# even part

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## 1—10 of 21 matching pages

##### 1: 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 $w\left(z\right)=\frac{i}{\sqrt{\pi}}\cfrac{1}{z-\cfrac{\frac{1}{2}}{z-\cfrac{1% }{z-\cfrac{\frac{3}{2}}{z-\cfrac{2}{z-\cdots}}}}},$ $\Im z>0$.
##### 2: 1.12 Continued Fractions
###### §1.12(iv) Contraction and Extension
If $C^{\prime}_{n}=C_{2n}$, $n=0,1,2,\dots$, then $C^{\prime}$ is called the even part of $C$. The even part of $C$ exists iff $b_{2k}\not=0$, $k=1,2,\dots$, and up to equivalence is given by … and the even and odd parts of the continued fraction converge to finite values. …
##### 3: 18.17 Integrals
18.17.41 $\int_{0}^{\infty}e^{-ax}\mathit{He}_{n}\left(x\right)x^{z-1}\,\mathrm{d}x=% \Gamma\left(z+n\right)a^{-n-2}{{}_{2}F_{2}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}% n+\tfrac{1}{2}\atop-\tfrac{1}{2}z-\tfrac{1}{2}n,-\tfrac{1}{2}z-\tfrac{1}{2}n+% \tfrac{1}{2}};-\tfrac{1}{2}a^{2}\right),$ $\Re a>0$. Also, $\Re z>0$, $n$ even; $\Re z>-1$, $n$ odd.
##### 4: 28.23 Expansions in Series of Bessel Functions
When $j=2,3,4$ the series in the even-numbered equations converge for $\Re z>0$ and $|\cosh z|>1$, and the series in the odd-numbered equations converge for $\Re z>0$ and $|\sinh z|>1$. …
##### 5: 20.1 Special Notation
 $m$, $n$ integers. … the nome, $q=e^{i\pi\tau}$, $0<\left|q\right|<1$. Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\Re\tau=0$, $\Im\tau>0$, so that $0 and $\tau$ and $q$ are uniquely related. …
##### 6: 19.38 Approximations
They are valid over parts of the complex $k$ and $\phi$ planes. The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\phi$ near $\pi/2$ with the improvements made in the 1970 reference. …
##### 7: 4.37 Inverse Hyperbolic Functions
4.37.11 $\operatorname{arccosh}\left(-z\right)=\pm\pi i+\operatorname{arccosh}z,$ $\Im z\gtrless 0$.
##### 8: 14.17 Integrals
14.17.15 $\int_{-1}^{1}\mathsf{P}_{\nu}\left(x\right)\mathsf{Q}_{\nu}\left(x\right)\,% \mathrm{d}x=-\frac{\sin\left(2\nu\pi\right)\psi'\left(\nu+1\right)}{\pi(2\nu+1% )},$ $\Re\nu>0$.
(When $l+m+n$ is even the condition $\left|m-n\right| is not needed.) …
14.17.18 $\int_{1}^{\infty}P_{\nu}\left(x\right)Q_{\lambda}\left(x\right)\,\mathrm{d}x=% \frac{1}{(\lambda-\nu)(\nu+\lambda+1)},$ $\Re\lambda>\Re\nu>0$.
14.17.19 $\int_{1}^{\infty}Q_{\nu}\left(x\right)Q_{\lambda}\left(x\right)\,\mathrm{d}x=% \frac{\psi\left(\lambda+1\right)-\psi\left(\nu+1\right)}{(\lambda-\nu)(\lambda% +\nu+1)},$ $\Re\left(\lambda+\nu\right)>-1$, $\lambda\neq\nu$, $\lambda$ and $\nu\neq-1,-2,-3,\dots$.
14.17.20 $\int_{1}^{\infty}(Q_{\nu}\left(x\right))^{2}\,\mathrm{d}x=\frac{\psi'\left(\nu% +1\right)}{2\nu+1},$ $\Re\nu>-\tfrac{1}{2}$.
##### 9: 20.2 Definitions and Periodic Properties
For fixed $\tau$, each $\theta_{j}\left(z\middle|\tau\right)$ is an entire function of $z$ with period $2\pi$; $\theta_{1}\left(z\middle|\tau\right)$ is odd in $z$ and the others are even. For fixed $z$, each of $\ifrac{\theta_{1}\left(z\middle|\tau\right)}{\sin z}$, $\ifrac{\theta_{2}\left(z\middle|\tau\right)}{\cos z}$, $\theta_{3}\left(z\middle|\tau\right)$, and $\theta_{4}\left(z\middle|\tau\right)$ is an analytic function of $\tau$ for $\Im\tau>0$, with a natural boundary $\Im\tau=0$, and correspondingly, an analytic function of $q$ for $\left|q\right|<1$ with a natural boundary $\left|q\right|=1$. …
##### 10: 9.13 Generalized Airy Functions
9.13.10 $A_{n}\left(-z\right)=\begin{cases}2\sqrt{p/\pi}\cos\left(\tfrac{1}{2}p\pi% \right)z^{-n/4}\left(\cos\left(\zeta-\tfrac{1}{4}\pi\right)+e^{|\Im\zeta|}O% \left(\zeta^{-1}\right)\right),&\text{|\operatorname{ph}z|\leq 2p\pi-\delta,% n odd},\\ \sqrt{p/\pi}z^{-n/4}e^{\zeta}\left(1+O\left(\zeta^{-1}\right)\right),&\text{|% \operatorname{ph}z|\leq p\pi-\delta, n even},\end{cases}$
9.13.12 $B_{n}\left(-z\right)=\begin{cases}-(\ifrac{2}{\sqrt{\pi}})\sin\left(\tfrac{1}{% 2}p\pi\right)z^{-n/4}\left(\sin\left(\zeta-\tfrac{1}{4}\pi\right)+e^{\left|\Im% \zeta\right|}O\left(\zeta^{-1}\right)\right),&\left|\operatorname{ph}z\right|% \leq 2p\pi-\delta,n\text{ odd},\\ (\ifrac{1}{\sqrt{\pi}})\sin\left(p\pi\right)z^{-n/4}e^{-\zeta}\left(1+O\left(% \zeta^{-1}\right)\right),&\left|\operatorname{ph}z\right|\leq 3p\pi-\delta,n% \text{ even}.\end{cases}$