case m=2

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21: 32.8 Rational Solutions

… ►In the general case assume $\delta \ne 0$, so that as in §32.2(ii) we may set $\delta =-\frac{1}{2}$. … ►

$\alpha =\frac{1}{2}{n}^2$ and $\beta =-\frac{1}{2}{(m+\epsilon \gamma )}^2$, where $n>0$, $m+n$ is odd, and $\beta \ne 0$ when .

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$\alpha =\frac{1}{2}{a}^2$, $\beta =-\frac{1}{2}{(a+n)}^2$, and $\gamma =m$, with $m+n$ even.

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$\alpha =\frac{1}{2}{(b+n)}^2$, $\beta =-\frac{1}{2}{b}^2$, and $\gamma =m$, with $m+n$ even.

… ►For the case $\delta =0$ see Airault (1979) and Lukaševič (1968). …

22: 36.6 Scaling Relations

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Indices for $k$-Scaling of Coordinates $x_m$

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$\text{cuspoids: }{\gamma }_{mK}=1-{\displaystyle \frac{m}{K+2}},$

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$\text{cuspoids: }{\gamma }_K={\displaystyle \sum _{m=1}^K}{\gamma }_{mK}={\displaystyle \frac{K(K+3)}{2(K+2)}},$

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$\text{umbilics: }{\gamma }^{(\mathrm{U})}={\displaystyle \sum _{m=1}^3}{\gamma }_m^{(\mathrm{U})}=\frac{5}{3}.$

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Table 36.6.1: Special cases of scaling exponents for cuspoids.

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singularity	$K$	${\beta }_K$	${\gamma }_{1K}$	${\gamma }_{2K}$	${\gamma }_{3K}$	${\gamma }_K$
…

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…

23: 14.16 Zeros

… ►For all cases concerning ${𝖯}_{\nu }^{\mu }\left(x\right)$ and $P_{\nu }^{\mu }\left(x\right)$ we assume that $\nu \ge -\frac{1}{2}$ without loss of generality (see (14.9.5) and (14.9.11)). … ►

$\mu >0$, $n\ge m$, and ${\delta }_{\nu }>{\delta }_{\mu }$.

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$\nu =0,1,2,3,\mathrm{\dots }$.

… ►In the special case $\mu =0$ and $\nu =n=0,1,2,3,\mathrm{\dots }$, ${𝖰}_n\left(x\right)$ has $n+1$ zeros in the interval . ►For uniform asymptotic approximations for the zeros of ${𝖯}_n^{-m}\left(x\right)$ in the interval when $n\to \mathrm{\infty }$ with $m$ $(\ge 0)$ fixed, see Olver (1997b, p. 469). …

24: 13.14 Definitions and Basic Properties

… ►except that $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$. … ►Although $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$, many formulas containing $M_{\kappa ,\mu }\left(z\right)$ continue to apply in their limiting form. … ►In cases when $\frac{1}{2}-\kappa \pm \mu =-n$, where $n$ is a nonnegative integer, …In all other cases … ►Except when $\mu -\kappa =-\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$ (polynomial cases), …

25: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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§34.3(i) Special Cases

►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). … ►For the polynomials $P_l$ see §18.3, and for the function $Y_{l,m}$ see §14.30. …Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to $\mathit{3}j$ symbols, for which see Edmonds (1974, Chapter 4). …

26: 19.5 Maclaurin and Related Expansions

… ►where ${}_2{}^{}F_1^{}$ is the Gauss hypergeometric function (§§15.1 and 15.2(i)). … ►Coefficients of terms up to ${\lambda }^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\le k^2\le 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). … ►where $k_0=k$ and … ►Series expansions of $F(\varphi ,k)$ and $E(\varphi ,k)$ are surveyed and improved in Van de Vel (1969), and the case of $F(\varphi ,k)$ is summarized in Gautschi (1975, §1.3.2). For series expansions of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when see Erdélyi et al. (1953b, §13.6(9)). …

27: 4.13 Lambert $W$-Function

… ►where ${\ln }_k(z)=\ln \left(z\right)+2\pi \mathrm{i}k$. …The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. … ►where ${\xi }_k=\ln \left(z\right)+2\pi \mathrm{i}k$. …In the case of $k=0$ and real $z$ the series converges for $z\ge \mathrm{e}$. … ►For integrals of $W\left(z\right)$ use the substitution $w=W\left(z\right)$, $z=w{\mathrm{e}}^w$ and $dz=(w+1){\mathrm{e}}^{w}dw$. …

28: 29.5 Special Cases and Limiting Forms

§29.5 Special Cases and Limiting Forms

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${\mathrm{𝐸𝑐}}_{\nu }^m(z,0)=\cos \left(m(\frac{1}{2}\pi -z)\right),$ $m\ge 1$,

… ►Let $\mu =\max (\nu -m,0)$. … ►If $k\to 0+$ and $\nu \to \mathrm{\infty }$ in such a way that $k^2\nu (\nu +1)=4\theta$ (a positive constant), then …where ${\mathrm{ce}}_m(z,\theta )$ and ${\mathrm{se}}_m(z,\theta )$ are Mathieu functions; see §28.2(vi).

29: 2.11 Remainder Terms; Stokes Phenomenon

… ►Taking $m=10$ in (2.11.2), the first three terms give us the approximation … … ►In the transition through $\theta =\pi$, $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case $\rho =100$. … ►with $m=0,1,2,\mathrm{\dots }$, and $C_1,C_2$ as in (2.7.17). … ►uniformly with respect to $\mathrm{ph}z$ in each case. …

30: 28.4 Fourier Series

… ►For $n=0,1,2,3,\mathrm{\dots }$, … ►

§28.4(iv) Case $q=0$

… ►For fixed $s=1,2,3,\mathrm{\dots }$ and fixed $m=1,2,3,\mathrm{\dots }$, … ►

§28.4(vii) Asymptotic Forms for Large $m$

►As $m\to \mathrm{\infty }$, with fixed $q$ ($\ne 0$) and fixed $n$, …