generalized sine and cosine integrals

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1: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

… ►From here on it is assumed that unless indicated otherwise the functions $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ have their principal values. … ►

§8.21(iv) Interrelations

… ►For the corresponding expansions for $\mathrm{si}(a,z)$ and $\mathrm{ci}(a,z)$ apply (8.21.20) and (8.21.21).

2: 8.1 Special Notation

… ►Unless otherwise indicated, primes denote derivatives with respect to the argument. ►The functions treated in this chapter are the incomplete gamma functions $\gamma (a,z)$, $\mathrm{\Gamma }(a,z)$, ${\gamma }^{\ast }(a,z)$, $P(a,z)$, and $Q(a,z)$; the incomplete beta functions ${\mathrm{B}}_x(a,b)$ and $I_x(a,b)$; the generalized exponential integral $E_p\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$. ►Alternative notations include: Prym’s functions $P_z(a)=\gamma (a,z)$, $Q_z(a)=\mathrm{\Gamma }(a,z)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma (a+1,z)$, $[a,z]!=\mathrm{\Gamma }(a+1,z)$, Dingle (1973); $B(a,b,x)={\mathrm{B}}_x(a,b)$, $I(a,b,x)=I_x(a,b)$, Magnus et al. (1966); $\mathrm{Si}(a,x)\to \mathrm{Si}(1-a,x)$, $\mathrm{Ci}(a,x)\to \mathrm{Ci}(1-a,x)$, Luke (1975).

3: 6.4 Analytic Continuation

§6.4 Analytic Continuation

… ►The general value of $E_1\left(z\right)$ is given by … ►The general values of the other functions are defined in a similar manner, and … ►Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_1\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.

4: 6.2 Definitions and Interrelations

… ►

§6.2(i) Exponential and Logarithmic Integrals

… ►

§6.2(ii) Sine and Cosine Integrals

… ►

Values at Infinity

… ►

Hyperbolic Analogs of the Sine and Cosine Integrals

… ►

§6.2(iii) Auxiliary Functions

…

5: 6 Exponential, Logarithmic, Sine, and
Cosine Integrals

Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals

…

6: 10.15 Derivatives with Respect to Order

… ►For the notations $\mathrm{Ci}$ and $\mathrm{Si}$ see §6.2(ii). … ►

10.15.6 $${\frac{\partial J_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\sin x-\mathrm{Si}\left(2x\right)\cos x\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.41 (corrected)

Referenced by:

§10.15, §10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.15.E6

Encodings:

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Annotations for §10.15, §10.15 and Ch.10

►

10.15.7 $${\frac{\partial J_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =-\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\cos x+\mathrm{Si}\left(2x\right)\sin x\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.42 (corrected)

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

►

10.15.8 $${\frac{\partial Y_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\cos x+\left(\mathrm{Si}\left(2x\right)-\pi \right)\sin x\right),$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.43 (corrected)

Referenced by:

§10.15

Permalink:

http://dlmf.nist.gov/10.15.E8

Encodings:

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See also:

Annotations for §10.15, §10.15 and Ch.10

►

10.15.9 $${\frac{\partial Y_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =-\frac{1}{2}}=-\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\sin x-\left(\mathrm{Si}\left(2x\right)-\pi \right)\cos x\right).$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.44 (corrected)

Referenced by:

§10.15, Ch.10

Permalink:

http://dlmf.nist.gov/10.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

…

7: 7.14 Integrals

§7.14 Integrals

… ►

Fourier Transform

… ►

§7.14(ii) Fresnel Integrals

►

Laplace Transforms

… ►In a series of ten papers Hadži (1968, 1969, 1970, 1972, 1973, 1975a, 1975b, 1976a, 1976b, 1978) gives many integrals containing error functions and Fresnel integrals, also in combination with the hypergeometric function, confluent hypergeometric functions, and generalized hypergeometric functions.

8: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
6 Exponential, Logarithmic, Sine, and Cosine Integrals
6.21(ii) $E_1\left(x\right)$, $\mathrm{Ei}\left(x\right)$, $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $\mathrm{Shi}\left(x\right)$, $\mathrm{Chi}\left(x\right)$, $x\in ℝ$	✓	✓	✓	✓	✓	✓	✓		✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
6.21(iii) $E_1\left(z\right)$, $\mathrm{Si}\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Shi}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $z\in ℂ$	✓	✓				✓			✓			✓	✓	✓							✓	✓	✓
…
7.25(iv) $C\left(x\right)$, $S\left(x\right)$, $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in ℝ$	✓		✓						✓			a	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
7.25(v) $C\left(z\right)$, $S\left(z\right)$, $z\in ℂ$	✓											a	✓								✓	✓	✓
…

…

9: 7.11 Relations to Other Functions

… ►

Incomplete Gamma Functions and Generalized Exponential Integral

… ►

Confluent Hypergeometric Functions

… ►

7.11.6 $$C\left(z\right)+\mathrm{i}S\left(z\right)=zM(\frac{1}{2},\frac{3}{2},\frac{1}{2}\pi \mathrm{i}{z}^2)=z{\mathrm{e}}^{\pi \mathrm{i}{z}^2/2}M(1,\frac{3}{2},-\frac{1}{2}\pi \mathrm{i}{z}^2).$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

►

Generalized Hypergeometric Functions

… ►

7.11.7 $$C\left(z\right)=z{}_1{}^{}F_2^{}(\frac{1}{4};\frac{5}{4},\frac{1}{2};-\frac{1}{16}{\pi }^2{z}^4),$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

…

10: 6.10 Other Series Expansions

… ►

§6.10(i) Inverse Factorial Series

… ►For a more general result (incomplete gamma function), and also for a result for the logarithmic integral, see Nielsen (1906a, p. 283: Formula (3) is incorrect). ►

§6.10(ii) Expansions in Series of Spherical Bessel Functions

… ►

6.10.4 $$\mathrm{Si}\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{\left({𝗃}_n\left(\frac{1}{2}z\right)\right)}^2,$$

ⓘ

Symbols:

$\mathrm{Si}\left(z\right)$: sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.10(ii)

Permalink:

http://dlmf.nist.gov/6.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

… ►An expansion for $E_1\left(z\right)$ can be obtained by combining (6.2.4) and (6.10.8).