Problem 14.5d1.2.2

A First Course in String Theory

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The generating function is an infinite product:

\displaystyle{ \begin{aligned} \alpha' M_L^2: \end{aligned}}

\displaystyle{\begin{aligned} &f_{L, NS+}(x) \\ &= a_{NS+} (r) x^r \\ &= \frac{1}{x} \prod_{r=1}^\infty \frac{(1 + x^{r-\frac{1}{2}})^{32}}{(1 - x^r)^8} \\ \end{aligned}}

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To evaluate the infinite product, you can use Mathematica (or its official free version Wolfram Engine) with the following commands:

TeXForm[
    HoldForm[
        (1/x)*Product[
                 (1+x^(r-1/2))^32/(1-x^r)^8,
                 {r, 1, Infinity}]]]

f[x_] := (1/x)*Product[
                 (1+x^(r-1/2))^32/(1-x^r)^8,
                 {r, 1, Infinity}]

Print[f[x]]

TeXForm[f[x]]

TeXForm[Series[f[x], {x,0,3}]]
\displaystyle{\frac{1}{x}\prod _{r=1}^{\infty } \frac{\left(1+x^{r-\frac{1}{2}}\right)^{32}}{\left(1-x^r\right)^8}}


                     1        32
    QPochhammer[-(-------), x]
                  Sqrt[x]
------------------------------------
        1    32                    8
(1 + -------)   x QPochhammer[x, x]
     Sqrt[x]


\displaystyle{\frac{\left(-\frac{1}{\sqrt{x}};x\right)_{\infty }^{32}}{\left(\frac{1}{\sqrt{x}}+1\right)^{32} x (x;x)_{\infty }^8}}


\displaystyle{\frac{1}{x}+\frac{32}{\sqrt{x}}+504+5248 \sqrt{x}+40996 x+258624 x^{3/2}+1384320 x^2+O\left(x^{5/2}\right)}

\displaystyle{ \begin{aligned}  &f_{L, NS+}(x) \\  \end{aligned}}

\displaystyle{  \approx \frac{1}{x}+\frac{32}{\sqrt{x}}+504+5248 \, \sqrt{x}+40996 \, x+258624 \, x^{\frac{3}{2}}+1384320 \, x^{2}+6512384 \, x^{\frac{5}{2}} + ...}

— Me@2022-11-23 04:40:28 PM

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2022.11.23 Wednesday (c) All rights reserved by ACHK