I'm completely drawing a blank for some reason.
The function tanh(x) has poles when -I*z/Pi-1/2 is equal to an integer. That is, in the sequence
$i/2\pi\, \left( 2\,n+1 \right)$
There must be some standard trig function that has as roots (like the way sin(2*Pin) has roots at each n) at the points (I(1/2))Pi(2*n+1) so that
tanh(x)*f(x) would be an entire function rather than merely meromorphic