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1: 4.1 Special Notation

… ► ►►

$k,m,n$	integers.
…

… ►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. …

2: 4.32 Inequalities

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4.32.1 $$\cosh x\le {\left(\frac{\sinh x}{x}\right)}^3,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E1

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See also:

Annotations for §4.32 and Ch.4

►

4.32.2 $x>0$,

ⓘ

Symbols:

$\cos z$: cosine function, $\tanh z$: hyperbolic tangent function, $\sin z$: sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E2

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See also:

Annotations for §4.32 and Ch.4

►

4.32.3 $$|\cosh x-\cosh y|\ge |x-y|\sqrt{\sinh x\sinh y},$$ $x>0$, $y>0$,

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E3

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See also:

Annotations for §4.32 and Ch.4

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3: 4.35 Identities

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4.35.11 $${\cosh }^{2}z-{\sinh }^{2}z=1,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.16

Permalink:

http://dlmf.nist.gov/4.35.E11

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

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4.35.20 $$\sinh \frac{z}{2}={\left(\frac{\cosh z-1}{2}\right)}^{1/2},$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.28

Permalink:

http://dlmf.nist.gov/4.35.E20

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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

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

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4.35.35 $$\cosh z=\cosh x\cos y+\mathrm{i}\sinh x\sin y,$$

ⓘ

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:

4.5.50

Permalink:

http://dlmf.nist.gov/4.35.E35

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

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4.35.38 $$|\sinh z|={({\sinh }^{2}x+{\sin }^{2}y)}^{1/2}={\left(\frac{1}{2}(\cosh \left(2x\right)-\cos \left(2y\right))\right)}^{1/2},$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.54

Permalink:

http://dlmf.nist.gov/4.35.E38

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See also:

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

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4: 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

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Annotations for §4.14 and Ch.4

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4.14.4 $$\tan z=\frac{\sin z}{\cos z},$$

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Defines:

$\tan z$: tangent function

Symbols:

$\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.3

Referenced by:

§4.14, §4.45(i)

Permalink:

http://dlmf.nist.gov/4.14.E4

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Annotations for §4.14 and Ch.4

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

ⓘ

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

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4.14.7 $$\cot z=\frac{\cos z}{\sin z}=\frac{1}{\tan z}.$$

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Defines:

$\cot z$: cotangent function

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.6

Referenced by:

§4.14, §4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.14.E7

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Annotations for §4.14 and Ch.4

►The functions $\sin z$ and $\cos z$ are entire. …

5: 4.21 Identities

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4.21.12 $${\sin }^{2}z+{\cos }^{2}z=1,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.10

Permalink:

http://dlmf.nist.gov/4.21.E12

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

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4.21.21 $$\sin \frac{z}{2}=\pm {\left(\frac{1-\cos z}{2}\right)}^{1/2},$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.20

Referenced by:

§4.21(iii)

Permalink:

http://dlmf.nist.gov/4.21.E21

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See also:

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

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4.21.37 $$\sin z=\sin x\cosh y+\mathrm{i}\cos x\sinh y,$$

ⓘ

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:

4.3.55

Permalink:

http://dlmf.nist.gov/4.21.E37

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

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4.21.38 $$\cos z=\cos x\cosh y-\mathrm{i}\sin x\sinh y,$$

ⓘ

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:

4.3.56

Permalink:

http://dlmf.nist.gov/4.21.E38

Encodings:

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See also:

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

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4.21.41 $$|\sin z|={({\sin }^{2}x+{\sinh }^{2}y)}^{1/2}={\left(\frac{1}{2}\left(\cosh \left(2y\right)-\cos \left(2x\right)\right)\right)}^{1/2},$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.59

Permalink:

http://dlmf.nist.gov/4.21.E41

Encodings:

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See also:

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

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6: 4.18 Inequalities

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Jordan’s Inequality

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4.18.3 $$\cos x\le \frac{\sin x}{x}\le 1,$$ $0\le x\le \pi$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.81

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E3

Encodings:

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See also:

Annotations for §4.18, §4.18 and Ch.4

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4.18.5 $$|\sinh y|\le |\sin z|\le \cosh y,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: sine function, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.83

