# for ratios

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##### 1: 7.8 Inequalities
###### §7.8 Inequalities
Let $\mathsf{M}\left(x\right)$ denote Mills’ ratio: …
7.8.2 $\frac{1}{x+\sqrt{x^{2}+2}}<\mathsf{M}\left(x\right)\leq\frac{1}{x+\sqrt{x^{2}+% (4/\pi)}},$ $x\geq 0$,
7.8.3 $\frac{\sqrt{\pi}}{2\sqrt{\pi}x+2}\leq\mathsf{M}\left(x\right)<\frac{1}{x+1},$ $x\geq 0$,
7.8.4 $\mathsf{M}\left(x\right)<\frac{2}{3x+\sqrt{x^{2}+4}},$ $x>-\tfrac{1}{2}\sqrt{2}$,
##### 2: 12.6 Continued Fraction
For a continued-fraction expansion of the ratio $\ifrac{U\left(a,x\right)}{U\left(a-1,x\right)}$ see Cuyt et al. (2008, pp. 340–341).
##### 3: 4.22 Infinite Products and Partial Fractions
4.22.5 $\csc z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}-n^{2}\pi^{2}}.$
##### 4: 4.36 Infinite Products and Partial Fractions
4.36.4 ${\operatorname{csch}^{2}}z=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}},$
##### 5: 3.12 Mathematical Constants
3.12.1 $\pi=3.14159\;26535\;89793\;23846\;\ldots$
##### 6: 6.15 Sums
6.15.1 $\sum_{n=1}^{\infty}\mathrm{Ci}\left(\pi n\right)=\tfrac{1}{2}(\ln 2-\gamma),$
6.15.2 $\sum_{n=1}^{\infty}\frac{\mathrm{si}\left(\pi n\right)}{n}=\tfrac{1}{2}\pi(\ln% \pi-1),$
6.15.3 $\sum_{n=1}^{\infty}(-1)^{n}\mathrm{Ci}\left(2\pi n\right)=1-\ln 2-\tfrac{1}{2}\gamma,$
6.15.4 $\sum_{n=1}^{\infty}(-1)^{n}\frac{\mathrm{si}\left(2\pi n\right)}{n}=\pi(\tfrac% {3}{2}\ln 2-1).$
##### 7: 21.8 Abelian Functions
For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …
##### 8: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
22.12.1 $\tau=i{K^{\prime}}\left(k\right)/K\left(k\right),$
22.12.2 $2Kk\operatorname{sn}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{% \sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(% \sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-(n+\frac{1}{2})\tau}\right),$
22.12.8 $2K\operatorname{dc}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin% \left(\pi(t+\frac{1}{2}-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=% -\infty}^{\infty}\frac{(-1)^{m}}{t+\frac{1}{2}-m-n\tau}\right),$
22.12.11 $2K\operatorname{ns}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin% \left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{% \infty}\frac{(-1)^{m}}{t-m-n\tau}\right),$
22.12.12 $2K\operatorname{ds}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}% \pi}{\sin\left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-% \infty}^{\infty}\frac{(-1)^{m+n}}{t-m-n\tau}\right),$
##### 10: 7.9 Continued Fractions
7.9.1 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{z}{z^{2}+\cfrac{\frac{1}{2}}{1+% \cfrac{1}{z^{2}+\cfrac{\frac{3}{2}}{1+\cfrac{2}{z^{2}+\cdots}}}}},$ $\Re z>0$,
7.9.2 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{2z}{2z^{2}+1-\cfrac{1\cdot 2}{2% z^{2}+5-\cfrac{3\cdot 4}{2z^{2}+9-\cdots}}},$ $\Re z>0$,
7.9.3 $w\left(z\right)=\frac{i}{\sqrt{\pi}}\cfrac{1}{z-\cfrac{\frac{1}{2}}{z-\cfrac{1% }{z-\cfrac{\frac{3}{2}}{z-\cfrac{2}{z-\cdots}}}}},$ $\Im z>0$.