# variational operator

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## 8 matching pages

##### 1: 10.17 Asymptotic Expansions for Large Argument
10.17.14 $\left|R_{\ell}^{\pm}(\nu,z)\right|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\pm i% \infty}\left(t^{-\ell}\right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,% \pm i\infty}\left(t^{-1}\right)\right),$
where $\mathcal{V}$ denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that $|\Im t|$ changes monotonically. Bounds for $\mathcal{V}_{z,i\infty}\left(t^{-\ell}\right)$ are given by
10.17.15 $\mathcal{V}_{z,i\infty}\left(t^{-\ell}\right)\leq\begin{cases}|z|^{-\ell},&0% \leq\operatorname{ph}z\leq\pi,\\ \chi(\ell)|z|^{-\ell},&\parbox[t]{224.03743pt}{-\tfrac{1}{2}\pi\leq% \operatorname{ph}z\leq 0 or \pi\leq\operatorname{ph}z\leq\tfrac{3}{2}\pi,}\\ 2\chi(\ell)|\Im z|^{-\ell},&\parbox[t]{224.03743pt}{-\pi<\operatorname{ph}z% \leq-\tfrac{1}{2}\pi or \tfrac{3}{2}\pi\leq\operatorname{ph}z<2\pi,}\end{cases}$
The bounds (10.17.15) also apply to $\mathcal{V}_{z,-i\infty}\left(t^{-\ell}\right)$ in the conjugate sectors. …
##### 2: 10.40 Asymptotic Expansions for Large Argument
10.40.11 $|R_{\ell}(\nu,z)|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\infty}\left(t^{-\ell}% \right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,\infty}\left(t^{-1}% \right)\right),$
where $\mathcal{V}$ denotes the variational operator2.3(i)), and the paths of variation are subject to the condition that $|\Re t|$ changes monotonically. Bounds for $\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)$ are given by
10.40.12 $\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)\leq\begin{cases}|z|^{-\ell},&|% \operatorname{ph}z|\leq\tfrac{1}{2}\pi,\\ \chi(\ell)|z|^{-\ell},&\tfrac{1}{2}\pi\leq|\operatorname{ph}z|\leq\pi,\\ 2\chi(\ell)|\Re z|^{-\ell},&\pi\leq|\operatorname{ph}z|<\tfrac{3}{2}\pi,\end{cases}$
##### 3: 1.4 Calculus of One Variable
1.4.33 $\mathcal{V}_{a,b}\left(f\right)=\sup\sum^{n}_{j=1}|f(x_{j})-f(x_{j-1})|,$
If $\mathcal{V}_{a,b}\left(f\right)<\infty$, then $f(x)$ is of bounded variation on $(a,b)$. In this case, $g(x)=\mathcal{V}_{a,x}\left(f\right)$ and $h(x)=\mathcal{V}_{a,x}\left(f\right)-f(x)$ are nondecreasing bounded functions and $f(x)=g(x)-h(x)$. …
1.4.34 $\mathcal{V}_{a,b}\left(f\right)=\int^{b}_{a}\left|f^{\prime}(x)\right|\,% \mathrm{d}x,$
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. …
##### 4: 2.7 Differential Equations
2.7.23 $|\epsilon_{j}(x)|,\;\;\tfrac{1}{2}f^{-1/2}(x)|\epsilon_{j}^{\prime}(x)|\leq% \exp\left(\tfrac{1}{2}\mathcal{V}_{a_{j},x}\left(F\right)\right)-1,$ $j=1,2$,
provided that $\mathcal{V}_{a_{j},x}\left(F\right)<\infty$. …and $\mathcal{V}$ denotes the variational operator2.3(i)). …
2.7.25 $\mathcal{V}_{a_{j},x}\left(F\right)=\left|\int_{a_{j}}^{x}\left|\frac{1}{f^{1/% 4}(t)}\frac{{\mathrm{d}}^{2}}{{\mathrm{d}t}^{2}}\left(\frac{1}{f^{1/4}(t)}% \right)-\frac{g(t)}{f^{1/2}(t)}\right|\,\mathrm{d}t\right|.$
Assuming also $\mathcal{V}_{a_{1},a_{2}}\left(F\right)<\infty$, we have …
##### 5: 2.8 Differential Equations with a Parameter
In addition, $\mathcal{V}_{\mathscr{Q}_{j}}\left(A_{1}\right)$ and $\mathcal{V}_{\mathscr{Q}_{j}}\left(A_{n}\right)$ must be bounded on $\mathbf{\Delta}_{j}(\alpha_{j})$. … These results are valid when $\mathcal{V}_{\alpha_{1},\alpha_{2}}\left(|\xi|^{1/2}B_{0}\right)$ and $\mathcal{V}_{\alpha_{1},\alpha_{2}}\left(|\xi|^{1/2}B_{n-1}\right)$ are finite. … These results are valid when $\mathcal{V}_{0,\alpha_{2}}\left(\xi^{1/2}B_{0}\right)$ and $\mathcal{V}_{0,\alpha_{2}}\left(\xi^{1/2}B_{n-1}\right)$ are finite. … These results are valid when $\mathcal{V}_{\alpha_{1},0}\left(|\xi|^{1/2}B_{0}\right)$ and $\mathcal{V}_{\alpha_{1},0}\left(|\xi|^{1/2}B_{n-1}\right)$ are finite. …
##### 6: 2.3 Integrals of a Real Variable
In both cases the $n$th error term is bounded in absolute value by $x^{-n}\mathcal{V}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator $\mathcal{V}_{a,b}$ is defined by
2.3.6 $\mathcal{V}_{a,b}\left(f(t)\right)=\int_{a}^{b}\left|f^{\prime}(t)\right|\,% \mathrm{d}t;$
##### 7: Errata
• Equation (2.7.25)
2.7.25 $\mathcal{V}_{a_{j},x}\left(F\right)=\left|\int_{a_{j}}^{x}\left|\frac{1}{f^{1/% 4}(t)}\frac{{\mathrm{d}}^{2}}{{\mathrm{d}t}^{2}}\left(\frac{1}{f^{1/4}(t)}% \right)-\frac{g(t)}{f^{1/2}(t)}\right|\,\mathrm{d}t\right|$

The integrand was corrected so that the absolute value does not include the differential. Also an absolute value was introduced on the right-hand side to ensure a non-negative value for $\mathcal{V}_{a_{j},x}\left(F\right)$.

• Equation (2.3.6)
2.3.6 $\mathcal{V}_{a,b}\left(f(t)\right)=\int_{a}^{b}\left|f^{\prime}(t)\right|\,% \mathrm{d}t$

The integrand has been corrected so that the absolute value does not include the differential.

Reported by Juan Luis Varona on 2021-02-08

• Equation (1.4.34)
1.4.34 $\mathcal{V}_{a,b}\left(f\right)=\int^{b}_{a}\left|f^{\prime}(x)\right|\,% \mathrm{d}x$

The integrand has been corrected so that the absolute value does not include the differential.

Reported by Tran Quoc Viet on 2020-08-11

• Equation (10.17.14)
10.17.14 $\left|R_{\ell}^{\pm}(\nu,z)\right|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\pm% \mathrm{i}\infty}\left(t^{-\ell}\right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|% \mathcal{V}_{z,\pm\mathrm{i}\infty}\left(t^{-1}\right)\right)$

Originally the factor $\mathcal{V}_{z,\pm\mathrm{i}\infty}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as $\mathcal{V}_{z,\pm\mathrm{i}\infty}\left(t^{-\ell}\right)$.

Reported 2014-09-27 by Gergő Nemes.