DLMF: Untitled Document

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32.4.10

zu_{0}^{\prime}=\theta_{\infty}u_{0}-zv_{0}v_{1},

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Symbols:: $z$ : real and $\theta_{\infty}$ : constant
Referenced by:: §32.4(iv)
Permalink:: http://dlmf.nist.gov/32.4.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §32.4(iv), §32.4 and Ch.32

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32.4.12

zv_{0}^{\prime}=2v_{0}u_{1}v_{1}+v_{0}+(u_{0}(2v_{0}-z)/v_{1}),

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Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.4.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §32.4(iv), §32.4 and Ch.32

… ►If

w=-u_{0}/(v_{0}v_{1})

, then …where ►

32.4.16

\theta_{0}=\frac{4v_{0}}{z}\left(\theta_{\infty}\left(1-\frac{z}{4v_{0}}\right% )+\frac{z-2v_{0}}{2v_{0}v_{1}}u_{0}+u_{1}v_{1}\right).

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Symbols:: $z$ : real, $\theta_{\infty}$ : constant and $\theta_{0}$
Permalink:: http://dlmf.nist.gov/32.4.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §32.4(iv), §32.4 and Ch.32

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Abramowitz and Stegun (1964, Chapter 8) tabulates $\mathsf{P}_{n}\left(x\right)$ for $n=0(1)3,9,10$ , $x=0(.01)1$ , 5–8D; $\mathsf{P}_{n}'\left(x\right)$ for $n=1(1)4,9,10$ , $x=0(.01)1$ , 5–7D; $\mathsf{Q}_{n}\left(x\right)$ and $\mathsf{Q}_{n}'\left(x\right)$ for $n=0(1)3,9,10$ , $x=0(.01)1$ , 6–8D; $P_{n}\left(x\right)$ and $P_{n}'\left(x\right)$ for $n=0(1)5,9,10$ , $x=1(.2)10$ , 6S; $Q_{n}\left(x\right)$ and $Q_{n}'\left(x\right)$ for $n=0(1)3,9,10$ , $x=1(.2)10$ , 6S. (Here primes denote derivatives with respect to $x$ .)

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Zhang and Jin (1996, Chapter 4) tabulates $\mathsf{P}_{n}\left(x\right)$ for $n=2(1)5,10$ , $x=0(.1)1$ , 7D; $\mathsf{P}_{n}\left(\cos\theta\right)$ for $n=1(1)4,10$ , $\theta=0(5^{\circ})90^{\circ}$ , 8D; $\mathsf{Q}_{n}\left(x\right)$ for $n=0(1)2,10$ , $x=0(.1)0.9$ , 8S; $\mathsf{Q}_{n}\left(\cos\theta\right)$ for $n=0(1)3,10$ , $\theta=0(5^{\circ})90^{\circ}$ , 8D; $\mathsf{P}^{m}_{n}\left(x\right)$ for $m=1(1)4$ , $n-m=0(1)2$ , $n=10$ , $x=0,0.5$ , 8S; $\mathsf{Q}^{m}_{n}\left(x\right)$ for $m=1(1)4$ , $n=0(1)2,10$ , 8S; $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ for $m=0(1)3$ , $\nu=0(.25)5$ , $\theta=0(15^{\circ})90^{\circ}$ , 5D; $P_{n}\left(x\right)$ for $n=2(1)5,10$ , $x=1(1)10$ , 7S; $Q_{n}\left(x\right)$ for $n=0(1)2,10$ , $x=2(1)10$ , 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 $\nu$ -zeros of $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ and of its derivative for $m=0(1)4$ , $\theta=10^{\circ},30^{\circ},150^{\circ}$ .

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Belousov (1962) tabulates $\mathsf{P}^{m}_{n}\left(\cos\theta\right)$ (normalized) for $m=0(1)36$ , $n-m=0(1)56$ , $\theta=0(2.5^{\circ})90^{\circ}$ , 6D.

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Žurina and Karmazina (1964, 1965) tabulate the conical functions $\mathsf{P}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)50$ , $x=-0.9(.1)0.9$ , 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)50$ , $x=1.1(.1)2(.2)5(.5)10(10)60$ , 7D. Auxiliary tables are included to facilitate computation for larger values of $\tau$ when $-1<x<1$ .

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Žurina and Karmazina (1963) tabulates the conical functions $\mathsf{P}^{1}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)25$ , $x=-0.9(.1)0.9$ , 7S; $P^{1}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)25$ , $x=1.1(.1)2(.2)5(.5)10(10)60$ , 7S. Auxiliary tables are included to assist computation for larger values of $\tau$ when $-1<x<1$ .

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15.15.1

\mathbf{F}\left({a,b\atop c};\frac{1}{z}\right)=\left(1-\frac{z_{0}}{z}\right)% ^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{s!}\*\mathbf{F}\left({-s,b% \atop c};\frac{1}{z_{0}}\right)\left(1-\frac{z}{z_{0}}\right)^{-s}.

