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1: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

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§8.21(iv) Interrelations

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§8.21(v) Special Values

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§8.21(viii) Asymptotic Expansions

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2: 6.2 Definitions and Interrelations

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§6.2(ii) Sine and Cosine Integrals

… ► $\mathrm{Si}\left(z\right)$ is an odd entire function. … ►

Values at Infinity

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Hyperbolic Analogs of the Sine and Cosine Integrals

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§6.2(iii) Auxiliary Functions

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3: 4.21 Identities

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4.21.15 $$2\sin u\sin v=\cos \left(u-v\right)-\cos \left(u+v\right),$$

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Symbols:

$\cos z$: cosine function and $\sin z$: sine function

A&S Ref:

4.3.31

Permalink:

http://dlmf.nist.gov/4.21.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(ii), §4.21 and Ch.4

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4.21.17 $$2\sin u\cos v=\sin \left(u-v\right)+\sin \left(u+v\right).$$

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Symbols:

$\cos z$: cosine function and $\sin z$: sine function

A&S Ref:

4.3.33

Permalink:

http://dlmf.nist.gov/4.21.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(ii), §4.21 and Ch.4

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4.21.18 $${\sin }^{2}u-{\sin }^{2}v=\sin \left(u+v\right)\sin \left(u-v\right),$$

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Symbols:

$\sin z$: sine function

A&S Ref:

4.3.40

Permalink:

http://dlmf.nist.gov/4.21.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(ii), §4.21 and Ch.4

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4.21.24 $$\sin \left(-z\right)=-\sin z,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.13

Permalink:

http://dlmf.nist.gov/4.21.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(iii), §4.21 and Ch.4

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4.21.30 $$\sin \left(3z\right)=3\sin z-4{\sin }^{3}z,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.27

Permalink:

http://dlmf.nist.gov/4.21.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(iii), §4.21 and Ch.4

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4: 4.35 Identities

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4.35.16 $$2\sinh u\cosh v=\sinh \left(u+v\right)+\sinh \left(u-v\right).$$

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Symbols:

$\cosh z$: hyperbolic cosine function and $\sinh z$: hyperbolic sine function

A&S Ref:

4.5.40

Permalink:

http://dlmf.nist.gov/4.35.E16

Encodings:

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See also:

Annotations for §4.35(ii), §4.35 and Ch.4

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4.35.17 $${\sinh }^{2}u-{\sinh }^{2}v=\sinh \left(u+v\right)\sinh \left(u-v\right),$$

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Symbols:

$\sinh z$: hyperbolic sine function

A&S Ref:

4.5.47

Permalink:

http://dlmf.nist.gov/4.35.E17

Encodings:

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See also:

Annotations for §4.35(ii), §4.35 and Ch.4

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4.35.23 $$\sinh \left(-z\right)=-\sinh z,$$

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Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.21

Permalink:

http://dlmf.nist.gov/4.35.E23

Encodings:

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See also:

Annotations for §4.35(iii), §4.35 and Ch.4

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4.35.29 $$\sinh \left(3z\right)=3\sinh z+4{\sinh }^{3}z,$$

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Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.34

Permalink:

http://dlmf.nist.gov/4.35.E29

Encodings:

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See also:

Annotations for §4.35(iii), §4.35 and Ch.4

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4.35.34 $$\sinh z=\sinh x\cos y+\mathrm{i}\cosh x\sin y,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.49

Permalink:

http://dlmf.nist.gov/4.35.E34

Encodings:

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See also:

Annotations for §4.35(iv), §4.35 and Ch.4

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5: 6.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the argument. ►The main functions treated in this chapter are the exponential integrals $\mathrm{Ei}\left(x\right)$, $E_1\left(z\right)$, and $\mathrm{Ein}\left(z\right)$; the logarithmic integral $\mathrm{li}\left(x\right)$; the sine integrals $\mathrm{Si}\left(z\right)$ and $\mathrm{si}\left(z\right)$; the cosine integrals $\mathrm{Ci}\left(z\right)$ and $\mathrm{Cin}\left(z\right)$.

