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## 1—10 of 673 matching pages

##### 1: 4.17 Special Values and Limits
4.17.3 $\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.$
##### 2: 18.13 Continued Fractions
18.13.1 $\cfrac{-1}{x+\cfrac{-1}{2x+\cfrac{-1}{2x+}}}\cdots,$
18.13.2 $\cfrac{-1}{2x+\cfrac{-1}{2x+\cfrac{-1}{2x+}}}\cdots.$
18.13.3 $\cfrac{a_{1}}{x+\cfrac{-\frac{1}{2}}{\frac{3}{2}x+\cfrac{-\frac{2}{3}}{\frac{5% }{3}x+\cfrac{-\frac{3}{4}}{\frac{7}{4}x+}}}}\cdots,$
18.13.4 $\cfrac{a_{1}}{1-x+\cfrac{-\frac{1}{2}}{\frac{1}{2}(3-x)+\cfrac{-\frac{2}{3}}{% \frac{1}{3}(5-x)+\cfrac{-\frac{3}{4}}{\frac{1}{4}(7-x)+}}}}\cdots,$
18.13.5 $\cfrac{1}{2x+\cfrac{-2}{2x+\cfrac{-4}{2x+\cfrac{-6}{2x+}}}}\cdots.$
##### 4: 12.7 Relations to Other Functions
12.7.8 $U\left(-2,z\right)=\frac{z^{5/2}}{4\sqrt{2\pi}}\left(2K_{\frac{1}{4}}\left(% \tfrac{1}{4}z^{2}\right)+3K_{\frac{3}{4}}\left(\tfrac{1}{4}z^{2}\right)-K_{% \frac{5}{4}}\left(\tfrac{1}{4}z^{2}\right)\right),$
For these, the corresponding results for $U\left(a,z\right)$ with $a=2$, $\pm 3$, $-\tfrac{1}{2}$, $-\tfrac{3}{2}$, $-\tfrac{5}{2}$, and the corresponding results for $V\left(a,z\right)$ with $a=0$, $\pm 1$, $\pm 2$, $\pm 3$, $\tfrac{1}{2}$, $\tfrac{3}{2}$, $\tfrac{5}{2}$, see Miller (1955, pp. 42–43 and 77–79). …
12.7.12 $u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2% },\tfrac{1}{2}z^{2}\right)=e^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac{1% }{4},\tfrac{1}{2},-\tfrac{1}{2}z^{2}\right),$
12.7.13 $u_{2}(a,z)=ze^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{% 2},\tfrac{1}{2}z^{2}\right)=ze^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac% {3}{4},\tfrac{3}{2},-\tfrac{1}{2}z^{2}\right).$
12.7.14 $U\left(a,z\right)=2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}}U\left(% \tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=2^{-\frac{3}{% 4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}U\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac% {3}{2},\tfrac{1}{2}z^{2}\right)=2^{-\frac{1}{2}a}z^{-\frac{1}{2}}W_{-\frac{1}{% 2}a,\pm\frac{1}{4}}\left(\tfrac{1}{2}z^{2}\right).$
##### 5: 26.3 Lattice Paths: Binomial Coefficients
$\genfrac{(}{)}{0.0pt}{}{m}{n}$ is the number of ways of choosing $n$ objects from a collection of $m$ distinct objects without regard to order. $\genfrac{(}{)}{0.0pt}{}{m+n}{n}$ is the number of lattice paths from $(0,0)$ to $(m,n)$. …The number of lattice paths from $(0,0)$ to $(m,n)$, $m\leq n$, that stay on or above the line $y=x$ is $\genfrac{(}{)}{0.0pt}{}{m+n}{m}-\genfrac{(}{)}{0.0pt}{}{m+n}{m-1}.$For numerical values of $\genfrac{(}{)}{0.0pt}{}{m}{n}$ and $\genfrac{(}{)}{0.0pt}{}{m+n}{n}$ see Tables 26.3.1 and 26.3.2.
##### 6: 13.15 Recurrence Relations and Derivatives
13.15.3 $(\kappa-\mu-\tfrac{1}{2})M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)+% (1+2\mu)\sqrt{z}M_{\kappa,\mu}\left(z\right)-(\kappa+\mu+\tfrac{1}{2})M_{% \kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,$
13.15.12 $(\kappa-\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z% \right)+2\mu W_{\kappa,\mu}\left(z\right)-(\kappa+\mu-\tfrac{1}{2})\sqrt{z}W_{% \kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)=0,$
13.15.16 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z^{-\mu-\frac{% 1}{2}}M_{\kappa,\mu}\left(z\right)\right)=\frac{{\left(\frac{1}{2}+\mu-\kappa% \right)_{n}}}{{\left(1+2\mu\right)_{n}}}e^{\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+% 1)}M_{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}\left(z\right),$
13.15.24 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}z^{-\mu-\frac% {1}{2}}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}e^{-\frac{1}{2}z}z^{-\mu-% \frac{1}{2}(n+1)}W_{\kappa+\frac{1}{2}n,\mu+\frac{1}{2}n}\left(z\right),$
13.15.25 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}z^{\mu-\frac{% 1}{2}}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}e^{-\frac{1}{2}z}z^{\mu-% \frac{1}{2}(n+1)}W_{\kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z\right),$
##### 7: 12.13 Sums
12.13.2 $U\left(a,x+y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{-a-\tfrac{1}{2}}{m}y^{m}U\left(a+m,x\right),$
12.13.3 $V\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{a-\tfrac{1}{2}}{m}y^{m}V\left(a-m,x\right),$
12.13.4 $V\left(a,x+y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \frac{y^{m}}{m!}V\left(a+m,x\right).$
12.13.5 $U\left(a,x\cos t+y\sin t\right)\\ =e^{\frac{1}{4}(x\sin t-y\cos t)^{2}}\*\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt% }{}{-a-\tfrac{1}{2}}{m}(\tan t)^{m}U\left(m+a,x\right)U\left(-m-\tfrac{1}{2},y% \right),$ $\Re a\leq-\tfrac{1}{2},0\leq t\leq\tfrac{1}{4}\pi$.
