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11: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

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12: 11.11 Asymptotic Expansions of Anger–Weber Functions

… ►Let

F_{0}(\nu)=G_{0}(\nu)=1

, and for

k=1,2,3,\dots

, …For sharp error bounds and exponentially-improved extensions, see Nemes (2018). … ►where … ►and …with the

U_{k}

defined in §10.41(ii). …

13: 18.40 Methods of Computation

… ►The quadrature abscissas

x_{n}

and weights

w_{n}

then follow from the discussion of §3.5(vi). … ►Results of low (

~{}2

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

20

. … ►where the coefficients are defined recursively via

a_{1}=\frac{x_{1,N}}{x_{2,N}}-1

, and … ►The example chosen is inversion from the

\alpha_{n},\beta_{n}

for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50). … ►Further, exponential convergence in

N

, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate

w(x)

for these OP systems on

x\in[-1,1]

and

(-\infty,\infty)

respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

14: 18.32 OP’s with Respect to Freud Weights

… ►The special case

Q(x)=\frac{1}{4}x^{4}-tx^{2}

is of particular interest, see Clarkson and Jordaan (2018). …

15: Philip J. Davis

… ► 2018) received an undergraduate degree in mathematics from Harvard College in 1943. …

16: 8.26 Tables

… ►

•

Zhang and Jin (1996, Table 3.8) tabulates $\gamma\left(a,x\right)$ for $a=0.5,1,3,5,10,25,50,100$ , $x=0(.1)1(1)3,5(5)30,50,100$ to 8D or 8S.

… ►

•

Pearson (1968) tabulates $I_{x}\left(a,b\right)$ for $x=0.01(.01)1$ , $a,b=0.5(.5)11(1)50$ , with $b\leq a$ , to 7D.

►

•

Zhang and Jin (1996, Table 3.9) tabulates $I_{x}\left(a,b\right)$ for $x=0(.05)1$ , $a=0.5,1,3,5,10$ , $b=1,10$ to 8D.

… ►

•

Stankiewicz (1968) tabulates $E_{n}\left(x\right)$ for $n=1(1)10$ , $x=0.01(.01)5$ to 7D.

►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

17: Errata

… ►

Paragraph Prime Number Theorem (in §27.12)

The largest known prime, which is a Mersenne prime, was updated from $2^{43,112,609}-1$ (2009) to $2^{82,589,933}-1$ (2018).

… ►

Version 1.0.21 (December 15, 2018)

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Version 1.0.20 (September 15, 2018)

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Version 1.0.19 (June 22, 2018)

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Version 1.0.18 (March 27, 2018)

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18: 13.30 Tables

… ►

•

Slater (1960) tabulates $M\left(a,b,x\right)$ for $a=-1(.1)1$ , $b=0.1(.1)1$ , and $x=0.1(.1)10$ , 7–9S; $M\left(a,b,1\right)$ for $a=-11(.2)2$ and $b=-4(.2)1$ , 7D; the smallest positive $x$ -zero of $M\left(a,b,x\right)$ for $a=-4(.1){-}0.1$ and $b=0.1(.1)2.5$ , 7D.

►

•

Abramowitz and Stegun (1964, Chapter 13) tabulates $M\left(a,b,x\right)$ for $a=-1(.1)1$ , $b=0.1(.1)1$ , and $x=0.1(.1)1(1)10$ , 8S. Also the smallest positive $x$ -zero of $M\left(a,b,x\right)$ for $a=-1(.1){-}0.1$ and $b=0.1(.1)1$ , 7D.

►

•

Zhang and Jin (1996, pp. 411–423) tabulates $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$ for $a=-5(.5)5$ , $b=0.5(.5)5$ , and $x=0.1,1,5,10,20,30$ , 8S (for $M\left(a,b,x\right)$ ) and 7S (for $U\left(a,b,x\right)$ ).

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