M417

1.

Find

\textp\.v\. ⌠ ⌡ ∞ 0  \dfractz−1t−1 dt, 0 < \operatornameRez < 1.

2.

Let C be a simple closed contour and C_i be the interior of C. Suppose that f(z) is analytic and nonzero on C, meromorphic in C_i, and that in C_i, f has zeroes at a₁, ..., a_N, and poles at b₁, …, b_M. Let H(z) be analytic on C and C_i. Then

\frac12πi ⌠ (⎜) ⌡ C  H(z)\fracf′(z)f(z) dz = N ∑ j=1  H(aj) − M ∑ k=1  H(bk),

where each zero and pole occurs as often in the sum as is required by its multiplicity.