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11: 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:

pMML, png, TeX

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

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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12: 4.42 Solution of Triangles

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4.42.2 $$\cos A=\frac{b}{c}=\frac{1}{\sec A},$$

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Symbols:

$\cos z$: cosine function, $\sec z$: secant function, $A$: angle, $b$: base and $c$: hypotenuse

A&S Ref:

4.3.147

Permalink:

http://dlmf.nist.gov/4.42.E2

Encodings:

pMML, png, TeX

See also:

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

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13: 4.34 Derivatives and Differential Equations

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4.34.3 $$\frac{d}{dz}\tanh z={\mathrm{sech}}^{2}z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.73

Permalink:

http://dlmf.nist.gov/4.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.5 $$\frac{d}{dz}\mathrm{sech}z=-\mathrm{sech}z\tanh z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.75

Permalink:

http://dlmf.nist.gov/4.34.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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14: 4.37 Inverse Hyperbolic Functions

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4.37.5 $$\mathrm{Arcsech}z=\mathrm{Arccosh}\left(1/z\right),$$

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Defines:

$\mathrm{Arcsech}z$: general inverse hyperbolic secant function

Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(i), §4.37 and Ch.4

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4.37.8 $$\mathrm{arcsech}z=\mathrm{arccosh}\left(1/z\right).$$

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Defines:

$\mathrm{arcsech}z$: inverse hyperbolic secant function

Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.6.5

Permalink:

http://dlmf.nist.gov/4.37.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(ii), §4.37 and Ch.4

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4.37.14 $$\mathrm{arcsech}\left(-z\right)=\mp \pi \mathrm{i}+\mathrm{arcsech}z,$$ $\mathrm{\Im }z\gtrless 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iii), §4.37 and Ch.4

… ►For the corresponding results for $\mathrm{arccsch}z$, $\mathrm{arcsech}z$, and $\mathrm{arccoth}z$, use (4.37.7)–(4.37.9); compare §4.23(iv). …

15: 4.23 Inverse Trigonometric Functions

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4.23.17 $$\mathrm{arcsec}z=\frac{1}{2}\pi -\mathrm{arccsc}z.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccsc}z$: arccosecant function, $\mathrm{arcsec}z$: arcsecant function and $z$: complex variable

A&S Ref:

4.4.9

Permalink:

http://dlmf.nist.gov/4.23.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iii), §4.23 and Ch.4

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4.23.39 $$\mathrm{gd}\left(x\right)={\int }_0^x\mathrm{sech}tdt,$$ .

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Defines:

$\mathrm{gd}x$: Gudermannian function

Symbols:

$dx$: differential of $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral and $x$: real variable

Referenced by:

§4.40(ii)

Permalink:

http://dlmf.nist.gov/4.23.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(viii), §4.23 and Ch.4

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4.23.40 $$\mathrm{gd}\left(x\right)=2\arctan \left({\mathrm{e}}^x\right)-\frac{1}{2}\pi =\arcsin \left(\tanh x\right)=\mathrm{arccsc}\left(\coth x\right)=\arccos \left(\mathrm{sech}x\right)=\mathrm{arcsec}\left(\cosh x\right)=\arctan \left(\sinh x\right)=\mathrm{arccot}\left(\mathrm{csch}x\right).$$

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Symbols:

$\mathrm{gd}x$: Gudermannian function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\mathrm{arccsc}z$: arccosecant function, $\arccos z$: arccosine function, $\mathrm{arccot}z$: arccotangent function, $\mathrm{arcsec}z$: arcsecant function, $\arcsin z$: arcsine function, $\arctan z$: arctangent function and $x$: real variable

Referenced by:

§4.23(viii), §4.40(ii)

Permalink:

http://dlmf.nist.gov/4.23.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(viii), §4.23 and Ch.4

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4.23.41 $${\mathrm{gd}}^{-1}\left(x\right)={\int }_0^x\sec tdt,$$ .

