# modulus and phase functions

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## 1—10 of 134 matching pages

##### 2: 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). …
##### 3: 10.18 Modulus and Phase Functions
###### §10.18(iii) Asymptotic Expansions for Large Argument
In (10.18.17) and (10.18.18) the remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n>\nu-\frac{1}{2}$ for (10.18.17) and $-\frac{3}{2}\leq\nu\leq\frac{3}{2}$ for (10.18.18).
##### 4: 12.19 Tables
• Miller (1955) includes $W\left(a,x\right)$, $W\left(a,-x\right)$, and reduced derivatives for $a=-10(1)10$, $x=0(.1)10$, 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

• Fox (1960) includes modulus and phase functions for $W\left(a,x\right)$ and $W\left(a,-x\right)$, and several auxiliary functions for $x^{-1}=0(.005)0.1$, $a=-10(1)10$, 8S.

• ##### 5: 10.3 Graphics
###### §10.3(i) Real Order and Variable
For the modulus and phase functions $M_{\nu}\left(x\right)$, $\theta_{\nu}\left(x\right)$, $N_{\nu}\left(x\right)$, and $\phi_{\nu}\left(x\right)$ see §10.18. …
##### 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: 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. …
##### 9: 9.11 Products
9.11.19 $\int_{0}^{\infty}\frac{\,\mathrm{d}t}{{\operatorname{Ai}}^{2}\left(t\right)+{% \operatorname{Bi}}^{2}\left(t\right)}=\int_{0}^{\infty}\frac{t\,\mathrm{d}t}{{% \operatorname{Ai}'}^{2}\left(t\right)+{\operatorname{Bi}'}^{2}\left(t\right)}=% \frac{\pi^{2}}{6}.$
##### 10: 12.2 Differential Equations
###### §12.2(vi) Solution $\overline{U}\left(a,x\right)$; Modulus and PhaseFunctions
12.2.22 $U\left(a,x\right)+i\overline{U}\left(a,x\right)=F(a,x)e^{i\theta(a,x)},$
12.2.23 $U'\left(a,x\right)+i\overline{U}'\left(a,x\right)=-G(a,x)e^{i\psi(a,x)},$
For properties of the modulus and phase functions, including differential equations, see Miller (1955, pp. 72–73). …