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

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Eugene Chen
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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?
309674106_809953983537296_1863235808189370018_n.jpg
 
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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.