Revisions to Is the asymptotic expansion of this table of values $\log_{2}(x!)-.5$? If not, what is the expansion?

Fixed an issue in the definition of `entropyAsymptotic2`

edited Sep 25, 2024 at 5:37

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constK=1/4 (-16 ArcSinh[1]+16 Sqrt[2] ArcSinh[1]-8 Log[2]-6 Sqrt[2] Log[2]-2 EulerGamma Zeta[1/2]-\[Pi] Zeta[1/2]+2 Log[2] Zeta[1/2]-2 Log[\[Pi]] Zeta[1/2]); entropyAsymptotic2[r_] :=1= 1/(4Log[2]Sqrt[r4 Log[2] Sqrt[r!]) (constK - HurwitzZeta[1/2,1+r 1 + r!] Log[4]+2Log[4] + 2 Sqrt[r!] Log[r!]+(Zeta^(1] + Derivative[1,0))[10][Zeta][1/2,1+r!])

constK=1/4 (-16 ArcSinh[1]+16 Sqrt[2] ArcSinh[1]-8 Log[2]-6 Sqrt[2] Log[2]-2 EulerGamma Zeta[1/2]-\[Pi] Zeta[1/2]+2 Log[2] Zeta[1/2]-2 Log[\[Pi]] Zeta[1/2]); entropyAsymptotic2[r_]:=1/(4Log[2]Sqrt[r!]) (constK-HurwitzZeta[1/2,1+r!] Log[4]+2 Sqrt[r!] Log[r!]+(Zeta^(1,0))[1/2,1+r!])

constK=1/4 (-16 ArcSinh[1]+16 Sqrt[2] ArcSinh[1]-8 Log[2]-6 Sqrt[2] Log[2]-2 EulerGamma Zeta[1/2]-\[Pi] Zeta[1/2]+2 Log[2] Zeta[1/2]-2 Log[\[Pi]] Zeta[1/2]); entropyAsymptotic2[r_] := 1/(4 Log[2] Sqrt[r!]) (constK - HurwitzZeta[1/2, 1 + r!] Log[4] + 2 Sqrt[r!] Log[r!] + Derivative[1,0][Zeta][1/2,1+r!])

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edited Sep 23, 2024 at 4:03

ydd

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(small add-on note: entropyRecursive stores exact values, which can get huge in terms of LeafCount. If you want approximate values change the initial term to entropyRecursive[1] = 0.. You can also replace Sum with NSum to get really fast computation but you will need to Check each output or something similar as eventually NSum will run into trouble.)

(small add-on note: entropyRecursive stores exact values, which can get huge in terms of LeafCount. If you want approximate values change the initial term to entropyRecursive[1] = 0.)

(small add-on note: entropyRecursive stores exact values, which can get huge in terms of LeafCount. If you want approximate values change the initial term to entropyRecursive[1] = 0.. You can also replace Sum with NSum to get really fast computation but you will need to Check each output or something similar as eventually NSum will run into trouble.)

added 192 characters in body

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edited Sep 23, 2024 at 3:52

ydd

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A good simple approximation is $$ \text{entropy}(r) \approx log_2{r!} - \log_2(e) + 1 $$$$ \text{entropy}(r) \approx \log_2{r!} - \log_2(e) + 1 $$

$$ \text{entropy}(r) \approx log_2{r!} - \log_2(e) + 1 $$$$ \text{entropy}(r) \approx \log_2{r!} - \log_2(e) + 1 $$

(small add-on note: entropyRecursive stores exact values, which can get huge in terms of LeafCount. If you want approximate values change the initial term to entropyRecursive[1] = 0.)

A good simple approximation is $$ \text{entropy}(r) \approx log_2{r!} - \log_2(e) + 1 $$

$$ \text{entropy}(r) \approx log_2{r!} - \log_2(e) + 1 $$

A good simple approximation is $$ \text{entropy}(r) \approx \log_2{r!} - \log_2(e) + 1 $$

$$ \text{entropy}(r) \approx \log_2{r!} - \log_2(e) + 1 $$

(small add-on note: entropyRecursive stores exact values, which can get huge in terms of LeafCount. If you want approximate values change the initial term to entropyRecursive[1] = 0.)