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Defines:

${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sec z$: secant function and $x$: real variable

Referenced by:

§19.9(ii), §4.26(ii)

Permalink:

http://dlmf.nist.gov/4.23.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(viii), §4.23 and Ch.4

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4.23.42 $${\mathrm{gd}}^{-1}\left(x\right)=\ln \tan \left(\frac{1}{2}x+\frac{1}{4}\pi \right)=\ln \left(\sec x+\tan x\right)=\mathrm{arcsinh}\left(\tan x\right)=\mathrm{arccsch}\left(\cot x\right)=\mathrm{arccosh}\left(\sec x\right)=\mathrm{arcsech}\left(\cos x\right)=\mathrm{arctanh}\left(\sin x\right)=\mathrm{arccoth}\left(\csc x\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $\cot z$: cotangent function, $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{arccoth}z$: inverse hyperbolic cotangent function, $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\ln z$: principal branch of logarithm function, $\sec z$: secant function, $\sin z$: sine function, $\tan z$: tangent function and $x$: real variable

Referenced by:

§19.9(ii), §4.23(viii), §4.26(ii)

Permalink:

http://dlmf.nist.gov/4.23.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(viii), §4.23 and Ch.4

16: 24.7 Integral Representations

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24.7.1 $$B_{2n}={(-1)}^{n+1}\frac{4n}{1-2^{1-2n}}{\int }_0^{\mathrm{\infty }}\frac{t^{2n-1}}{{\mathrm{e}}^{2\pi t}+1}dt={(-1)}^{n+1}\frac{2n}{1-2^{1-2n}}{\int }_0^{\mathrm{\infty }}{t}^{2n-1}{\mathrm{e}}^{-\pi t}\mathrm{sech}\left(\pi t\right)dt,$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(i), §24.7 and Ch.24

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24.7.3 $$B_{2n}={(-1)}^{n+1}\frac{\pi }{1-2^{1-2n}}{\int }_0^{\mathrm{\infty }}{t}^{2n}{\mathrm{sech}}^2\left(\pi t\right)dt,$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(i), §24.7 and Ch.24

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24.7.6 $$E_{2n}={(-1)}^n{2}^{2n+1}{\int }_0^{\mathrm{\infty }}{t}^{2n}\mathrm{sech}\left(\pi t\right)dt.$$

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral, $n$: integer and $t$: real or complex

Referenced by:

§24.7(i)

Permalink:

http://dlmf.nist.gov/24.7.E6

Encodings:

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See also:

Annotations for §24.7(i), §24.7 and Ch.24

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17: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

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§10.19(ii) Debye’s Expansions

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§10.19(iii) Transition Region

… ►See also §10.20(i).

18: 22.11 Fourier and Hyperbolic Series

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22.11.10 $$\mathrm{dc}(z,k)-\frac{\pi }{2K}\sec \zeta =\frac{2\pi }{K}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{2n+1}\cos \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

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Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $q$: nome, $\sec z$: secant function, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.7

Permalink:

http://dlmf.nist.gov/22.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.11 $$\mathrm{nc}(z,k)-\frac{\pi }{2K{k}^{\prime }}\sec \zeta =-\frac{2\pi }{K{k}^{\prime }}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{2n+1}\cos \left((2n+1)\zeta \right)}{1+q^{2n+1}},$$

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Symbols:

$\mathrm{nc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $q$: nome, $\sec z$: secant function, $z$: complex, $k$: modulus, $k^{\prime }$: complementary modulus and $\zeta$: change of variable

A&S Ref:

16.23.8

Permalink:

http://dlmf.nist.gov/22.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.14 $$k^2{\mathrm{sn}}^2(z,k)=\frac{E^{\prime }}{K^{\prime }}-{\left(\frac{\pi }{2K^{\prime }}\right)}^2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left({\mathrm{sech}}^2\left(\frac{\pi }{2K^{\prime }}(z-2nK)\right)\right),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{sech}z$: hyperbolic secant function, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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19: 4.40 Integrals

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4.40.5 $$\int \mathrm{sech}xdx=\mathrm{gd}\left(x\right).$$

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Symbols:

$\mathrm{gd}x$: Gudermannian function, $dx$: differential of $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.81

Permalink:

http://dlmf.nist.gov/4.40.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.15 $$\int \mathrm{arcsech}xdx=x\mathrm{arcsech}x+\arcsin x,$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\int$: integral, $\arcsin z$: arcsine function and $x$: real variable

A&S Ref:

4.6.47 (modified. The first printing had an error.)

Permalink:

http://dlmf.nist.gov/4.40.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iv), §4.40 and Ch.4

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20: 22.5 Special Values

§22.5 Special Values

… ►For the other nine functions ratios can be taken; compare (22.2.10). … ►

§22.5(ii) Limiting Values of $k$

… ►In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. … ►