Theory and Modern Applications

# Table 2 Values of discrete and continual fluxes

Degree of freedom (k) $\sum _{n=0}^{\infty }HS\left(k,n,r\right)=$ $\sum _{n=0}^{\infty }HS\left(k,n,1\right)\approx$ $\underset{0}{\overset{+\infty }{\int }}HS\left(k,n,1\right)dn\approx$
0 e r 2.71828182846 2.89982256317
1 2er 5.43656365692 5.24809906025
2 2πer 17.0794684453 17.6417407306
3 8πer 68.3178737814 56.964225268
4 12π2(er-1) 203.505142758 139.918441638
5 64π2(er-r-1) 453.706079704 271.32230045
6 60π3[2er-(r+1)2-1] 812.172812098 437.960809928
7 $\sum _{n=0}^{\infty }\frac{768{\pi }^{3}{r}^{n+4}}{\Gamma \left(n+5\right)}$ 1229.10258235 611.722905550
8 $\sum _{n=0}^{\infty }\frac{1680{\pi }^{4}{r}^{n+5}}{\Gamma \left(n+6\right)}$ 1628.04409715 759.633692941
9 $\sum _{n=0}^{\infty }\frac{12288{\pi }^{4}{r}^{n+6}}{\Gamma \left(n+7\right)}$ 1933.28876014 855.051695653
10 $\sum _{n=0}^{\infty }\frac{30240{\pi }^{5}{r}^{n+7}}{\Gamma \left(n+8\right)}$ 2093.93742907 884.975895298
11 $\sum _{n=0}^{\infty }\frac{245760{\pi }^{5}{r}^{n+8}}{\Gamma \left(n+9\right)}$ 2095.29352414 851.441651487
12 $\sum _{n=0}^{\infty }\frac{665280{\pi }^{6}{r}^{n+9}}{\Gamma \left(n+10\right)}$ 1956.27052708 768.011397877
13 $\sum _{n=0}^{\infty }\frac{5898240{\pi }^{6}{r}^{n+10}}{\Gamma \left(n+11\right)}$ 1717.51550066 653.938458847
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50 $\sum _{n=0}^{\infty }HS\left(50,n,r\right)$ 0.00000002078 0.00000000526
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$\underset{k\to \infty }{\text{lim}}\sum _{n=0}^{\infty }HS\left(k,n,r\right)=0$ 0 $\underset{k\to \infty }{\text{lim}}\underset{0}{\overset{\infty }{\int }}HS\left(k,n,r\right)dn=0$
$\underset{k}{\Sigma }$ 19427.858848843922 -