# as z→∞

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## 1—10 of 725 matching pages

##### 1: 4.14 Definitions and Periodicity
4.14.4 $\tan z=\frac{\sin z}{\cos z},$
4.14.5 $\csc z=\frac{1}{\sin z},$
The functions $\sin z$ and $\cos z$ are entire. In $\mathbb{C}$ the zeros of $\sin z$ are $z=k\pi$, $k\in\mathbb{Z}$; the zeros of $\cos z$ are $z=\left(k+\tfrac{1}{2}\right)\pi$, $k\in\mathbb{Z}$. The functions $\tan z$, $\csc z$, $\sec z$, and $\cot z$ are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). …
##### 2: 10.29 Recurrence Relations and Derivatives
With $\mathscr{Z}_{\nu}\left(z\right)$ defined as in §10.25(ii),
$\mathscr{Z}_{\nu-1}\left(z\right)-\mathscr{Z}_{\nu+1}\left(z\right)=(2\nu/z)% \mathscr{Z}_{\nu}\left(z\right),$
$\mathscr{Z}_{\nu}'\left(z\right)=\mathscr{Z}_{\nu-1}\left(z\right)-(\nu/z)% \mathscr{Z}_{\nu}\left(z\right),$
$\mathscr{Z}_{\nu}'\left(z\right)=\mathscr{Z}_{\nu+1}\left(z\right)+(\nu/z)% \mathscr{Z}_{\nu}\left(z\right).$
For results on modified quotients of the form $\ifrac{z\mathscr{Z}_{\nu\pm 1}\left(z\right)}{\mathscr{Z}_{\nu}\left(z\right)}$ see Onoe (1955) and Onoe (1956). …
##### 3: 4.28 Definitions and Periodicity
4.28.2 $\cosh z=\frac{e^{z}+e^{-z}}{2},$
The functions $\sinh z$ and $\cosh z$ have period $2\pi i$, and $\tanh z$ has period $\pi i$. The zeros of $\sinh z$ and $\cosh z$ are $z=ik\pi$ and $z=i\left(k+\frac{1}{2}\right)\pi$, respectively, $k\in\mathbb{Z}$.
##### 4: 10.51 Recurrence Relations and Derivatives
Let $f_{n}(z)$ denote any of $\mathsf{j}_{n}\left(z\right)$, $\mathsf{y}_{n}\left(z\right)$, ${\mathsf{h}^{(1)}_{n}}\left(z\right)$, or ${\mathsf{h}^{(2)}_{n}}\left(z\right)$. …
$\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{n+1}f_{n}(z))=z^% {n-m+1}f_{n-m}(z),$ $m=0,1,\dots,n$,
$\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{m}(z^{-n}f_{n}(z))=(-1% )^{m}z^{-n-m}f_{n+m}(z),$ $m=0,1,\dots.$
Let $g_{n}(z)$ denote ${\mathsf{i}^{(1)}_{n}}\left(z\right)$, ${\mathsf{i}^{(2)}_{n}}\left(z\right)$, or $(-1)^{n}$ $\mathsf{k}_{n}\left(z\right)$. Then …
##### 5: 10.36 Other Differential Equations
The quantity $\lambda^{2}$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-\lambda^{2}$ if at the same time the symbol $\mathscr{C}$ in the given solutions is replaced by $\mathscr{Z}$. …
10.36.1 $z^{2}(z^{2}+\nu^{2})w^{\prime\prime}+z(z^{2}+3\nu^{2})w^{\prime}-\left((z^{2}+% \nu^{2})^{2}+z^{2}-\nu^{2}\right)w=0,$ $w=\mathscr{Z}_{\nu}'\left(z\right)$,
10.36.2 ${z^{2}w^{\prime\prime}+z(1\pm 2z)w^{\prime}+(\pm z-\nu^{2})w=0},$ $w=e^{\mp z}\mathscr{Z}_{\nu}\left(z\right)$.
Differential equations for products can be obtained from (10.13.9)–(10.13.11) by replacing $z$ by $iz$.
##### 6: 22.13 Derivatives and Differential Equations
22.13.4 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cd}\left(z,k\right)\right)^{% 2}=\left(1-{\operatorname{cd}}^{2}\left(z,k\right)\right)\left(1-k^{2}{% \operatorname{cd}}^{2}\left(z,k\right)\right),$
22.13.7 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dc}\left(z,k\right)\right)^{% 2}=\left({\operatorname{dc}}^{2}\left(z,k\right)-1\right)\left({\operatorname{% dc}}^{2}\left(z,k\right)-k^{2}\right),$
22.13.10 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{ns}\left(z,k\right)\right)^{% 2}=\left({\operatorname{ns}}^{2}\left(z,k\right)-k^{2}\right)\left({% \operatorname{ns}}^{2}\left(z,k\right)-1\right),$
22.13.19 $\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{dc}\left(z,k\right)=-(% 1+k^{2})\operatorname{dc}\left(z,k\right)+2{\operatorname{dc}}^{3}\left(z,k% \right),$
22.13.22 $\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{ns}\left(z,k\right)=-(% 1+k^{2})\operatorname{ns}\left(z,k\right)+2{\operatorname{ns}}^{3}\left(z,k% \right),$
##### 7: 7.4 Symmetry
7.4.1 $\operatorname{erf}\left(-z\right)=-\operatorname{erf}\left(z\right),$
7.4.2 $\operatorname{erfc}\left(-z\right)=2-\operatorname{erfc}\left(z\right),$
7.4.4 $F\left(-z\right)=-F\left(z\right).$
$C\left(-z\right)=-C\left(z\right),$
##### 8: 22.10 Maclaurin Series
###### §22.10(i) Maclaurin Series in $z$
The full expansions converge when $|z|<\min\left(K\left(k\right),{K^{\prime}}\left(k\right)\right)$. …
22.10.8 $\operatorname{cn}\left(z,k\right)=\operatorname{sech}z+\frac{{k^{\prime}}^{2}}% {4}(z-\sinh z\cosh z)\tanh z\operatorname{sech}z+O\left({k^{\prime}}^{4}\right),$
##### 10: 4.33 Maclaurin Series and Laurent Series
4.33.3 $\tanh z=z-\frac{z^{3}}{3}+\frac{2}{15}z^{5}-\frac{17}{315}z^{7}+\cdots+\frac{2% ^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$ $|z|<\frac{1}{2}\pi$.
For expansions that correspond to (4.19.4)–(4.19.9), change $z$ to $iz$ and use (4.28.8)–(4.28.13).