12.13.6 $n!U\left(n+\tfrac{1}{2},z\right)=i^{n}e^{-\frac{1}{2}z^{2}}\operatorname{erfc}% (z/\sqrt{2})U\left(-n-\tfrac{1}{2},iz\right)+\sum_{m=1}^{\left\lfloor\frac{1}{% 2}n+\frac{1}{2}\right\rfloor}U\left(2m-n-\tfrac{1}{2},z\right),$ $n=0,1,2,\dots.$
##### 8: 12.4 Power-Series Expansions
12.4.3 $u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}\left(1+(a+\tfrac{1}{2})\frac{z^{2}}{2!}+(a+% \tfrac{1}{2})(a+\tfrac{5}{2})\frac{z^{4}}{4!}+\cdots\right),$
12.4.4 $u_{2}(a,z)=e^{-\tfrac{1}{4}z^{2}}\left(z+(a+\tfrac{3}{2})\frac{z^{3}}{3!}+(a+% \tfrac{3}{2})(a+\tfrac{7}{2})\frac{z^{5}}{5!}+\cdots\right).$
12.4.5 $u_{1}(a,z)=e^{\tfrac{1}{4}z^{2}}\left(1+(a-\tfrac{1}{2})\frac{z^{2}}{2!}+(a-% \tfrac{1}{2})(a-\tfrac{5}{2})\frac{z^{4}}{4!}+\cdots\right),$
12.4.6 $u_{2}(a,z)=e^{\tfrac{1}{4}z^{2}}\left(z+(a-\tfrac{3}{2})\frac{z^{3}}{3!}+(a-% \tfrac{3}{2})(a-\tfrac{7}{2})\frac{z^{5}}{5!}+\cdots\right).$
##### 9: 10.57 Uniform Asymptotic Expansions for Large Order
Asymptotic expansions for $\mathsf{j}_{n}\left((n+\tfrac{1}{2})z\right)$, $\mathsf{y}_{n}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{h}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{h}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{i}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)$, and $\mathsf{k}_{n}\left((n+\tfrac{1}{2})z\right)$ as $n\to\infty$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). Subsequently, for ${\mathsf{i}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)$ the connection formula (10.47.11) is available. For the corresponding expansion for $\mathsf{j}_{n}'\left((n+\tfrac{1}{2})z\right)$ use
10.57.1 $\mathsf{j}_{n}'\left((n+\tfrac{1}{2})z\right)=\frac{\pi^{\frac{1}{2}}}{((2n+1)% z)^{\frac{1}{2}}}J_{n+\frac{1}{2}}'\left((n+\tfrac{1}{2})z\right)-\frac{\pi^{% \frac{1}{2}}}{((2n+1)z)^{\frac{3}{2}}}J_{n+\frac{1}{2}}\left((n+\tfrac{1}{2})z% \right).$
##### 10: 28.6 Expansions for Small $q$
28.6.1 $a_{0}\left(q\right)=-\tfrac{1}{2}q^{2}+\tfrac{7}{128}q^{4}-\tfrac{29}{2304}q^{% 6}+\tfrac{68687}{188\;74368}q^{8}+\cdots,$
28.6.2 $a_{1}\left(q\right)=1+q-\tfrac{1}{8}q^{2}-\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^% {4}+\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}+\tfrac{55}{94\;37184}q^{7% }-\tfrac{83}{353\;89440}q^{8}+\cdots,$
28.6.3 $b_{1}\left(q\right)=1-q-\tfrac{1}{8}q^{2}+\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^% {4}-\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}-\tfrac{55}{94\;37184}q^{7% }-\tfrac{83}{353\;89440}q^{8}+\cdots,$
28.6.6 $a_{3}\left(q\right)=9+\tfrac{1}{16}q^{2}+\tfrac{1}{64}q^{3}+\tfrac{13}{20480}q% ^{4}-\tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{6}-\tfrac{609}{1048\;576% 00}q^{7}+\cdots,$
28.6.7 $b_{3}\left(q\right)=9+\tfrac{1}{16}q^{2}-\tfrac{1}{64}q^{3}+\tfrac{13}{20480}q% ^{4}+\tfrac{5}{16384}q^{5}-\tfrac{1961}{235\;92960}q^{6}+\tfrac{609}{1048\;576% 00}q^{7}+\cdots,$