Example
The simplest examples of meromorphic functions with an infinite number of poles are the non-entire trigonometric functions, so take the function tan(z). tan(z) is meromorphic with poles at (n + 1/2)π, n = 0, ±1, ±2, ... The contours Γk will be squares with vertices at ±πk ± πki traversed counterclockwise, k > 1, which are easily seen to satisfy the necessary conditions.
On the horizontal sides of Γk,
so
sinh(x) < cosh(x) for all real x, which yields
For x > 0, coth(x) is continuous, decreasing, and bounded below by 1, so it follows that on the horizontal sides of Γk, |tan(z)| < coth(π). Similarly, it can be shown that |tan(z)| < 1 on the vertical sides of Γk.
With this bound on |tan(z)| we can see that
(The maximum of |1/z| on Γk occurs at the minimum of |z|, which is kπ).
Therefore p = 0, and the partial fraction expansion of tan(z) looks like
The principal parts and residues are easy enough to calculate, as all the poles of tan(z) are simple and have residue -1:
We can ignore λ0 = 0, since both tan(z) and tan(z)/z are analytic at 0, so there is no contribution to the sum, and ordering the poles λk so that λ1 = π/2, λ2 = -π/2, λ3 = 3π/2, etc., gives
Read more about this topic: Partial Fractions In Complex Analysis
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