Graduate How do we derive the number of string excitation modes for large N?

[Eugene Chen]

On page 52 in Becker, Becker, Schwarz, there is an equation (2.148) for the number of open string excitation modes.
I tried to Tayler expand eq 2.145, but couldn't reproduce 2.148. Plus, one gets 2.145 by setting w close to 1; even if I use the 2.146 and try to analyze it around 0, I am still very far from getting 2.148
Does anyone know any trick to do this?

 

[Paul Colby]

It looks like one applies the residue theorem to 2.144 using right hand expression in 2.145. $\omega=1$ is an isolated essential singularity of this expression.