# variation of real or complex functions

(0.023 seconds)

## 1—10 of 596 matching pages

##### 1: 1.4 Calculus of One Variable
###### Functions of Bounded Variation
Lastly, whether or not the real numbers $a$ and $b$ satisfy $a, and whether or not they are finite, we define $\mathcal{V}_{a,b}\left(f\right)$ by (1.4.34) whenever this integral exists. This definition also applies when $f(x)$ is a complex function of the real variable $x$. …
##### 2: 10.40 Asymptotic Expansions for Large Argument
###### §10.40(iii) Error Bounds for Complex Argument and Order
where $\mathcal{V}$ denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that $|\Re t|$ changes monotonically. …
10.40.12 $\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)\leq\begin{cases}|z|^{-\ell},&|% \operatorname{ph}z|\leq\tfrac{1}{2}\pi,\\ \chi(\ell)|z|^{-\ell},&\tfrac{1}{2}\pi\leq|\operatorname{ph}z|\leq\pi,\\ 2\chi(\ell)|\Re z|^{-\ell},&\pi\leq|\operatorname{ph}z|<\tfrac{3}{2}\pi,\end{cases}$
##### 5: 14.31 Other Applications
###### §14.31(iii) Miscellaneous
Legendre functions $P_{\nu}\left(x\right)$ of complex degree $\nu$ appear in the application of complex angular momentum techniques to atomic and molecular scattering (Connor and Mackay (1979)). …
##### 6: 30.6 Functions of Complex Argument
###### §30.6 Functions of Complex Argument
The solutions …
30.6.3 $\mathscr{W}\left\{\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right),\mathit{Qs}^{m}% _{n}\left(z,\gamma^{2}\right)\right\}=\frac{(-1)^{m}(n+m)!}{(1-z^{2})(n-m)!}A_% {n}^{m}(\gamma^{2})A_{n}^{-m}(\gamma^{2}),$
30.6.4 $\mathit{Ps}^{m}_{n}\left(x\pm\mathrm{i}0,\gamma^{2}\right)=(\mp\mathrm{i})^{m}% \mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),$
##### 7: 12.8 Recurrence Relations and Derivatives
12.8.1 $zU\left(a,z\right)-U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0,$
12.8.2 $U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)+(a+\tfrac{1}{2})U\left(a+1,z% \right)=0,$
12.8.3 $U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)+U\left(a-1,z\right)=0,$
12.8.4 $2U'\left(a,z\right)+U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0.$
12.8.5 $zV\left(a,z\right)-V\left(a+1,z\right)+(a-\tfrac{1}{2})V\left(a-1,z\right)=0,$
##### 9: 22.17 Moduli Outside the Interval [0,1]
22.17.1 $\operatorname{pq}\left(z,k\right)=\operatorname{pq}\left(z,-k\right),$
22.17.2 $\operatorname{sn}\left(z,1/k\right)=k\operatorname{sn}\left(z/k,k\right),$
##### 10: 4.3 Graphics
###### §4.3(ii) Complex Arguments: Conformal Maps Figure 4.3.3: ln ⁡ ( x + i ⁢ y ) (principal value). … Magnify 3D Help Figure 4.3.4: e x + i ⁢ y . Magnify 3D Help