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Derivation

1. Initial program 0.4

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

2. Initial simplification0.3

   \[\leadsto \frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}\]
3. Using strategy rm

4. Applied *-un-lft-identity0.3

   \[\leadsto \frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\color{blue}{1 \cdot \sqrt{k}}}\]

5. Applied unpow-prod-down0.5

   \[\leadsto \frac{\color{blue}{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)} \cdot {\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{1 \cdot \sqrt{k}}\]

6. Applied times-frac0.5

   \[\leadsto \color{blue}{\frac{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{1} \cdot \frac{{\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}}\]

7. Simplified0.5

   \[\leadsto \color{blue}{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{{\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}\]
8. Using strategy rm

9. Applied pow-to-exp0.4

   \[\leadsto \color{blue}{e^{\log \pi \cdot \left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot \frac{{\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}}\]

10. Final simplification0.4

    \[\leadsto \frac{{\left(2 \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{k}} \cdot e^{\left(\frac{1}{2} - \frac{k}{2}\right) \cdot \log \pi}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

Details

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