triple%20integrals

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11: 34.13 Methods of Computation

… ►For

\mathit{9j}

symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). …

12: 1.5 Calculus of Two or More Variables

… ►

Finite Integrals

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Infinite Integrals

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Double Integrals

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Triple Integrals

… ►In case of triple integrals the

(x,y,z)

sets are of the form …

13: 22.4 Periods, Poles, and Zeros

… ►The other poles are at congruent points, which is the set of points obtained by making translations by

2mK+2niK^{\prime}

, where

m,n\in\mathbb{Z}

. … ►Figure 22.4.1 illustrates the locations in the

z

-plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices

0

2K

2K+2iK^{\prime}

2iK^{\prime}

. … ►Figure 22.4.2 depicts the fundamental unit cell in the

z

-plane, with vertices

\mbox{s}=0

\mbox{c}=K

\mbox{d}=K+iK^{\prime}

\mbox{n}=iK^{\prime}

. The set of points

z=mK+niK^{\prime}

m,n\in\mathbb{Z}

, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by

mK+niK^{\prime}

, where again

m,n\in\mathbb{Z}

. … ►This half-period will be plus or minus a member of the triple

{K,iK^{\prime},K+iK^{\prime}}

; the other two members of this triple are quarter periods of

\operatorname{pq}\left(z,k\right)

. …

14: 8.26 Tables

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•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

… ►

§8.26(iv) Generalized Exponential Integral

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•

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 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.

15: 6.19 Tables

… ►

§6.19(ii) Real Variables

►

•

Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\operatorname{Si}\left(x\right)$ , $-x^{-2}\operatorname{Cin}\left(x\right)$ , $x^{-1}\operatorname{Ein}\left(x\right)$ , $-x^{-1}\operatorname{Ein}\left(-x\right)$ , $x=0(.01)0.5$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=0.5(.01)2$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x=2(.1)10$ ; $x\mathrm{f}\left(x\right)$ , $x^{2}\mathrm{g}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x^{-1}=0(.005)0.1$ ; $\operatorname{Si}\left(\pi x\right)$ , $\operatorname{Cin}\left(\pi x\right)$ , $x=0(.1)10$ . Accuracy varies but is within the range 8S–11S.

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•

Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

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•

Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $ze^{z}E_{1}\left(z\right)$ , $x=-19(1)20$ , $y=0(1)20$ , 6D; $e^{z}E_{1}\left(z\right)$ , $x=-4(.5)-2$ , $y=0(.2)1$ , 6D; $E_{1}\left(z\right)+\ln z$ , $x=-2(.5)2.5$ , $y=0(.2)1$ , 6D.

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•

Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.

16: 36 Integrals with Coalescing Saddles

Chapter 36 Integrals with Coalescing Saddles

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17: 6.20 Approximations

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•

Hastings (1955) gives several minimax polynomial and rational approximations for $E_{1}\left(x\right)+\ln x$ , $xe^{x}E_{1}\left(x\right)$ , and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ . These are included in Abramowitz and Stegun (1964, Ch. 5).

►

•

Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.

►

•

Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.

►

•

MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$ , with accuracies up to 20S.

… ►

•

Luke and Wimp (1963) covers $\operatorname{Ei}\left(x\right)$ for $x\leq-4$ (20D), and $\operatorname{Si}\left(x\right)$ and $\operatorname{Ci}\left(x\right)$ for $x\geq 4$ (20D).

…

18: 1.6 Vectors and Vector-Valued Functions

… ►

§1.6(iv) Path and Line Integrals

… ►The path integral of a continuous function

f(x,y,z)

is …The line integral of a vector-valued function

\mathbf{F}=F_{1}\mathbf{i}+F_{2}\mathbf{j}+F_{3}\mathbf{k}

along

\mathbf{c}

is given by … ►

§1.6(v) Surfaces and Integrals over Surfaces

… ►The integral of a continuous function

f(x,y,z)

over a surface

S

is …

19: 7.13 Zeros

… ►

§7.13(iii) Zeros of the Fresnel Integrals

►At

z=0

C\left(z\right)

has a simple zero and

S\left(z\right)

has a triple zero. …Similarly for

S\left(z\right)

. … ►

§7.13(iv) Zeros of $\mathcal{F}\left(z\right)$

… ►

20: 20 Theta Functions

Chapter 20 Theta Functions

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