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21: 23.6 Relations to Other Functions
22: 8.27 Approximations
DiDonato (1978) gives a simple approximation for the function $F(p,x)={x}^{p}{\mathrm{e}}^{{x}^{2}/2}{\int}_{x}^{\mathrm{\infty}}{\mathrm{e}}^{{t}^{2}/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3({x}^{2}p)+\sqrt{{({x}^{2}p)}^{2}+8({x}^{2}+p)}$ and is shown to produce an absolute error $O\left({x}^{7}\right)$ as $x\to \mathrm{\infty}$.
23: 9.5 Integral Representations
24: 27.2 Functions
25: 19.25 Relations to Other Functions
26: Errata

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In Paragraph Steed’s Algorithm in §3.10(iii), a sentence was added to inform the reader of alternatives to Steed’s algorithm, namely the Lentz algorithm (see e.g., Lentz (1976)) and the modified Lentz algorithm (see e.g., Thompson and Barnett (1986)).

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In Subsection 19.11(i), a sentence and unnumbered equation
$${R}_{C}(\gamma \delta ,\gamma )=\frac{1}{\sqrt{\delta}}\mathrm{arctan}\left(\frac{\sqrt{\delta}\mathrm{sin}\theta \mathrm{sin}\varphi \mathrm{sin}\psi}{{\alpha}^{2}1{\alpha}^{2}\mathrm{cos}\theta \mathrm{cos}\varphi \mathrm{cos}\psi}\right),$$were added which indicate that care must be taken with the multivalued functions in (19.11.5). See (Cayley, 1961, pp. 103106). This was suggested by Albert Groenenboom.

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In Subsection 33.14(iv), there were several modifications. Just below (33.14.9), the constraint described in the text “$$ when $$,” was removed. In Equation (33.14.13), the constraint ${\u03f5}_{1},{\u03f5}_{2}>0$ was added. In the line immediately below (33.14.13), it was clarified that $s(\u03f5,\mathrm{\ell};r)$ is $\mathrm{exp}\left(r/n\right)$ times a polynomial in $r/n$, instead of simply a polynomial in $r$. In Equation (33.14.14), a second equality was added which relates ${\varphi}_{n,\mathrm{\ell}}(r)$ to Laguerre polynomials. A sentence was added immediately below (33.14.15) indicating that the functions ${\varphi}_{n,\mathrm{\ell}}$, $n=\mathrm{\ell},\mathrm{\ell}+1,\mathrm{\dots}$, do not form a complete orthonormal system.

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The factor on the righthand side of Equation (10.9.26) containing $\mathrm{cos}(\mu \nu )\theta $ has been been replaced with $\mathrm{cos}\left((\mu \nu )\theta \right)$ to clarify the meaning.

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In Paragraph Confluent Hypergeometric Functions in §10.16, several Whittaker confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

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In Equation (15.6.9), it was clarified that $\lambda \in \u2102$.

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Originally Equation (19.16.9) had the constraint $a,{a}^{\prime}>0$. This constraint was replaced with ${b}_{1}+\mathrm{\cdots}+{b}_{n}>a>0$, ${b}_{j}\in \mathbb{R}$. It therefore follows from Equation (19.16.10) that ${a}^{\prime}>0$. The last sentence of Subsection 19.16(ii) was elaborated to mention that generalizations may also be found in Carlson (1977b). These were suggested by Bastien Roucariès.

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In Section 19.25(vi), the Weierstrass lattice roots ${e}_{j},$ were labeled inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots ${e}_{j}$, and lattice invariants ${g}_{2}$, ${g}_{3}$, now link to their respective definitions (see §§23.2(i), 23.3(i)). This was reported by Felix Ospald.

