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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

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

5. Applied associate-/l*0.4

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

7. Applied unpow-prod-down0.5

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

8. Applied *-un-lft-identity0.5

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

9. Applied times-frac0.4

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

11. Applied associate-*l/0.5

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

12. Applied associate-/r/0.5

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

14. Applied unpow-prod-down0.4

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

15. Final simplification0.4

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

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

Details

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