# complementary

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##### 2: 29.10 Lamé Functions with Imaginary Periods
29.10.2 $z^{\prime}=\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),$
$\mathit{Ec}^{2m}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{\prime}% }^{2}\right),$
$\mathit{Ec}^{2m+1}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{% \prime}}^{2}\right),$
$\mathit{Es}^{2m+1}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{% \prime}}^{2}\right),$
The first and the fourth functions have period $2\mathrm{i}{K^{\prime}}$; the second and the third have period $4\mathrm{i}{K^{\prime}}$. …
##### 3: 7.22 Methods of Computation
###### §7.22(iii) Repeated Integrals of the Complementary Error Function
The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(x\right)$. … The computation of these functions can be based on algorithms for the complementary error function with complex argument; compare (7.19.3). …
##### 4: 29.17 Other Solutions
29.17.1 $F(z)=E(z)\int_{\mathrm{i}{K^{\prime}}}^{z}\frac{\,\mathrm{d}u}{(E(u))^{2}}.$
Lamé–Wangerin functions are solutions of (29.2.1) with the property that $(\operatorname{sn}\left(z,k\right))^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}{K^{\prime}}$ to $2K+\mathrm{i}{K^{\prime}}$. …
##### 5: 7.1 Special Notation
The main functions treated in this chapter are the error function $\operatorname{erf}z$; the complementary error functions $\operatorname{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $\mathcal{F}\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)$; the Voigt functions $\mathsf{U}\left(x,t\right)$ and $\mathsf{V}\left(x,t\right)$. Alternative notations are $Q(z)=\tfrac{1}{2}\operatorname{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\Phi(z)=\tfrac{1}{2}\operatorname{erfc}\left(-z/\sqrt{2}\right)$, $\operatorname{Erf}z=\tfrac{1}{2}\sqrt{\pi}\operatorname{erf}z$, $\operatorname{Erfi}z=e^{z^{2}}F\left(z\right)$, $C_{1}(z)=C\left(\sqrt{2/\pi}z\right)$, $S_{1}(z)=S\left(\sqrt{2/\pi}z\right)$, $C_{2}(z)=C\left(\sqrt{2z/\pi}\right)$, $S_{2}(z)=S\left(\sqrt{2z/\pi}\right)$. …
##### 6: 22.4 Periods, Poles, and Zeros
The other poles are at congruent points, which is the set of points obtained by making translations by $2mK+2niK^{\prime}$, where $m,n\in\mathbb{Z}$. … Figure 22.4.1 illustrates the locations in the $z$-plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices $0$, $2K$, $2K+2iK^{\prime}$, $2iK^{\prime}$. … Figure 22.4.2 depicts the fundamental unit cell in the $z$-plane, with vertices $\mbox{s}=0$, $\mbox{c}=K$, $\mbox{d}=K+iK^{\prime}$, $\mbox{n}=iK^{\prime}$. The set of points $z=mK+niK^{\prime}$, $m,n\in\mathbb{Z}$, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by $mK+niK^{\prime}$, where again $m,n\in\mathbb{Z}$. … This half-period will be plus or minus a member of the triple ${K,iK^{\prime},K+iK^{\prime}}$; the other two members of this triple are quarter periods of $\operatorname{pq}\left(z,k\right)$. …
##### 9: 7.9 Continued Fractions
7.9.1 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{z}{z^{2}+\cfrac{\frac{1}{2}}{1+% \cfrac{1}{z^{2}+\cfrac{\frac{3}{2}}{1+\cfrac{2}{z^{2}+\cdots}}}}},$ $\Re z>0$,
7.9.2 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{2z}{2z^{2}+1-\cfrac{1\cdot 2}{2% z^{2}+5-\cfrac{3\cdot 4}{2z^{2}+9-\cdots}}},$ $\Re z>0$,
##### 10: 7.21 Physical Applications
###### §7.21 Physical Applications
Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)$. …