# with respect to modulus

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##### 2: 22.13 Derivatives and Differential Equations
22.13.2 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cn}\left(z,k\right)\right)^{% 2}={\left(1-{\operatorname{cn}}^{2}\left(z,k\right)\right)}{\left({k^{\prime}}% ^{2}+k^{2}{\operatorname{cn}}^{2}\left(z,k\right)\right)},$
22.13.3 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dn}\left(z,k\right)\right)^{% 2}=\left(1-{\operatorname{dn}}^{2}\left(z,k\right)\right)\left({\operatorname{% dn}}^{2}\left(z,k\right)-{k^{\prime}}^{2}\right).$
22.13.5 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sd}\left(z,k\right)\right)^{% 2}={\left(1-{k^{\prime}}^{2}{\operatorname{sd}}^{2}\left(z,k\right)\right)}{% \left(1+k^{2}{\operatorname{sd}}^{2}\left(z,k\right)\right)},$
22.13.6 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{nd}\left(z,k\right)\right)^{% 2}=\left({\operatorname{nd}}^{2}\left(z,k\right)-1\right)\left(1-{k^{\prime}}^% {2}{\operatorname{nd}}^{2}\left(z,k\right)\right),$
22.13.8 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{nc}\left(z,k\right)\right)^{% 2}={\left(k^{2}+{k^{\prime}}^{2}{\operatorname{nc}}^{2}\left(z,k\right)\right)% }{\left({\operatorname{nc}}^{2}\left(z,k\right)-1\right)},$
##### 3: 10.68 Modulus and Phase Functions
###### §10.68(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 $x$ and $\nu$, with the branches of $\theta_{\nu}\left(x\right)$ and $\phi_{\nu}\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to\infty$. … Equations (10.68.8)–(10.68.14) also hold with the symbols $\operatorname{ber}$, $\operatorname{bei}$, $M$, and $\theta$ replaced throughout by $\operatorname{ker}$, $\operatorname{kei}$, $N$, and $\phi$, respectively. … 10)), and $\lim_{x\to\infty}(\phi_{1}\left(x\right)+(x/\sqrt{2}))=-\tfrac{5}{8}\pi$ (Eqs. …
##### 4: 19.4 Derivatives and Differential Equations
19.4.3 $\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}k}^{2}}=-\frac{1}{k}\frac{% \mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{{k^{\prime}}^{2}K\left(k\right)-% E\left(k\right)}{k^{2}{k^{\prime}}^{2}},$
19.4.4 $\frac{\partial\Pi\left(\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{\prime}}^% {2}(k^{2}-\alpha^{2})}(E\left(k\right)-{k^{\prime}}^{2}\Pi\left(\alpha^{2},k% \right)).$
19.4.5 $\frac{\partial F\left(\phi,k\right)}{\partial k}={\frac{E\left(\phi,k\right)-{% k^{\prime}}^{2}F\left(\phi,k\right)}{k{k^{\prime}}^{2}}-\frac{k\sin\phi\cos% \phi}{{k^{\prime}}^{2}\sqrt{1-k^{2}{\sin}^{2}\phi}}},$
19.4.6 $\frac{\partial E\left(\phi,k\right)}{\partial k}=\frac{E\left(\phi,k\right)-F% \left(\phi,k\right)}{k},$
19.4.7 $\frac{\partial\Pi\left(\phi,\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{% \prime}}^{2}(k^{2}-\alpha^{2})}\left({E\left(\phi,k\right)-{k^{\prime}}^{2}\Pi% \left(\phi,\alpha^{2},k\right)}-\frac{k^{2}\sin\phi\cos\phi}{\sqrt{1-k^{2}{% \sin}^{2}\phi}}\right).$
##### 5: 7.23 Tables
• Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$, $x\in[0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\,\mathrm{d}t$, $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\,\mathrm{d}t$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

• ##### 6: 3.11 Approximation Techniques
The functions $\phi_{k}(x)$ are orthogonal on the set $x_{0},x_{1},\dots,x_{n-1}$, $x_{j}=2\pi j/n$, with respect to the weight function $w(x)=1$, in the sense that …
##### 7: 8.1 Special Notation
 $x$ real variable. … $\Gamma'\left(z\right)/\Gamma\left(z\right)$.
Unless otherwise indicated, primes denote derivatives with respect to the argument. … Alternative notations include: Prym’s functions $P_{z}(a)=\gamma\left(a,z\right)$, $Q_{z}(a)=\Gamma\left(a,z\right)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma\left(a+1,z\right)$, $[a,z]!=\Gamma\left(a+1,z\right)$, Dingle (1973); $B(a,b,x)=\mathrm{B}_{x}\left(a,b\right)$, $I(a,b,x)=I_{x}\left(a,b\right)$, Magnus et al. (1966); $\operatorname{Si}\left(a,x\right)\to\operatorname{Si}\left(1-a,x\right)$, $\operatorname{Ci}\left(a,x\right)\to\operatorname{Ci}\left(1-a,x\right)$, Luke (1975).
##### 8: 15.1 Special Notation
 $x$ real variable. … $\ifrac{\Gamma'\left(z\right)}{\Gamma\left(z\right)}$.
Unless indicated otherwise primes denote derivatives with respect to the variable. …
##### 9: 13.8 Asymptotic Approximations for Large Parameters
For asymptotic approximations to $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$ as $a\to-\infty$ that hold uniformly with respect to $x\in(0,\infty)$ and bounded positive values of $(b-1)/\left|a\right|$, combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii). …
##### 10: 7.7 Integral Representations
7.7.7 $\int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\,\mathrm{d}t=\frac{\sqrt{\pi}}{% 4a}\left(e^{2ab}\operatorname{erfc}\left(ax+(b/x)\right)+e^{-2ab}\operatorname% {erfc}\left(ax-(b/x)\right)\right),$ $x>0$, $|\operatorname{ph}a|<\tfrac{1}{4}\pi$.
7.7.8 $\int_{0}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\,\mathrm{d}t=\frac{\sqrt{\pi}}{% 2a}e^{-2ab},$ $|\operatorname{ph}a|<\tfrac{1}{4}\pi$, $|\operatorname{ph}b|<\tfrac{1}{4}\pi$.
7.7.12 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=e^{-\pi iz^{2}/2}\int_{z}^{% \infty}e^{\pi it^{2}/2}\,\mathrm{d}t.$