# sign function

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##### 1: 32.11 Asymptotic Approximations for Real Variables
32.11.9 $\theta_{0}=\tfrac{3}{2}d^{2}\ln 2+\operatorname{ph}\Gamma\left(1-\tfrac{1}{2}% id^{2}\right)+\tfrac{1}{4}\pi(1-2\operatorname{sign}\left(k\right)),$
32.11.17 $d^{2}=\pi^{-1}\ln\left(1+k^{2}\right),$ $\operatorname{sign}\left(k\right)=(-1)^{n}$.
##### 2: 1.16 Distributions
1.16.44 $\operatorname{sign}\left(x\right)=2H\left(x\right)-1,$ $x\neq 0$,
1.16.46 $\mathscr{F}\left({\operatorname{sign}}^{\prime}\right)=\mathscr{F}\left(2H'% \right)=2\mathscr{F}\left(\delta\right)=\sqrt{\frac{2}{\pi}},$
1.16.49 $\left\langle\mathscr{F}\left(\operatorname{sign}\right),\phi\right\rangle=% \mathrm{i}\sqrt{\frac{2}{\pi}}\pvint^{\infty}_{-\infty}\frac{\phi(x)}{x}% \mathrm{d}x.$
##### 4: 4.24 Inverse Trigonometric Functions: Further Properties
4.24.4 $\operatorname{arctan}z=\pm\frac{\pi}{2}-\frac{1}{z}+\frac{1}{3z^{3}}-\frac{1}{% 5z^{5}}+\cdots,$ $\Re z\gtrless 0$, $|z|\geq 1$.
##### 5: 10.21 Zeros
For sign properties of the forward differences that are defined by … The zeros of the functions
##### 6: Bibliography S
• J. Steinig (1972) The sign of Lommel’s function. Trans. Amer. Math. Soc. 163, pp. 123–129.
• ##### 7: 28.12 Definitions and Basic Properties
###### §28.12 Definitions and Basic Properties
For change of signs of $\nu$ and $q$, …
###### §28.12(ii) Eigenfunctions $\mathrm{me}_{\nu}\left(z,q\right)$
For changes of sign of $\nu$, $q$, and $z$, … For change of signs of $\nu$ and $z$, …
##### 8: 10.44 Sums
If $\mathscr{Z}=I$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary. …
##### 9: 4.37 Inverse Hyperbolic Functions
###### §4.37 Inverse Hyperbolic Functions
the upper/lower signs corresponding to the right/left sides. … the upper/lower sign corresponding to the upper/lower side. … the upper/lower sign corresponding to the upper/lower side. … the upper/lower sign corresponding to the upper/lower sides. …
##### 10: 33.2 Definitions and Basic Properties
The function $F_{\ell}\left(\eta,\rho\right)$ is recessive (§2.7(iii)) at $\rho=0$, and is defined by …The choice of ambiguous signs in (33.2.3) and (33.2.4) is immaterial, provided that either all upper signs are taken, or all lower signs are taken. … The functions ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ are defined by … …