A simple pole is a singularity in a complex function, where the function approaches infinity as the variable approaches a specific point. In other words, the function has a pole at that point, but it is not a branch point.
A simple pole affects the behavior of a function by causing it to have a singularity at that point. This means that the function is undefined at the point of the pole, and the function's value becomes infinitely large as the variable approaches the pole.
This function has a simple pole at $z = im$, where $i$ is the imaginary unit and $m$ is a constant. This is because when $z = im$, the denominator becomes 0, and the function becomes undefined at that point.
To solve for the simple pole in this function, we set the denominator equal to 0 and solve for $z$. In this case, we get $z = \pm im$, which are the poles of the function.
Yes, we can use the residue theorem to solve for the simple pole in this function. The residue at a simple pole is equal to the limit of the function as $z$ approaches the pole. In this case, the residue is equal to $e^{-mk}$.