Heine integral

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21: 6.17 Physical Applications

§6.17 Physical Applications

►Geller and Ng (1969) cites work with applications from diffusion theory, transport problems, the study of the radiative equilibrium of stellar atmospheres, and the evaluation of exchange integrals occurring in quantum mechanics. …Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.

22: 6.5 Further Interrelations

§6.5 Further Interrelations

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6.5.2 $$\mathrm{Ei}\left(x\right)=-\frac{1}{2}(E_1\left(-x+\mathrm{i}0\right)+E_1\left(-x-\mathrm{i}0\right)),$$

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Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $x$: real variable

A&S Ref:

5.1.7

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.3 $$\frac{1}{2}(\mathrm{Ei}\left(x\right)+E_1\left(x\right))=\mathrm{Shi}\left(x\right)=-\mathrm{i}\mathrm{Si}\left(\mathrm{i}x\right),$$

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Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{Shi}\left(z\right)$: hyperbolic sine integral, $\mathrm{i}$: imaginary unit, $\mathrm{Si}\left(z\right)$: sine integral and $x$: real variable

A&S Ref:

5.2.22 (in modified form)

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E3

Encodings:

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See also:

Annotations for §6.5 and Ch.6

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6.5.6 $$\mathrm{Ci}\left(z\right)=-\frac{1}{2}(E_1\left(\mathrm{i}z\right)+E_1\left(-\mathrm{i}z\right)),$$

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Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

5.2.23

Referenced by:

§6.12(ii), §6.5

Permalink:

http://dlmf.nist.gov/6.5.E6

Encodings:

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Annotations for §6.5 and Ch.6

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6.5.7 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=E_1\left(\mp \mathrm{i}z\right){\mathrm{e}}^{\mp \mathrm{i}z}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, §6.5, §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.5.E7

Encodings:

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Annotations for §6.5 and Ch.6

23: 6.3 Graphics

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Figure 6.3.1: The exponential integrals $E_1\left(x\right)$ and $\mathrm{Ei}\left(x\right)$, . Magnify

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Figure 6.3.2: The sine and cosine integrals $\mathrm{Si}\left(x\right),\mathrm{Ci}\left(x\right)$, $0\le x\le 15$. Magnify

►For a graph of $\mathrm{li}\left(x\right)$ see Figure 6.16.2. … ►

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Figure 6.3.3: $|E_1\left(x+\mathrm{i}y\right)|$, $-4\le x\le 4$, $-4\le y\le 4$. … Magnify 3D Help

24: 19.13 Integrals of Elliptic Integrals

§19.13 Integrals of Elliptic Integrals

… ►For definite and indefinite integrals of complete elliptic integrals see Byrd and Friedman (1971, pp. 610–612, 615), Prudnikov et al. (1990, §§1.11, 2.16), Glasser (1976), Bushell (1987), and Cvijović and Klinowski (1999). ►For definite and indefinite integrals of incomplete elliptic integrals see Byrd and Friedman (1971, pp. 613, 616), Prudnikov et al. (1990, §§1.10.2, 2.15.2), and Cvijović and Klinowski (1994). … ►

§19.13(iii) Laplace Transforms

►For direct and inverse Laplace transforms for the complete elliptic integrals $K\left(k\right)$, $E\left(k\right)$, and $D\left(k\right)$ see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.

25: 6.20 Approximations

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Hastings (1955) gives several minimax polynomial and rational approximations for $E_1\left(x\right)+\ln x$, $x{\mathrm{e}}^x{E}_1\left(x\right)$, 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).

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Cody and Thacher (1968) provides minimax rational approximations for $E_1\left(x\right)$, with accuracies up to 20S.

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

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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.

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

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26: 19.21 Connection Formulas

§19.21 Connection Formulas

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§19.21(i) Complete Integrals

… ►The complete case of $R_J$ can be expressed in terms of $R_F$ and $R_D$: … ►

§19.21(ii) Incomplete Integrals

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§19.21(iii) Change of Parameter of $R_J$

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27: 6.19 Tables

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§6.19(ii) Real Variables

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Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\mathrm{Si}\left(x\right)$, $-x^{-2}\mathrm{Cin}\left(x\right)$, $x^{-1}\mathrm{Ein}\left(x\right)$, $-x^{-1}\mathrm{Ein}\left(-x\right)$, $x=0(.01)0.5$; $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $\mathrm{Ei}\left(x\right)$, $E_1\left(x\right)$, $x=0.5(.01)2$; $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x{\mathrm{e}}^{-x}\mathrm{Ei}\left(x\right)$, $x{\mathrm{e}}^x{E}_1\left(x\right)$, $x=2(.1)10$; $x\mathrm{f}\left(x\right)$, $x^2\mathrm{g}\left(x\right)$, $x{\mathrm{e}}^{-x}\mathrm{Ei}\left(x\right)$, $x{\mathrm{e}}^x{E}_1\left(x\right)$, $x^{-1}=0(.005)0.1$; $\mathrm{Si}\left(\pi x\right)$, $\mathrm{Cin}\left(\pi x\right)$, $x=0(.1)10$. Accuracy varies but is within the range 8S–11S.

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Zhang and Jin (1996, pp. 652, 689) includes $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x=0(.5)20(2)30$, 8D; $\mathrm{Ei}\left(x\right)$, $E_1\left(x\right)$, $x=[0,100]$, 8S.

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Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $z{\mathrm{e}}^z{E}_1\left(z\right)$, $x=-19(1)20$, $y=0(1)20$, 6D; ${\mathrm{e}}^z{E}_1\left(z\right)$, $x=-4(.5)-2$, $y=0(.2)1$, 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.

28: 7.22 Methods of Computation

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§7.22(i) Main Functions

►The methods available for computing the main functions in this chapter are analogous to those described in §§6.18(i)–6.18(iv) for the exponential integral and sine and cosine integrals, and similar comments apply. … ►

§7.22(ii) Goodwin–Staton Integral

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§7.22(iii) Repeated Integrals of the Complementary Error Function

►The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$. …

29: 19.15 Advantages of Symmetry

§19.15 Advantages of Symmetry

… ►Symmetry in $x,y,z$ of $R_F(x,y,z)$, $R_G(x,y,z)$, and $R_J(x,y,z,p)$ replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). … … ►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …

30: 19.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. ►The first set of main functions treated in this chapter are Legendre’s complete integrals …of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals … ►However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF. … ►The first three functions are incomplete integrals of the first, second, and third kinds, and the $\mathrm{cel}$ function includes complete integrals of all three kinds.