ANSWERS 599lk. i (! + n) n, where n is an integer.
Section 6.1. Complex Integrals: page 197
la. 2 - 3i.le. 1.le. J21T' /8 + VZ/2 - 1 + i ( VZ/2 - v'zn /8).
- Using (6-8), roo e-•tdt = Lim rT e- '^1 dt = Jim (-le-•T + le-•(Ol) =
Jo T-oo Jo T - oo z z
l z + T-oo Lim ( -l e-zTz ). Show that Re( z) > 0 implies this last limit equals zero. - ThL~ follows from (6-8), and the fact that if 1L and v are different iable, then
f is differentiable, and ft [~U(t))
2
] =f(t)f'(t).
Section 6.2. Contours and Contour Integrals: page 211
la. Ci : z1 (t) = 2eit, 0 ~ t ~ ~· C2: z 2 (t) = - t + i (2-t), 0 ~ t ~ 2.
Sa. The approximation simplifies to -2J2 + 2 :::l - 0.828427.
3b. -~.
5a. -32i.5b. - 8ni.7a. O.7c. -2ni.7e. i-2.
7g. -4- in.
9a. 2ni.9b. 0.
- 1 + 2i 3 ·
1 3. -2e.- exp (1 + i) - 1.
17. sin (1 + i).