# modulus and phase

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##### 2: 36.3 Visualizations of Canonical Integrals
###### §36.3(ii) Canonical Integrals: Phase
In Figure 36.3.13(a) points of confluence of phase contours are zeros of $\Psi_{2}\left(x,y\right)$; similarly for other contour plots in this subsection. …
##### 3: 10.68 Modulus and Phase Functions
###### §10.68(iii) Asymptotic Expansions for Large Argument
Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …
##### 4: 10.18 Modulus and Phase Functions
###### §10.18(i) Definitions
where $M_{\nu}\left(x\right)$ $(>0)$, $N_{\nu}\left(x\right)$ $(>0)$, $\theta_{\nu}\left(x\right)$, and $\phi_{\nu}\left(x\right)$ are continuous real functions of $\nu$ and $x$, with the branches of $\theta_{\nu}\left(x\right)$ and $\phi_{\nu}\left(x\right)$ fixed by …
###### §10.18(iii) Asymptotic Expansions for Large Argument
In doing this, however, we would like to place the mathematically significant phase values, specifically the multiples of $\pi/2$ correponding to the real and imaginary axes, at more immediately recognizable colors. … We therefore use a piecewise linear mapping as illustrated below, that takes phase $0$ to red, $\pi/2$ to yellow, $\pi$ to cyan and $3\pi/2$ to blue. …
##### 6: 33.13 Complex Variable and Parameters
33.13.1 $C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),$
##### 7: 33.11 Asymptotic Expansions for Large $\rho$
where ${\theta_{\ell}}\left(\eta,\rho\right)$ is defined by (33.2.9), and $a$ and $b$ are defined by (33.8.3). …
$F_{\ell}\left(\eta,\rho\right)=g(\eta,\rho)\cos{\theta_{\ell}}+f(\eta,\rho)% \sin{\theta_{\ell}},$
$G_{\ell}\left(\eta,\rho\right)=f(\eta,\rho)\cos{\theta_{\ell}}-g(\eta,\rho)% \sin{\theta_{\ell}},$
$F_{\ell}'\left(\eta,\rho\right)=\widehat{g}(\eta,\rho)\cos{\theta_{\ell}}+% \widehat{f}(\eta,\rho)\sin{\theta_{\ell}},$
Here $f_{0}=1$, $g_{0}=0$, $\widehat{f}_{0}=0$, $\widehat{g}_{0}=1-(\eta/\rho)$, and for $k=0,1,2,\dots$, …
##### 8: 10.27 Connection Formulas
10.27.6 $I_{\nu}\left(z\right)=e^{\mp\nu\pi i/2}J_{\nu}\left(ze^{\pm\pi i/2}\right),$ $-\pi\leq\pm\operatorname{ph}z\leq\tfrac{1}{2}\pi$,
10.27.7 $I_{\nu}\left(z\right)=\tfrac{1}{2}e^{\mp\nu\pi i/2}\left({H^{(1)}_{\nu}}\left(% ze^{\pm\pi i/2}\right)+{H^{(2)}_{\nu}}\left(ze^{\pm\pi i/2}\right)\right),$ $-\pi\leq\pm\operatorname{ph}z\leq\tfrac{1}{2}\pi$.
10.27.9 $\pi iJ_{\nu}\left(z\right)=e^{-\nu\pi i/2}K_{\nu}\left(ze^{-\pi i/2}\right)-e^% {\nu\pi i/2}K_{\nu}\left(ze^{\pi i/2}\right),$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi$.
10.27.10 $-\pi Y_{\nu}\left(z\right)=e^{-\nu\pi i/2}K_{\nu}\left(ze^{-\pi i/2}\right)+e^% {\nu\pi i/2}K_{\nu}\left(ze^{\pi i/2}\right),$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi$.
10.27.11 $Y_{\nu}\left(z\right)=e^{\pm(\nu+1)\pi i/2}I_{\nu}\left(ze^{\mp\pi i/2}\right)% -(2/\pi)e^{\mp\nu\pi i/2}K_{\nu}\left(ze^{\mp\pi i/2}\right),$ $-\tfrac{1}{2}\pi\leq\pm\operatorname{ph}z\leq\pi$.
##### 10: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$
$F_{\ell}\left(\eta,\rho\right)=\sin\left({\theta_{\ell}}\left(\eta,\rho\right)% \right)+o\left(1\right),$
$G_{\ell}\left(\eta,\rho\right)=\cos\left({\theta_{\ell}}\left(\eta,\rho\right)% \right)+o\left(1\right),$
where ${\theta_{\ell}}\left(\eta,\rho\right)$ is defined by (33.2.9). …
${\sigma_{0}}\left(\eta\right)=\eta(\ln\eta-1)+\tfrac{1}{4}\pi+o\left(1\right),$
${\sigma_{0}}\left(\eta\right)=\eta(\ln\left(-\eta\right)-1)-\tfrac{1}{4}\pi+o% \left(1\right),$