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.19(i)
Permalink:: http://dlmf.nist.gov/15.15.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.1.2):: The Pochhammer symbol now links to its definition.
See also:: Annotations for §15.15 and Ch.15

►Here

z_{0}

(

{\neq 0}

) is an arbitrary complex constant and the expansion converges when

|z-z_{0}|>\max(|z_{0}|,|z_{0}-1|)

. …

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•

Luke (1975, pp. 416–421) gives Chebyshev-series expansions for $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{L}_{n}\left(x\right)$ , $0\leq\left|x\right|\leq 8$ , and $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , $x\geq 8$ , for $n=0,1$ ; $\int_{0}^{x}t^{-m}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}t^{-m}\mathbf{L}_{0}\left(t\right)\,\mathrm{d}t$ , $0\leq\left|x\right|\leq 8$ , $m=0,1$ and $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ , $x\geq 8$ ; the coefficients are to 20D.

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MacLeod (1993) gives Chebyshev-series expansions for $\mathbf{L}_{0}\left(x\right)$ , $\mathbf{L}_{1}\left(x\right)$ , $0\leq x\leq 16$ , and $I_{0}\left(x\right)-\mathbf{L}_{0}\left(x\right)$ , $I_{1}\left(x\right)-\mathbf{L}_{1}\left(x\right)$ , $x\geq 16$ ; the coefficients are to 20D.

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Newman (1984) gives polynomial approximations for $\mathbf{H}_{n}\left(x\right)$ for $n=0,1$ , $0\leq x\leq 3$ , and rational-fraction approximations for $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ for $n=0,1$ , $x\geq 3$ . The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.

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See accompanying text — Figure 28.3.1: $\operatorname{ce}_{2n}\left(x,1\right)$ for $0\leq x\leq\pi/2$ , $n=0,1,2,3$ . Magnify

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… ►If

\gamma=0

and

\alpha\delta\neq 0

, then set

\alpha=1

and

\delta=-1

, without loss of generality. …Similar results hold for

\mbox{P}_{\mbox{\scriptsize III}}

with

\delta=0

and

\beta\gamma\neq 0

. … ►Let

W_{0}=W(z;\alpha_{0},\beta_{0},\gamma_{0},-\tfrac{1}{2})

and

W_{1}=W(z;\alpha_{1},\beta_{1},\gamma_{1},-\tfrac{1}{2})

be solutions of

\mbox{P}_{\mbox{\scriptsize V}}

, where …and assume

\Phi\neq 0

. … ►for

j=0,1,2,\infty

, where …

… ►Tabulated for

k=0(.01)1

to 10D by Fettis and Caslin (1964), and for

k=0(.02)1

to 7D by Zhang and Jin (1996, p. 673). … ►Tabulated for

k^{2}=0(.001)1

to 8D by Beli͡akov et al. (1962). … ►Tabulated for

\phi=0(5^{\circ})90^{\circ}

,

k^{2}=0(.01)1

to 10D by Fettis and Caslin (1964). ►Tabulated for

\phi=0(1^{\circ})90^{\circ}

,

k^{2}=0(.01)1

to 7S by Beli͡akov et al. (1962). … ►Tabulated for

\phi=0(5^{\circ})90^{\circ}

,

k=0(.01)1

to 10D by Fettis and Caslin (1964). …

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Abramowitz and Stegun (1964, Chapter 19) includes $U\left(a,x\right)$ and $V\left(a,x\right)$ for $\pm a=0(.1)1(.5)5$ , $x=0(.1)5$ , 5S; $W\left(a,\pm x\right)$ for $\pm a=0(.1)1(1)5$ , $x=0(.1)5$ , 4-5D or 4-5S.

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Miller (1955) includes $W\left(a,x\right)$ , $W\left(a,-x\right)$ , and reduced derivatives for $a=-10(1)10$ , $x=0(.1)10$ , 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

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Kireyeva and Karpov (1961) includes $D_{p}\left(x(1+i)\right)$ for $\pm x=0(.1)5$ , $p=0(.1)2$ , and $\pm x=5(.01)10$ , $p=0(.5)2$ , 7D.

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Karpov and Čistova (1964) includes $D_{p}\left(x\right)$ for $p=-2(.1)0$ , $\pm x=0(.01)5$ ; $p=-2(.05)0$ , $\pm x=5(.01)10$ , 6D.

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Karpov and Čistova (1968) includes $e^{-\frac{1}{4}x^{2}}D_{p}\left(-x\right)$ and $e^{-\frac{1}{4}x^{2}}D_{p}\left(ix\right)$ for $x=0(.01)5$ and $x^{-1}$ = 0(.001 or .0001)5, $p=-1(.1)1$ , 7D or 8S.

…

… ►where

\xi_{0}

and

\xi_{1}\in I

. ►For the values of

n_{0}

and

n_{1}

used in the formulas below … ►For corresponding formulas for second, third, and fourth derivatives, with

t=0

, see Collatz (1960, Table III, pp. 538–539). … ►

f(z)=e^{z}

,

x_{0}=0

. … ►For partial derivatives we use the notation

u_{t,s}=u(x_{0}+th,y_{0}+sh)

. …

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Lauricella%0Afunction

11—20 of 699 matching pages

11: 32.4 Isomonodromy Problems

12: 14.33 Tables

13: 15.15 Sums

14: 11.15 Approximations

15: 28.3 Graphics

16: 32.7 Bäcklund Transformations

17: 19.37 Tables

18: 12.19 Tables

19: 3.4 Differentiation

20: 14.4 Graphics