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

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SUMMARY

The discussion centers on deriving the number of open string excitation modes as presented in Becker, Becker, Schwarz, specifically focusing on equation (2.148). The user attempts to Taylor expand equation (2.145) but fails to reproduce the results of (2.148). They note that setting ω close to 1 leads to equation (2.145), and even analyzing equation (2.146) around zero does not yield satisfactory results. The conversation suggests that applying the residue theorem to equation (2.144) may be a viable approach to resolve the issue.

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  • Understanding of string theory and excitation modes
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  • Knowledge of residue theorem in complex analysis
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  • Study the application of the residue theorem in complex analysis
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The discussion is beneficial for theoretical physicists, graduate students in string theory, and mathematicians interested in complex analysis applications within physics.

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.
 

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