# shifted

(0.001 seconds)

## 1—10 of 151 matching pages

##### 1: 33.25 Approximations
###### §33.25 Approximations
Cody and Hillstrom (1970) provides rational approximations of the phase shift ${\sigma_{0}}\left(\eta\right)=\operatorname{ph}\Gamma\left(1+\mathrm{i}\eta\right)$ (see (33.2.10)) for the ranges $0\leq\eta\leq 2$, $2\leq\eta\leq 4$, and $4\leq\eta\leq\infty$. …
##### 2: Possible Errors in DLMF
One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula. …
##### 3: 6.15 Sums
6.15.2 $\sum_{n=1}^{\infty}\frac{\operatorname{si}\left(\pi n\right)}{n}=\tfrac{1}{2}% \pi(\ln\pi-1),$
6.15.4 $\sum_{n=1}^{\infty}(-1)^{n}\frac{\operatorname{si}\left(2\pi n\right)}{n}=\pi(% \tfrac{3}{2}\ln 2-1).$
##### 4: 33.13 Complex Variable and Parameters
The quantities $C_{\ell}\left(\eta\right)$, ${\sigma_{\ell}}\left(\eta\right)$, and $R_{\ell}$, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that
33.13.1 $C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),$
##### 5: 6.14 Integrals
6.14.3 $\int_{0}^{\infty}e^{-at}\operatorname{si}\left(t\right)\,\mathrm{d}t=-\frac{1}% {a}\operatorname{arctan}a,$ $\Re a>0$.
6.14.5 $\int_{0}^{\infty}\cos t\operatorname{Ci}\left(t\right)\,\mathrm{d}t=\int_{0}^{% \infty}\sin t\operatorname{si}\left(t\right)\,\mathrm{d}t=-\tfrac{1}{4}\pi,$
6.14.6 $\int_{0}^{\infty}{\operatorname{Ci}}^{2}\left(t\right)\,\mathrm{d}t=\int_{0}^{% \infty}{\operatorname{si}}^{2}\left(t\right)\,\mathrm{d}t=\tfrac{1}{2}\pi,$
##### 7: 35.4 Partitions and Zonal Polynomials
Also, $|\kappa|$ denotes $k_{1}+\dots+k_{m}$, the weight of $\kappa$; $\ell(\kappa)$ denotes the number of nonzero $k_{j}$; $a+\kappa$ denotes the vector $(a+k_{1},\dots,a+k_{m})$. The partitional shifted factorial is given by
35.4.2 $Z_{\kappa}\left(\mathbf{I}\right)=|\kappa|!\,2^{2|\kappa|}\,{\left[m/2\right]_% {\kappa}}\frac{\prod\limits_{1\leq j
35.4.9 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|\mathbf{X}\right|^{a% -\frac{1}{2}(m+1)}\*\left|\mathbf{I}-\mathbf{X}\right|^{b-\frac{1}{2}(m+1)}Z_{% \kappa}\left(\mathbf{T}\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}=\frac{{\left[% a\right]_{\kappa}}}{{\left[a+b\right]_{\kappa}}}\mathrm{B}_{m}\left(a,b\right)% Z_{\kappa}\left(\mathbf{T}\right).$
##### 8: 18.1 Notation
###### Classical OP’s
• Shifted Chebyshev of first and second kinds: $T^{*}_{n}\left(x\right)$, $U^{*}_{n}\left(x\right)$.

• Shifted Legendre: $P^{*}_{n}\left(x\right)$.

• Nor do we consider the shifted Jacobi polynomials:
##### 9: 5.2 Definitions
5.2.5 ${\left(a\right)_{n}}=\Gamma\left(a+n\right)/\Gamma\left(a\right),$ $a\neq 0,-1,-2,\dots$.
5.2.6 ${\left(-a\right)_{n}}=(-1)^{n}{\left(a-n+1\right)_{n}},$
5.2.7 ${\left(-m\right)_{n}}=\begin{cases}\frac{(-1)^{n}m!}{(m-n)!},&0\leq n\leq m,\\ 0,&n>m,\end{cases}$
##### 10: 6.1 Special Notation
The main functions treated in this chapter are the exponential integrals $\operatorname{Ei}\left(x\right)$, $E_{1}\left(z\right)$, and $\operatorname{Ein}\left(z\right)$; the logarithmic integral $\operatorname{li}\left(x\right)$; the sine integrals $\operatorname{Si}\left(z\right)$ and $\operatorname{si}\left(z\right)$; the cosine integrals $\operatorname{Ci}\left(z\right)$ and $\operatorname{Cin}\left(z\right)$. …