general values

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21: 14.20 Conical (or Mehler) Functions

… ►Lastly, for the range

1<x<\infty

P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

is a real-valued solution of (14.20.1); in terms of

Q^{\mu}_{-\frac{1}{2}\pm\mathrm{i}\tau}\left(x\right)

(which are complex-valued in general): …

22: 9.17 Methods of Computation

… ►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing

\operatorname{Ai}\left(z\right)

in the complex plane, once values of this function can be generated on the positive real axis. …

23: 28.35 Tables

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Blanch and Clemm (1962) includes values of ${\operatorname{Mc}^{(1)}_{n}}\left(x,\sqrt{q}\right)$ and ${\operatorname{Mc}^{(1)}_{n}}'\left(x,\sqrt{q}\right)$ for $n=0(1)15$ with $q=0(.05)1$ , $x=0(.02)1$ . Also ${\operatorname{Ms}^{(1)}_{n}}\left(x,\sqrt{q}\right)$ and ${\operatorname{Ms}^{(1)}_{n}}'\left(x,\sqrt{q}\right)$ for $n=1(1)15$ with $q=0(.05)1$ , $x=0(.02)1$ . Precision is generally 7D.

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Blanch and Clemm (1965) includes values of ${\operatorname{Mc}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Mc}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for $n=0(1)7$ , $x=0(.02)1$ ; $n=8(1)15$ , $x=0(.01)1$ . Also ${\operatorname{Ms}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Ms}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for $n=1(1)7$ , $x=0(.02)1$ ; $n=8(1)15$ , $x=0(.01)1$ . In all cases $q=0(.05)1$ . Precision is generally 7D. Approximate formulas and graphs are also included.

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National Bureau of Standards (1967) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$ , and $n=4(1)15$ with $q=0(2)100$ ; Fourier coefficients for $\operatorname{ce}_{n}\left(x,q\right)$ and $\operatorname{se}_{n}\left(x,q\right)$ for $n=0(1)15$ , $n=1(1)15$ , respectively, and various values of $q$ in the interval $[0,100]$ ; joining factors $g_{\mathit{e},n}(\sqrt{q})$ , $f_{\mathit{e},n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\mbox{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$ . Precision is generally 8D.

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24: Philip J. Davis

… ►The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling. …

25: 3.7 Ordinary Differential Equations

… ►General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). …

26: 24.16 Generalizations

… ►For extensions of

B^{(\ell)}_{n}\left(x\right)

to complex values of

x

n

, and

\ell

, and also for uniform asymptotic expansions for large

x

and large

n

, see Temme (1995b) and López and Temme (1999b, 2010b). …

27: 9.19 Approximations

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Corless et al. (1992) describe a method of approximation based on subdividing $\mathbb{C}$ into a triangular mesh, with values of $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ stored at the nodes. $\operatorname{Ai}\left(z\right)$ and $\operatorname{Ai}'\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ at the node. Similarly for $\operatorname{Bi}\left(z\right)$ , $\operatorname{Bi}'\left(z\right)$ .

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