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1: 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. …

2: 4.32 Inequalities

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4.32.2 $x>0$,

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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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Annotations for §4.32 and Ch.4

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4.32.4 $$\arctan x\le \frac{1}{2}\pi \tanh x,$$ $x\ge 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tanh z$: hyperbolic tangent function, $\arctan z$: arctangent function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E4

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Annotations for §4.32 and Ch.4

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3: 4.14 Definitions and Periodicity

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

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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 $\tan z$, $\csc z$, $\sec z$, and $\cot z$ are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). … ►

4.14.10 $$\tan \left(z+k\pi \right)=\tan z.$$

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Symbols:

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

A&S Ref:

4.3.9

Permalink:

http://dlmf.nist.gov/4.14.E10

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Annotations for §4.14 and Ch.4

4: 4.28 Definitions and Periodicity

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4.28.4 $$\tanh z=\frac{\sinh z}{\cosh z},$$

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Defines:

$\tanh z$: hyperbolic tangent function

Symbols:

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

A&S Ref:

4.5.3

Permalink:

http://dlmf.nist.gov/4.28.E4

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Annotations for §4.28 and Ch.4

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4.28.7 $$\coth z=\frac{1}{\tanh z}.$$

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Defines:

$\coth z$: hyperbolic cotangent function

Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.6

Referenced by:

§4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.28.E7

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Annotations for §4.28 and Ch.4

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4.28.10 $$\tan \left(\mathrm{i}z\right)=\mathrm{i}\tanh z,$$

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Symbols:

$\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.5.9

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/4.28.E10

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 $\pi \mathrm{i}$. …

5: 4.31 Special Values and Limits

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4.31.2 $$\underset{z\to 0}{lim}\frac{\tanh z}{z}=1,$$

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Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E2

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See also:

Annotations for §4.31 and Ch.4

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6: 4.21 Identities

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4.21.10 $$\tan u\pm \tan v=\frac{\sin \left(u\pm v\right)}{\cos u\cos v},$$

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Symbols:

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

A&S Ref:

4.3.38

Permalink:

http://dlmf.nist.gov/4.21.E10

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

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4.21.13 $${\sec }^{2}z=1+{\tan }^{2}z,$$

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Symbols:

$\sec z$: secant function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.11

Permalink:

http://dlmf.nist.gov/4.21.E13

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

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4.21.26 $$\tan \left(-z\right)=-\tan z.$$

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Symbols:

$\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.15

Permalink:

http://dlmf.nist.gov/4.21.E26

Encodings:

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

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4.21.27 $$\sin \left(2z\right)=2\sin z\cos z=\frac{2\tan z}{1+{\tan }^{2}z},$$

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Symbols:

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

A&S Ref:

4.3.24

Permalink:

http://dlmf.nist.gov/4.21.E27

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

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4.21.29 $$\tan \left(2z\right)=\frac{2\tan z}{1-{\tan }^{2}z}=\frac{2\cot z}{{\cot }^{2}z-1}=\frac{2}{\cot z-\tan z}.$$

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Symbols:

$\cot z$: cotangent function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.26

Permalink:

http://dlmf.nist.gov/4.21.E29

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

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7: 4.20 Derivatives and Differential Equations

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4.20.3 $$\frac{d}{dz}\tan z={\sec }^{2}z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sec z$: secant function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.107

Permalink:

http://dlmf.nist.gov/4.20.E3

Encodings:

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See also:

Annotations for §4.20 and Ch.4

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4.20.5 $$\frac{d}{dz}\sec z=\sec z\tan z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sec z$: secant function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.109

Permalink:

http://dlmf.nist.gov/4.20.E5

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Annotations for §4.20 and Ch.4

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4.20.14 $$w=(1/a)\tan \left(az+c\right),$$

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Symbols:

$\tan z$: tangent function, $a$: real or complex constant, $z$: complex variable and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E14

Encodings:

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Annotations for §4.20 and Ch.4

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8: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.3: $\tanh x$ and $\coth x$. Magnify

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Figure 4.29.4: Principal values of $\mathrm{arctanh}x$ and $\mathrm{arccoth}x$. … Magnify

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§4.29(ii) Complex Arguments

… ►The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …

9: 4.35 Identities

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4.35.12 $${\mathrm{sech}}^{2}z=1-{\tanh }^{2}z,$$

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Symbols:

$\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.17

Permalink:

http://dlmf.nist.gov/4.35.E12

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

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4.35.25 $$\tanh \left(-z\right)=-\tanh z.$$

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Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.23

Permalink:

http://dlmf.nist.gov/4.35.E25

Encodings:

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See also:

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

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4.35.26 $$\sinh \left(2z\right)=2\sinh z\cosh z=\frac{2\tanh z}{1-{\tanh }^{2}z},$$

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Symbols:

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

A&S Ref:

4.5.31

Permalink:

http://dlmf.nist.gov/4.35.E26

Encodings:

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

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4.35.28 $$\tanh \left(2z\right)=\frac{2\tanh z}{1+{\tanh }^{2}z},$$

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Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.33

Permalink:

http://dlmf.nist.gov/4.35.E28

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

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4.35.36 $$\tanh z=\frac{\sinh \left(2x\right)+\mathrm{i}\sin \left(2y\right)}{\cosh \left(2x\right)+\cos \left(2y\right)},$$

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Symbols:

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

A&S Ref:

4.5.51

Permalink:

http://dlmf.nist.gov/4.35.E36

Encodings:

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See also:

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

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10: 22.10 Maclaurin Series

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

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

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

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

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22.10.9 $$\mathrm{dn}(z,k)=\mathrm{sech}z+\frac{k_{}^{\prime }{}_{}{}^2}{4}(z+\sinh z\cosh z)\tanh z\mathrm{sech}z+O\left(k_{}^{\prime }{}_{}{}^4\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: 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:

16.15.3

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/22.10.E9

Encodings:

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See also:

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

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