# in the complex plane

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## 1—10 of 123 matching pages

##### 1: 14.26 Uniform Asymptotic Expansions
The uniform asymptotic approximations given in §14.15 for $P^{-\mu}_{\nu}\left(x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ for $1 are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). …
##### 2: Sidebar 5.SB1: Gamma & Digamma Phase Plots
The color encoded phases of $\Gamma\left(z\right)$ (above) and $\psi\left(z\right)$ (below), are constrasted in the negative half of the complex plane. …
##### 3: 35.5 Bessel Functions of Matrix Argument
35.5.1 $A_{\nu}\left(\boldsymbol{{0}}\right)=\frac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(% m+1)\right)},$ $\nu\in\mathbb{C}$.
35.5.2 $A_{\nu}\left(\mathbf{T}\right)=A_{\nu}\left(\boldsymbol{{0}}\right)\sum_{k=0}^% {\infty}\frac{(-1)^{k}}{k!}\sum_{|\kappa|=k}\frac{1}{{\left[\nu+\frac{1}{2}(m+% 1)\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
35.5.3 $B_{\nu}\left(\mathbf{T}\right)=\int_{\boldsymbol{\Omega}}\operatorname{etr}% \left(-(\mathbf{T}\mathbf{X}+\mathbf{X}^{-1})\right)\left|\mathbf{X}\right|^{% \nu-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},$ $\nu\in\mathbb{C}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.5.5 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}A_{\nu_{1}}\left(\mathbf{S% }_{1}\mathbf{X}\right)\left|\mathbf{X}\right|^{\nu_{1}}\*A_{\nu_{2}}\left(% \mathbf{S}_{2}(\mathbf{T}-\mathbf{X})\right)\left|\mathbf{T}-\mathbf{X}\right|% ^{\nu_{2}}\,\mathrm{d}{\mathbf{X}}=\left|\mathbf{T}\right|^{\nu_{1}+\nu_{2}+% \frac{1}{2}(m+1)}A_{\nu_{1}+\nu_{2}+\frac{1}{2}(m+1)}\left((\mathbf{S}_{1}+% \mathbf{S}_{2})\mathbf{T}\right),$ $\nu_{j}\in\mathbb{C}$, $\Re\left(\nu_{j}\right)>-1$, $j=1,2$; $\mathbf{S}_{1},\mathbf{S}_{2}\in\boldsymbol{\mathcal{S}}$; $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.5.6 $B_{\nu}\left(\mathbf{T}\right)=\left|\mathbf{T}\right|^{-\nu}B_{-\nu}\left(% \mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
##### 4: 1.9 Calculus of a Complex Variable
Also, the union of $S$ and its limit points is the closure of $S$. …
##### 6: 4.37 Inverse Hyperbolic Functions
4.37.16 $\operatorname{arcsinh}z=\ln\left((z^{2}+1)^{1/2}+z\right),$ $z/i\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$;
4.37.19 $\operatorname{arccosh}z=\ln\left(\pm(z^{2}-1)^{1/2}+z\right),$ $z\in\mathbb{C}\setminus(-\infty,1)$,
4.37.21 $\operatorname{arccosh}z=2\ln\left(\left(\frac{z+1}{2}\right)^{1/2}+\left(\frac% {z-1}{2}\right)^{1/2}\right),$ $z\in\mathbb{C}\setminus(-\infty,1)$;
4.37.24 $\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right),$ $z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$;
##### 7: 4.5 Inequalities
4.5.16 $|e^{z}-1|\leq e^{|z|}-1\leq|z|e^{|z|},$ $z\in\mathbb{C}$.
##### 9: 18.24 Hahn Class: Asymptotic Approximations
When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to\infty$ ) of the Hahn polynomials $Q_{n}(z;\alpha,\beta,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. …
##### 10: 4.23 Inverse Trigonometric Functions
4.23.19 $\operatorname{arcsin}z=-i\ln\left((1-z^{2})^{1/2}+iz\right),$ $z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$;
4.23.22 $\operatorname{arccos}z=\tfrac{1}{2}\pi+i\ln\left((1-z^{2})^{1/2}+iz\right),$ $z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$;
4.23.23 $\operatorname{arccos}z=-2i\ln\left(\left(\frac{1+z}{2}\right)^{1/2}+i\left(% \frac{1-z}{2}\right)^{1/2}\right),$ $z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$;
4.23.26 $\operatorname{arctan}z=\frac{i}{2}\ln\left(\frac{i+z}{i-z}\right),$ $z/i\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$;