# argument ±1

(0.006 seconds)

## 1—10 of 135 matching pages

##### 1: DLMF Project News
error generating summary
##### 3: 35.10 Methods of Computation
See Yan (1992) for the ${{}_{1}F_{1}}$ and ${{}_{2}F_{1}}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $\mathbf{O}(m)$ applied to a generalization of the integral (35.5.8). …
##### 5: 4.29 Graphics
###### §4.29(i) Real Arguments Figure 4.29.4: Principal values of arctanh ⁡ x and arccoth ⁡ x . ( arctanh ⁡ x is complex when x < - 1 or x > 1 , and arccoth ⁡ x is complex when - 1 < x < 1 .) Magnify Figure 4.29.6: Principal values of arccsch ⁡ x and arcsech ⁡ x . … Magnify
###### §4.29(ii) Complex Arguments
The conformal mapping $w=\sinh z$ is obtainable from Figure 4.15.7 by rotating both the $w$-plane and the $z$-plane through an angle $\frac{1}{2}\pi$, compare (4.28.8). …
##### 6: 35.7 Gaussian Hypergeometric Function of Matrix Argument
35.7.1 ${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}% {k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}% {{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},$ $-c+\frac{1}{2}(j+1)\notin\mathbb{N}$, $1\leq j\leq m$; $\|\mathbf{T}\|<1$.
35.7.2 $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+% \nu+\frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*% {{}_{2}F_{1}}\left({-\nu,\gamma+\delta+\nu+\frac{1}{2}(m+1)\atop\gamma+\frac{1% }{2}(m+1)};\mathbf{T}\right),$ $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$; $\gamma,\delta,\nu\in\mathbb{C}$; $\Re\left(\gamma\right)>-1$.
35.7.3 ${{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).$
Let $f:{\boldsymbol{\Omega}}\to\mathbb{C}$ (a) be orthogonally invariant, so that $f(\mathbf{T})$ is a symmetric function of $t_{1},\dots,t_{m}$, the eigenvalues of the matrix argument $\mathbf{T}\in{\boldsymbol{\Omega}}$; (b) be analytic in $t_{1},\dots,t_{m}$ in a neighborhood of $\mathbf{T}=\boldsymbol{{0}}$; (c) satisfy $f(\boldsymbol{{0}})=1$. … Systems of partial differential equations for the ${{}_{0}F_{1}}$ (defined in §35.8) and ${{}_{1}F_{1}}$ functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). …
##### 7: 35.8 Generalized Hypergeometric Functions of Matrix Argument
The generalized hypergeometric function ${{}_{p}F_{q}}$ with matrix argument $\mathbf{T}\in\boldsymbol{\mathcal{S}}$, numerator parameters $a_{1},\dots,a_{p}$, and denominator parameters $b_{1},\dots,b_{q}$ is …
35.8.4 $A_{\nu}\left(\mathbf{T}\right)=\dfrac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(m+1)% \right)}{{}_{0}F_{1}}\left({-\atop\nu+\frac{1}{2}(m+1)};-\mathbf{T}\right),$ $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
35.8.8 ${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\boldsymbol{{0}}% \right)=1.$
Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions ${{}_{p}F_{q}}$ and ${{}_{p+1}F_{p}}$ of matrix argument. A similar result for the ${{}_{0}F_{1}}$ function of matrix argument is given in Faraut and Korányi (1994, p. 346). …
##### 8: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.1 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}{k!% }\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}}{{\left[b\right]_{\kappa}}}Z% _{\kappa}\left(\mathbf{T}\right).$
35.6.2 $\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m}\left(a\right)}\int_{% \boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}+\mathbf{X}\right|}^{% b-a-\frac{1}{2}(m+1)}\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right)>\frac{1}{2}(m-1)$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.6.3 $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+\nu+% \frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*{{}_% {1}F_{1}}\left({-\nu\atop\gamma+\frac{1}{2}(m+1)};\mathbf{T}\right),$ $\Re\left(\gamma\right),\Re\left(\gamma+\nu\right)>-1$.
35.6.7 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\mathrm{etr}\left(\mathbf{T}% \right){{}_{1}F_{1}}\left({b-a\atop b};-\mathbf{T}\right).$
35.6.9 $\lim_{a\to\infty}{{}_{1}F_{1}}\left({a\atop\nu+\frac{1}{2}(m+1)};-a^{-1}% \mathbf{T}\right)=\frac{A_{\nu}\left(\mathbf{T}\right)}{A_{\nu}\left(% \boldsymbol{{0}}\right)}.$
##### 9: 14.2 Differential Equations
Unless stated otherwise in §§14.214.20 it is assumed that the arguments of the functions $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ and $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ lie in the interval $(-1,1)$, and the arguments of the functions $P^{\mu}_{\nu}\left(x\right)$, $Q^{\mu}_{\nu}\left(x\right)$, and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ lie in the interval $(1,\infty)$. …
##### 10: 10.17 Asymptotic Expansions for Large Argument
10.17.4 $Y_{\nu}\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\*\left(% \sin\omega\sum_{k=0}^{\infty}(-1)^{k}\frac{a_{2k}(\nu)}{z^{2k}}+\cos\omega\sum% _{k=0}^{\infty}(-1)^{k}\frac{a_{2k+1}(\nu)}{z^{2k+1}}\right),$ $|\operatorname{ph}z|\leq\pi-\delta$,
10.17.12 ${H^{(2)}_{\nu}}'\left(z\right)\sim-i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}% e^{-i\omega}\sum_{k=0}^{\infty}(-i)^{k}\frac{b_{k}(\nu)}{z^{k}},$ $-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta$.
10.17.14 $\left|R_{\ell}^{\pm}(\nu,z)\right|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\pm i% \infty}\left(t^{-\ell}\right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,% \pm i\infty}\left(t^{-1}\right)\right),$