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21: Mathematical Introduction
As a consequence, in addition to providing more information about the special functions that were covered in AMS 55, the NIST Handbook includes several special functions that have appeared in the interim in applied mathematics, the physical sciences, and engineering, as well as in other areas. … All chapters went through several drafts (nine in some cases) before the authors, validators, and editors were fully satisfied. … Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. … Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. … This means that the variable $x$ ranges from 0 to 1 in intervals of 0. …
22: 15.7 Continued Fractions
15.7.1 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},$
15.7.3 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},$
15.7.5 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a+1,b+1;c+1;z\right)}={x% _{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},$
23: Mourad E. H. Ismail
His well-known book Classical and Quantum Orthogonal Polynomials in One Variable was published by Cambridge University Press in 2005 and reprinted with corrections in paperback in Ismail (2009). … Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. …
24: 13.15 Recurrence Relations and Derivatives
13.15.1 $(\kappa-\mu-\tfrac{1}{2})M_{\kappa-1,\mu}\left(z\right)+(z-2\kappa)M_{\kappa,% \mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})M_{\kappa+1,\mu}\left(z\right)=0,$
13.15.2 $2\mu(1+2\mu)\sqrt{z}M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-(z+2% \mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}M_{% \kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,$
13.15.3 $(\kappa-\mu-\tfrac{1}{2})M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)+% (1+2\mu)\sqrt{z}M_{\kappa,\mu}\left(z\right)-(\kappa+\mu+\tfrac{1}{2})M_{% \kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,$
13.15.4 $2\mu M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-2\mu M_{\kappa+\frac% {1}{2},\mu-\frac{1}{2}}\left(z\right)-\sqrt{z}M_{\kappa,\mu}\left(z\right)=0,$
Other versions of several of the identities in this subsection can be constructed by use of (13.3.29).
25: 19.7 Connection Formulas
This dichotomy of signs (missing in several references) is due to Fettis (1970). …
$\kappa=\frac{k}{\sqrt{1+k^{2}}},$
$\kappa^{\prime}=\frac{1}{\sqrt{1+k^{2}}},$
With $\sinh\phi=\tan\psi$, … The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\Pi\left(\phi,\alpha^{2},k\right)$ when $\alpha^{2}>{\csc}^{2}\phi$ (see (19.6.5) for the complete case). …
26: 15.19 Methods of Computation
Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. … In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …
27: 3.7 Ordinary Differential Equations
Write $\tau_{j}=z_{j+1}-z_{j}$, $j=0,1,\dots,P$, expand $w(z)$ and $w^{\prime}(z)$ in Taylor series (§1.10(i)) centered at $z=z_{j}$, and apply (3.7.2). …
3.7.10 $\mathbf{A}_{P}=\begin{bmatrix}-\mathbf{A}(\tau_{0},z_{0})&\mathbf{I}&% \boldsymbol{{0}}&\cdots&\boldsymbol{{0}}&\boldsymbol{{0}}\\ \boldsymbol{{0}}&-\mathbf{A}(\tau_{1},z_{1})&\mathbf{I}&\cdots&\boldsymbol{{0}% }&\boldsymbol{{0}}\\ \vdots&\vdots&\ddots&\ddots&\vdots&\vdots\\ \boldsymbol{{0}}&\boldsymbol{{0}}&\cdots&-\mathbf{A}(\tau_{P-2},z_{P-2})&% \mathbf{I}&\boldsymbol{{0}}\\ \boldsymbol{{0}}&\boldsymbol{{0}}&\cdots&\boldsymbol{{0}}&-\mathbf{A}(\tau_{P-% 1},z_{P-1})&\mathbf{I}\end{bmatrix}$
3.7.12 $\mathbf{b}=\left[b_{1}(\tau_{0},z_{0}),b_{2}(\tau_{0},z_{0}),b_{1}(\tau_{1},z_% {1}),b_{2}(\tau_{1},z_{1}),\ldots,b_{1}(\tau_{P-1},z_{P-1}),b_{2}(\tau_{P-1},z% _{P-1})\right]^{\rm T}.$
28: 7.20 Mathematical Applications
The spiral has several special properties (see Temme (1996b, p. 184)). …
7.20.1 $\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{x}e^{-(t-m)^{2}/(2\sigma^{2})}% \mathrm{d}t=\frac{1}{2}\operatorname{erfc}\left(\frac{m-x}{\sigma\sqrt{2}}% \right)=Q\left(\frac{m-x}{\sigma}\right)=P\left(\frac{x-m}{\sigma}\right).$
29: 6.18 Methods of Computation
However, this problem is less severe for the series of spherical Bessel functions because of their more rapid rate of convergence, and also (except in the case of (6.10.6)) absence of cancellation when $z=x$ ($>0$). …
30: 2.7 Differential Equations
The first of these references includes extensions to complex variables and reversions for zeros. … From the numerical standpoint, however, the pair $w_{3}(z)$ and $w_{4}(z)$ has the drawback that severe numerical cancellation can occur with certain combinations of $C$ and $D$, for example if $C$ and $D$ are equal, or nearly equal, and $z$, or $\Re z$, is large and negative. …