# Mill ratio for complementary error function

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##### 2: 7.2 Definitions
###### §7.2(i) ErrorFunctions
$\operatorname{erf}z$, $\operatorname{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection.
###### Values at Infinity
$\mathcal{F}\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. …
##### 3: 23.15 Definitions
###### §23.15 Definitions
In §§23.1523.19, $k$ and $k^{\prime}$ $(\in\mathbb{C})$ denote the Jacobi modulus and complementary modulus, respectively, and $q=e^{i\pi\tau}$ ($\Im\tau>0$) denotes the nome; compare §§20.1 and 22.1. …
##### 4: 9.12 Scorer Functions
###### §9.12 Scorer Functions
where … $-\operatorname{Gi}\left(x\right)$ is a numerically satisfactory companion to the complementary functions $\operatorname{Ai}\left(x\right)$ and $\operatorname{Bi}\left(x\right)$ on the interval $0\leq x<\infty$. …
##### 5: 5.12 Beta Function
###### Euler’s Beta Integral
5.12.2 $\int_{0}^{\pi/2}{\sin}^{2a-1}\theta{\cos}^{2b-1}\theta\,\mathrm{d}\theta=% \tfrac{1}{2}\mathrm{B}\left(a,b\right).$
5.12.5 $\int_{0}^{\pi/2}(\cos t)^{a-1}\cos\left(bt\right)\,\mathrm{d}t=\frac{\pi}{2^{a% }}\frac{1}{a\mathrm{B}\left(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1)\right)},$ $\Re a>0$.
##### 6: 11.9 Lommel Functions
###### §11.9(ii) Expansions in Series of Bessel Functions
For an error bound for (11.9.9) and an exponentially-improved extension see Nemes (2015b). …
##### 7: 20.2 Definitions and Periodic Properties
###### §20.2(ii) Periodicity and Quasi-Periodicity
The theta functions are quasi-periodic on the lattice: …
##### 8: 14.20 Conical (or Mehler) Functions
###### §14.20(ii) Graphics
For asymptotic expansions and explicit error bounds, see Olver (1997b, pp. 473–474). … For extensions to complex arguments (including the range $1), asymptotic expansions, and explicit error bounds, see Dunster (1991). …
##### 10: 5.15 Polygamma Functions
###### §5.15 Polygamma Functions
The functions $\psi^{(n)}\left(z\right)$, $n=1,2,\dots$, are called the polygamma functions. In particular, $\psi'\left(z\right)$ is the trigamma function; $\psi''$, $\psi^{(3)}$, $\psi^{(4)}$ are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … For $B_{2k}$ see §24.2(i). …