# §19.27 Asymptotic Approximations and Expansions

## §19.27(i) Notation

Throughout this section

 19.27.1 $\displaystyle a$ $\displaystyle=\tfrac{1}{2}(x+y),$ $\displaystyle b$ $\displaystyle=\tfrac{1}{2}(y+z),$ $\displaystyle c$ $\displaystyle=\tfrac{1}{3}(x+y+z),$ $\displaystyle f$ $\displaystyle=(xyz)^{1/3},$ $\displaystyle g$ $\displaystyle=(xy)^{1/2},$ $\displaystyle h$ $\displaystyle=(yz)^{1/2}.$ Symbols: $a$, $b$, $c$, $f$, $g$ and $h$ Permalink: http://dlmf.nist.gov/19.27.E1 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for 19.27(i)

## §19.27(ii) $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$

Assume $x$, $y$, and $z$ are real and nonnegative and at most one of them is 0. Then

 19.27.2 $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=\frac{1}{2\sqrt{z}}\left(\mathop% {\ln\/}\nolimits\frac{8z}{a+g}\right)\left(1+\mathop{O\/}\nolimits\!\left(% \frac{a}{z}\right)\right),$ $a/z\to 0$.
 19.27.3 $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=\mathop{R_{F}\/}\nolimits\!\left% (0,y,z\right)-\frac{1}{\sqrt{h}}\left(\sqrt{\frac{x}{h}}+\mathop{O\/}\nolimits% \!\left(\frac{x}{h}\right)\right),$ $x/h\to 0$.

## §19.27(iii) $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$

Assume $x$, $y$, and $z$ are real and nonnegative and at most one of them is 0. Then

 19.27.4 $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=\frac{\sqrt{z}}{2}\left(1+% \mathop{O\/}\nolimits\!\left(\frac{a}{z}\mathop{\ln\/}\nolimits\frac{z}{a}% \right)\right),$ $a/z\to 0$.
 19.27.5 $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=\mathop{R_{G}\/}\nolimits\!\left% (0,y,z\right)+\sqrt{x}\mathop{O\/}\nolimits\!\left(\sqrt{x/h}\right),$ $x/h\to 0$.
 19.27.6 $\mathop{R_{G}\/}\nolimits\!\left(0,y,z\right)={\frac{\sqrt{z}}{2}+\frac{y}{8% \sqrt{z}}\left(\mathop{\ln\/}\nolimits\left(\frac{16z}{y}\right)-1\right)\*% \left(1+\mathop{O\/}\nolimits\!\left(\frac{y}{z}\right)\right)},$ $y/z\to 0$.

## §19.27(iv) $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$

Assume $x$ and $y$ are real and nonnegative, at most one of them is 0, and $z>0$. Then

 19.27.7 $\displaystyle\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ $\displaystyle=\frac{3}{2z^{3/2}}\left(\mathop{\ln\/}\nolimits\left(\frac{8z}{a% +g}\right)-2\right)\left(1+\mathop{O\/}\nolimits\!\left(\frac{a}{z}\right)% \right),$ $a/z\to 0$. 19.27.8 $\displaystyle\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ $\displaystyle=\frac{3}{\sqrt{xyz}}-\frac{6}{xy}\mathop{R_{G}\/}\nolimits\!% \left(x,y,0\right)\left(1+\mathop{O\/}\nolimits\!\left(\frac{z}{g}\right)% \right),$ $z/g\to 0$. 19.27.9 $\displaystyle\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ $\displaystyle=\frac{3}{\sqrt{xz}(\sqrt{y}+\sqrt{z})}\left(1+\mathop{O\/}% \nolimits\!\left(\frac{b}{x}\mathop{\ln\/}\nolimits\frac{x}{b}\right)\right),$ $b/x\to 0$. 19.27.10 $\displaystyle\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ $\displaystyle=\mathop{R_{D}\/}\nolimits\!\left(0,y,z\right)-\frac{3\sqrt{x}}{% hz}\left(1+\mathop{O\/}\nolimits\!\left(\sqrt{\frac{x}{h}}\right)\right),$ $x/h\to 0$.

## §19.27(v) $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$

Assume $x$, $y$, and $z$ are real and nonnegative, at most one of them is 0, and $p>0$. Then

 19.27.11 $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ $\displaystyle={\frac{3}{p}\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)-\frac{% 3\pi}{2p^{3/2}}\left(1+\mathop{O\/}\nolimits\!\left(\sqrt{\frac{c}{p}}\right)% \right)},$ $c/p\to 0$. 19.27.12 $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ $\displaystyle={\frac{3}{2\sqrt{xyz}}\left(\mathop{\ln\/}\nolimits\left(\frac{4% f}{p}\right)-2\right)\left(1+\mathop{O\/}\nolimits\!\left(\frac{p}{f}\right)% \right)},$ $p/f\to 0$. 19.27.13 $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ $\displaystyle=\frac{3}{2\sqrt{z}p}\left(\mathop{\ln\/}\nolimits\left(\frac{8z}% {a+g}\right)-2\!\mathop{R_{C}\/}\nolimits\!\left(1,\frac{p}{z}\right)+\mathop{% O\/}\nolimits\!\left(\left(\frac{a}{z}+\frac{a}{p}\right)\mathop{\ln\/}% \nolimits\frac{p}{a}\right)\right),$ $\max(x,y)/\min(z,p)\to 0$. 19.27.14 $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ $\displaystyle=\frac{3}{\sqrt{yz}}\mathop{R_{C}\/}\nolimits\!\left(x,p\right)-% \frac{6}{yz}\mathop{R_{G}\/}\nolimits\!\left(0,y,z\right)+\mathop{O\/}% \nolimits\!\left(\frac{\sqrt{x+2p}}{yz}\right),$ $\max(x,p)/\min(y,z)\to 0$. 19.27.15 $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ $\displaystyle=\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)-\frac{3\sqrt{x}}% {hp}\left(1+\mathop{O\/}\nolimits\!\left(\left(\frac{b}{h}+\frac{h}{p}\right)% \sqrt{\frac{x}{h}}\right)\right),$ $x/\min(y,z,p)\to 0$. 19.27.16 $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ $\displaystyle=(3/\sqrt{x})\mathop{R_{C}\/}\nolimits\!\left((h+p)^{2},2(b+h)p% \right)+\mathop{O\/}\nolimits\!\left(\frac{1}{x^{3/2}}\mathop{\ln\/}\nolimits% \frac{x}{b+h}\right),$ $\max(y,z,p)/x\to 0$.

## §19.27(vi) Asymptotic Expansions

The approximations in §§19.27(i)19.27(v) are furnished with upper and lower bounds by Carlson and Gustafson (1994), sometimes with two or three approximations of differing accuracies. Although they are obtained (with some exceptions) by approximating uniformly the integrand of each elliptic integral, some occur also as the leading terms of known asymptotic series with error bounds (Wong (1983, §4), Carlson and Gustafson (1985), López (2000, 2001)). These series converge but not fast enough, given the complicated nature of their terms, to be very useful in practice.

A similar (but more general) situation prevails for $\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ when some of the variables $z_{1},\dots,z_{n}$ are smaller in magnitude than the rest; see Carlson (1985, (4.16)–(4.19) and (2.26)–(2.29)).