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E5

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Annotations for §4.18, §4.18 and Ch.4

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4.18.6 $$|\sinh y|\le |\cos z|\le \cosh y,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.84

Permalink:

http://dlmf.nist.gov/4.18.E6

Encodings:

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Annotations for §4.18, §4.18 and Ch.4

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4.18.9 $$|\sin z|\le \sinh |z|,$$

ⓘ

Symbols:

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

A&S Ref:

4.3.87

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E9

Encodings:

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See also:

Annotations for §4.18, §4.18 and Ch.4

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7: 4.28 Definitions and Periodicity

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4.28.1 $$\sinh z=\frac{{\mathrm{e}}^z-{\mathrm{e}}^{-z}}{2},$$

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Defines:

$\sinh z$: hyperbolic sine function

Symbols:

$\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

4.5.1

Referenced by:

§4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.28.E1

Encodings:

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Annotations for §4.28 and Ch.4

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4.28.3 $$\cosh z\pm \sinh z={\mathrm{e}}^{\pm z},$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.19 4.5.20

Permalink:

http://dlmf.nist.gov/4.28.E3

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Annotations for §4.28 and Ch.4

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4.28.5 $$\mathrm{csch}z=\frac{1}{\sinh z},$$

ⓘ

Defines:

$\mathrm{csch}z$: hyperbolic cosecant function

Symbols:

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

A&S Ref:

4.5.4

Permalink:

http://dlmf.nist.gov/4.28.E5

Encodings:

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Annotations for §4.28 and Ch.4

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4.28.8 $$\sin \left(\mathrm{i}z\right)=\mathrm{i}\sinh z,$$

ⓘ

Symbols:

$\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.5.7

Referenced by:

§22.10(ii), §4.29(ii), §4.29(ii), §4.33

Permalink:

http://dlmf.nist.gov/4.28.E8

Encodings:

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Annotations for §4.28, §4.28 and Ch.4

… ►The functions $\sinh z$ and $\cosh z$ have period $2\pi \mathrm{i}$, and $\tanh z$ has period $\pi \mathrm{i}$. …

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:

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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:

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See also:

Annotations for §10.64, §10.64 and Ch.10

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9: 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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4.42.8 $$\cos a=\cos b\cos c+\sin b\sin c\cos A,$$

ⓘ

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

ⓘ

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:

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See also:

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

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4.42.11 $$\cos a\cos C=\sin a\cot b-\sin C\cot B,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cot z$: cotangent function, $\sin z$: sine function, $B$: angle, $C$: angle, $a$: arc length and $b$: arc length

Permalink:

http://dlmf.nist.gov/4.42.E11

Encodings:

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See also:

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

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4.42.12 $$\cos A=-\cos B\cos C+\sin B\sin C\cos a.$$

ⓘ

Symbols:

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

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E12

Encodings:

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See also:

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

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10: 22.10 Maclaurin Series

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22.10.4 $$\mathrm{sn}(z,k)=\sin z-\frac{k^2}{4}(z-\sin z\cos z)\cos z+O\left(k^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.1

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E4

Encodings:

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Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.5 $$\mathrm{cn}(z,k)=\cos z+\frac{k^2}{4}(z-\sin z\cos z)\sin z+O\left(k^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.2

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E5

Encodings:

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See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.6 $$\mathrm{dn}(z,k)=1-\frac{k^2}{2}{\sin }^{2}z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.3

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E6

Encodings:

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See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.7 $$\mathrm{sn}(z,k)=\tanh z-\frac{k_{}^{\prime }{}_{}{}^2}{4}(z-\sinh z\cosh z){\mathrm{sech}}^{2}z+O\left(k_{}^{\prime }{}_{}{}^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.1

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/22.10.E7

Encodings:

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See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.8 $$\mathrm{cn}(z,k)=\mathrm{sech}z+\frac{k_{}^{\prime }{}_{}{}^2}{4}(z-\sinh z\cosh z)\tanh z\mathrm{sech}z+O\left(k_{}^{\prime }{}_{}{}^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.2

Permalink:

http://dlmf.nist.gov/22.10.E8

Encodings:

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See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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