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1: 19.2 Definitions

… ►Assume $1-{\sin }^2\varphi \in ℂ\setminus (-\mathrm{\infty },0]$ and $1-k^2{\sin }^2\varphi \in ℂ\setminus (-\mathrm{\infty },0]$, except that one of them may be 0, and $1-{\alpha }^2{\sin }^2\varphi \in ℂ\setminus \{0\}$. … ►The principal branch of $K\left(k\right)$ and $E\left(k\right)$ is $|\mathrm{ph}\left(1-k^2\right)|\le \pi$, that is, the branch-cuts are $(-\mathrm{\infty },-1]\cup [1,+\mathrm{\infty })$. … ►Let $k^{\prime }=\sqrt{1-k^2}$. … ►If , then $k_c$ is pure imaginary. … ►

§19.2(iv) A Related Function: $R_C(x,y)$

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2: Software Index

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	Open Source													With Book						Commercial
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8.28(vii) $E_p\left(z\right)$, $z,p\in ℂ$	✓											✓	✓								✓	✓	✓
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14.34(iv) ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$					✓									✓			✓
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15.20(ii) ${}_2{}^{}F_1^{}(a,b;c;x)$, $x,a,b,c\in ℝ$	✓		✓		✓	✓						a	✓	✓			✓		✓		✓	✓	a
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25.21(vii) Fermi–Dirac, Bose–Einstein					✓									✓			✓					✓
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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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3: 18.16 Zeros

… ►Then as $n\to \mathrm{\infty }$, with $\alpha$ ($>-\frac{1}{2}$) and $\beta$ ($\ge -1-\alpha$) fixed, … ►For $m=1,2,\mathrm{\dots },n$, and with $j_{\alpha ,m}$ as in §18.16(ii), … ►All zeros of $H_n\left(x\right)$ lie in the open interval $(-\sqrt{2n+1},\sqrt{2n+1})$. In view of the reflection formula, given in Table 18.6.1, we may consider just the positive zeros $x_{n,m}$, $m=1,2,\mathrm{\dots },\lfloor \frac{1}{2}n\rfloor$. … ►

§18.16(vii) Discriminants

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4: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►When any one of $j_1,j_2,j_3$ is equal to $0,\frac{1}{2}$, or $1$, the $\mathit{3}j$ symbol has a simple algebraic form. …For these and other results, and also cases in which any one of $j_1,j_2,j_3$ is $\frac{3}{2}$ or $2$, see Edmonds (1974, pp. 125–127). … ►Even permutations of columns of a $\mathit{3}j$ symbol leave it unchanged; odd permutations of columns produce a phase factor ${(-1)}^{j_1+j_2+j_3}$, for example, … ►

§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

►For the polynomials $P_l$ see §18.3, and for the function $Y_{l,m}$ see §14.30. …

5: 32.7 Bäcklund Transformations

… ►Let $W_0=W(z;{\alpha }_0,{\beta }_0,{\gamma }_0,-\frac{1}{2})$ and $W_1=W(z;{\alpha }_1,{\beta }_1,{\gamma }_1,-\frac{1}{2})$ be solutions of ${\text{P}}_{\text{V}}$, where …and ${\epsilon }_j=\pm 1$, $j=1,2,3$, independently. … ►

§32.7(vii) Sixth Painlevé Equation

… ►The quadratic transformation …transforms ${\text{P}}_{\text{VI}}$ with $\alpha =-\beta$ and $\gamma =\frac{1}{2}-\delta$ to ${\text{P}}_{\text{VI}}$ with $({\alpha }_1,{\beta }_1,{\gamma }_1,{\delta }_1)=(4\alpha ,-4\gamma ,0,\frac{1}{2})$. …

6: 10.74 Methods of Computation

… ►It should be noted, however, that there is a difficulty in evaluating the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, from the explicit expressions (10.20.10)–(10.20.13) when $z$ is close to $1$ owing to severe cancellation. … ►For $z\in ℂ$ the function $H_{\nu }^{(1)}\left(z\right)$, for example, can always be computed in a stable manner in the sector $0\le \mathrm{ph}z\le \pi$ by integrating along rays towards the origin. … ►For evaluation of the Hankel functions $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ for complex values of $\nu$ and $z$ based on the integral representations (10.9.18) see Remenets (1973). ►For applications of generalized Gauss–Laguerre quadrature (§3.5(v)) to the evaluation of the modified Bessel functions $K_{\nu }\left(z\right)$ for and see Gautschi (2002a). … ►

§10.74(vii) Integrals

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7: 18.30 Associated OP’s

… ►The inequality $A_n{A}_{n+1}{C}_{n+1}>0$, for $n\ge 0$ is replaced by … ►

§18.30(vii) Corecursive and Associated Monic Orthogonal Polynomials

… ►It is easily seen that ${\widehat{p}}_2^{(0)}(x)={\widehat{p}}_1(x;1)=x-{\alpha }_1$, and then …The usage of §18.2(x), where the monic associated polynomials, there denoted $p_n^{(1)}(x)$, instead of ${\widehat{p}}_n(x;1)$, are referred to as the first associated such polynomials in §18.2(x), is now evident. The ratio ${\widehat{p}}_{n-1}(x;1)/{\widehat{p}}_n(x)$ is then the $F_n(x)$ of (18.2.35), leading to Markov’s theorem as stated in (18.30.25). …

8: 8.21 Generalized Sine and Cosine Integrals

… ►Furthermore, $\mathrm{si}(a,z)$ and $\mathrm{ci}(a,z)$ are entire functions of $a$, and $\mathrm{Si}(a,z)$ and $\mathrm{Ci}(a,z)$ are meromorphic functions of $a$ with simple poles at $a=-1,-3,-5,\mathrm{\dots }$ and $a=0,-2,-4,\mathrm{\dots }$, respectively. … ►When $\mathrm{ph}z=0$ (and when $a\ne -1,-3,-5,\mathrm{\dots }$, in the case of $\mathrm{Si}(a,z)$, or $a\ne 0,-2,-4,\mathrm{\dots }$, in the case of $\mathrm{Ci}(a,z)$) the principal values of $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). … ►

§8.21(vii) Auxiliary Functions

… ►When and , … ►When , …

9: 8.17 Incomplete Beta Functions

… ►With $a>0$, $b>0$, and , …where and the branches of $s^{-a}$ and ${(1-s)}^{-b}$ are continuous on the path and assume their principal values when $s=c$. … ►The $4m$ and $4m+1$ convergents are less than $I_x(a,b)$, and the $4m+2$ and $4m+3$ convergents are greater than $I_x(a,b)$. … ►For $x>(a+1)/(a+b+2)$ or , more rapid convergence is obtained by computing $I_{1-x}(b,a)$ and using (8.17.4). … ►

§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

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10: 1.10 Functions of a Complex Variable

… ►Let $f_1(z)$ be analytic in a domain $D_1$. … ►Then … ►(Thus if $z_0$ is in the interval $(-1,1)$, then the logarithms are real.) … ►

§1.10(vii) Inverse Functions

… ►The product ${\prod }_{n=1}^{\mathrm{\infty }}(1+a_n)$, with $a_n\ne -1$ for all $n$, converges iff ${\sum }_{n=1}^{\mathrm{\infty }}\ln \left(1+a_n\right)$ converges; and it converges absolutely iff ${\sum }_{n=1}^{\mathrm{\infty }}\left|a_n\right|$ converges. …