6: 6.17 Physical Applications

§6.17 Physical Applications

… ►Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.

7: 4.42 Solution of Triangles

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4.42.1 $$\sin A=\frac{a}{c}=\frac{1}{\csc A},$$

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Symbols:

$\csc z$: cosecant function, $\sin z$: sine function, $A$: angle, $a$: height and $c$: hypotenuse

A&S Ref:

4.3.147

Permalink:

http://dlmf.nist.gov/4.42.E1

Encodings:

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See also:

Annotations for §4.42(i), §4.42 and Ch.4

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4.42.4 $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},$$

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Symbols:

$\sin z$: sine function, $a$: length, $b$: base, $c$: length, $A$: angle, $B$: angle and $C$: angle

A&S Ref:

4.3.148

Permalink:

http://dlmf.nist.gov/4.42.E4

Encodings:

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See also:

Annotations for §4.42(ii), §4.42 and Ch.4

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4.42.8 $$\cos a=\cos b\cos c+\sin b\sin c\cos A,$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $a$: arc length, $b$: arc length and $c$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E8

Encodings:

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See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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4.42.9 $$\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c},$$

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Symbols:

$\sin z$: sine function, $A$: angle, $B$: angle, $C$: angle, $a$: arc length, $b$: arc length and $c$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E9

Encodings:

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See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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4.42.10 $$\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $B$: angle, $a$: arc length, $b$: arc length and $c$: arc length

Permalink:

http://dlmf.nist.gov/4.42.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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8: 10.64 Integral Representations

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10.64.1 $${\mathrm{ber}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\cos \left(x\sin t-nt\right)\cosh \left(x\sin t\right)dt,$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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10.64.2 $${\mathrm{bei}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\sin \left(x\sin t-nt\right)\sinh \left(x\sin t\right)dt.$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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9: 4.1 Special Notation

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$k,m,n$	integers.
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… ►The main functions treated in this chapter are the logarithm $\ln z$, $\mathrm{Ln}z$; the exponential $\exp z$, ${\mathrm{e}}^z$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\arcsin z$, $\mathrm{Arcsin}z$, etc. ; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\mathrm{csch}z$, $\mathrm{sech}z$, $\coth z$; the inverse hyperbolic functions $\mathrm{arcsinh}z$, $\mathrm{Arcsinh}z$, etc. ►Sometimes in the literature the meanings of $\ln$ and $\mathrm{Ln}$ are interchanged; similarly for $\arcsin z$ and $\mathrm{Arcsin}z$, etc. …${\sin }^{-1}z$ for $\arcsin z$ and ${\mathrm{Sin}}^{-1}z$ for $\mathrm{Arcsin}z$.

10: 4.14 Definitions and Periodicity

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4.14.1 $$\sin z=\frac{{\mathrm{e}}^{\mathrm{i}z}-{\mathrm{e}}^{-\mathrm{i}z}}{2\mathrm{i}},$$

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Defines:

$\sin z$: sine function

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.3.1

Referenced by:

§4.45(ii), (9.5.3)

Permalink:

http://dlmf.nist.gov/4.14.E1

Encodings:

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See also:

Annotations for §4.14 and Ch.4

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4.14.5 $$\csc z=\frac{1}{\sin z},$$

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Defines:

$\csc z$: cosecant function

Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.4

Permalink:

http://dlmf.nist.gov/4.14.E5

Encodings:

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See also:

Annotations for §4.14 and Ch.4

… ►The functions $\sin z$ and $\cos z$ are entire. In $ℂ$ the zeros of $\sin z$ are $z=k\pi$, $k\in \mathbb{Z}$; the zeros of $\cos z$ are $z=\left(k+\frac{1}{2}\right)\pi$, $k\in \mathbb{Z}$. … ►

4.14.8 $$\sin \left(z+2k\pi \right)=\sin z,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $k$: integer and $z$: complex variable

A&S Ref:

4.3.7

Permalink:

http://dlmf.nist.gov/4.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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