rational function are poles. _____
5. The function f(z) = e^1/(z−1) has an essential singularity at z = 1. _____
6. The function f(z) = z/(e^z − 1) has a removable singularity at z = 0. _____
7. The function f(z) = z(e^z − 1) possesses a zero of order 2 at z = 0. _____
8. The function f(z) = (z + 5)/(z³ sin² z) has a pole of order _____ at z = 0.
9. If f(z) = cot π z, then Res(f(z), 0) = _____.
10. The Laurent series of f valid for 0 < |z − 1| is given by

    

    From this series we see that f has a pole of order _____ at z = 1 and Res(f(z), 1) = _____.

11. The circle of convergence of the power series is _____.
12. The power series converges at z = 2i. _____
13. Find a Maclaurin expansion of f(z) = e^z cos z. [Hint: Use the identity cos z = (e^iz + e^−iz)/2.]
14. Show that the function f(z) = 1/sin(π/z) has an infinite number of singular points. Are any of these isolated singular points?

In Problems 15–18, use known results as an aid in expanding the given function in a Laurent series valid for the indicated annular region.

15. f(z) = , 0 < |z|
16. f(z) = e^z/(z−2), 0 < |z − 2|
17. f(z) = (z − i)² sin , 0 < |z − i|
18. f(z) = , 0 < |z|
19. Expand f(z) = in an appropriate series valid for

    (a) |z| < 1

    (b) 1 < |z| < 3

    (c) |z| > 3

    (d) 0 < |z − 1| < 2.

20. Expand f(z) = in an appropriate series valid for

    (a) |z| < 5

    (b) |z| > 5

    (c) 0 < |z − 5|.

In Problems 21–30, use Cauchy’s residue theorem to evaluate the given integral along the indicated contour.

21. dz, C: |z + 2| =
22. dz, C is the ellipse x²/4 + y² = 1
23. dz, C: |z − | =
24. dz, C is the rectangle defined by x = −1, x = 1, y = 4, y = −1
25. dz, C: |z| = 4
26. dz, C is the square defined by x = −2, x = 2, y = 0, y = 1
27. dz, C: |z| = 1 [Hint: Use the Maclaurin series for z(e^z − 1).]
28. dz, C: |z − 1| = 3
29. dz, C: |z| = 6
30. dz, C is the rectangle defined by x = −, x = , y = −1, y = 1

In Problems 31 and 32, evaluate the Cauchy principal value of the given improper integral.

31. 
32. dx, a > 0 [Hint: Consider e^iz/(z − ai).]

In Problems 33 and 34, evaluate the given trigonometric integral.

33. 
34. 
35. Use an indented contour to show that

    

36. Show that by considering the complex integral dz along the contour C shown in FIGURE 19.R.1. Use the known result

    

37. The Laurent expansion of f(z) = e^{(u/2)(z−1/z)} valid for 0 < |z| can be shown to be f(z) = (u)z^k, where J_k(u) is the Bessel function of the first kind of order k. Use (4) of Section 19.3 and the contour C: |z| = 1 to show that the coefficients J_k(u) are given by