$\int_{|z|=n} z^m\tan z\;dz$

Question

Evaluate$$\int_{|z|=n} z^m\tan z\;dz$$

I am solving this example 17 about generalized argument formula in a book

Now i have some doubts

1. Author says singularities of f(z) are $(k+{1\over2})\pi$ . I think this should have been zeroes of f(z) since $\cos z$ is analytic and therefore have no singularities .

2. Before the formula above the example ( It's on last page ) author doesn't clearly mention that $a_i$ and $b_k$ s which are zeroes and poles of f(z) respectively lies inside or outside the contour $\gamma$ just states that they are not on contour . Please confirm me that these poles and zeroes are inside of contour !

3.If that is the case ( point 2) isn't answer is wrong and it should have been $$-\sum_{k=-\alpha}^{\alpha}(k+{1\over2})^m {\pi^m}$$ where $\alpha=[{n\over\pi}-{1\over2}]$ , here [.] is greatest integer function .

Answer 1

1. Yes, the points of the form $\left(k+\frac12\right)\pi$ are zeros of $\cos$; they are also singularities of $\tan$.
2. Yes, the $p_j$'s and the $q_k$'s should be on the region of $\Bbb C$ bounded by the contour.
3. That looks correct to me. Just a small detail: the sum should go from $-\alpha$ to $\alpha$, rather than in the opposite direction.