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In Equation (19.25.37), the Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

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In Equation (27.12.5), the term originally written as $\sqrt{\mathrm{ln}x}$ was rewritten as ${(\mathrm{ln}x)}^{1/2}$ to be consistent with other equations in the same subsection.
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Originally named as a complementary error function, (7.2.3) has been renamed as the Faddeeva (or Faddeyeva) function $w\left(z\right)$.
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Bounds have been sharpened in §9.7(iii). The second paragraph now reads, “The $n$th error term is bounded in magnitude by the first neglected term multiplied by $\chi (n+\sigma )+1$ where $\sigma =\frac{1}{6}$ for (9.7.7) and $\sigma =0$ for (9.7.8), provided that $n\ge 0$ in the first case and $n\ge 1$ in the second case.” Previously it read, “In (9.7.7) and (9.7.8) the $n$th error term is bounded in magnitude by the first neglected term multiplied by $2\chi (n)\mathrm{exp}\left(\sigma \pi /(72\zeta )\right)$ where $\sigma =5$ for (9.7.7) and $\sigma =7$ for (9.7.8), provided that $n\ge 1$ in both cases.” In Equation (9.7.16)
9.7.16$$\mathrm{Bi}\left(x\right)\le \frac{{\mathrm{e}}^{\xi}}{\sqrt{\pi}{x}^{1/4}}\left(1+\left(\chi (\frac{7}{6})+1\right)\frac{5}{72\xi}\right),$$$${\mathrm{Bi}}^{\prime}\left(x\right)\le \frac{{x}^{1/4}{\mathrm{e}}^{\xi}}{\sqrt{\pi}}\left(1+\left(\frac{\pi}{2}+1\right)\frac{7}{72\xi}\right),$$the bounds on the righthand sides have been sharpened. The factors $\left(\chi (\frac{7}{6})+1\right)\frac{5}{72\xi}$, $\left(\frac{\pi}{2}+1\right)\frac{7}{72\xi}$, were originally given by $\frac{5\pi}{72\xi}\mathrm{exp}\left(\frac{5\pi}{72\xi}\right)$, $\frac{7\pi}{72\xi}\mathrm{exp}\left(\frac{7\pi}{72\xi}\right)$, respectively.

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Bounds have been sharpened in §9.7(iv). The first paragraph now reads, “The $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by
9.7.17$$provided that $n\ge 0$, $\sigma =\frac{1}{6}$ for (9.7.5) and $n\ge 1$, $\sigma =0$ for (9.7.6).” Previously it read, “When $n\ge 1$ the $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by
9.7.17$$  •
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The second and the fourth lines of (19.7.2) containing ${k}^{\prime}/ik$ have both been replaced with $i{k}^{\prime}/k$ to clarify the meaning.

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Originally Equation (25.2.4) had the constraint $\mathrm{\Re}s>0$. This constraint was removed because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$, and therefore $\zeta \left(s\right){(s1)}^{1}$ is an entire function. This was suggested by John Harper.

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The title of §32.16 was changed from Physical to Physical Applications.
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To be consistent with the notation used in (8.12.16), Equation (8.12.5) was changed to read
8.12.5 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\mathrm{sin}(\pi a)}Q(a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\pm \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)\mathrm{i}T(a,\eta )$$ 
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Following a suggestion from James McTavish on 20170406, the recurrence relation ${u}_{k}=\frac{(6k5)(6k3)(6k1)}{(2k1)216k}{u}_{k1}$ was added to Equation (9.7.2).

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In §15.2(ii), the unnumbered equation
$$\underset{c\to n}{lim}\frac{F(a,b;c;z)}{\mathrm{\Gamma}\left(c\right)}=\mathbf{F}(a,b;n;z)=\frac{{\left(a\right)}_{n+1}{\left(b\right)}_{n+1}}{(n+1)!}{z}^{n+1}F(a+n+1,b+n+1;n+2;z),$$ $n=0,1,2,\mathrm{\dots}$was added in the second paragraph. An equation number will be assigned in an expanded numbering scheme that is under current development. Additionally, the discussion following (15.2.6) was expanded.
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A bibliographic citation was added in §11.13(i).
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The Wronskian (14.2.7) was generalized to include both associated Legendre and Ferrers functions.

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A crossreference has been added in §15.9(iv).
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An addition was made to the Software Index to reflect a multiple precision (MP) package written in C++ which uses a variety of different MP interfaces. See Kormanyos (2011).