Gudermannian function

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1: 4.46 Tables

§4.46 Tables

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2: 19.10 Relations to Other Functions

… ►In each case when $y=1$, the quantity multiplying $R_C$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. ►For relations to the Gudermannian function $\mathrm{gd}\left(x\right)$ and its inverse ${\mathrm{gd}}^{-1}\left(x\right)$ (§4.23(viii)), see (19.6.8) and ►

19.10.2 $$(\sinh \varphi )R_C(1,{\cosh }^2\varphi )=\mathrm{gd}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{gd}x$: Gudermannian function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.6(ii)

Permalink:

http://dlmf.nist.gov/19.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.10(ii), §19.10 and Ch.19

3: 19.6 Special Cases

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19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.10(ii)

Permalink:

http://dlmf.nist.gov/19.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(ii), §19.6 and Ch.19

►For the inverse Gudermannian function ${\mathrm{gd}}^{-1}\left(\varphi \right)$ see §4.23(viii). …

4: 4.23 Inverse Trigonometric Functions

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§4.23(viii) Gudermannian Function

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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

… ►The inverse Gudermannian function is given by ►

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

… ►

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

5: 22.16 Related Functions

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22.16.4 $$\mathrm{am}(x,0)=x,$$

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function and $x$: real

Permalink:

http://dlmf.nist.gov/22.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

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22.16.5 $$\mathrm{am}(x,1)=\mathrm{gd}\left(x\right).$$

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Symbols:

$\mathrm{gd}x$: Gudermannian function, $\mathrm{am}(x,k)$: Jacobi’s amplitude function and $x$: real

Permalink:

http://dlmf.nist.gov/22.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

►For the Gudermannian function $\mathrm{gd}\left(x\right)$ see §4.23(viii). … ►

22.16.8 $$\mathrm{am}(x,k)=\mathrm{gd}x-\frac{1}{4}k_{}^{\prime }{}_{}{}^2(x-\sinh x\cosh x)\mathrm{sech}x+O\left(k_{}^{\prime }{}_{}{}^4\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{gd}x$: Gudermannian function, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.4

Permalink:

http://dlmf.nist.gov/22.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

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6: 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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7: 4.26 Integrals

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4.26.5 $$\int \sec xdx={\mathrm{gd}}^{-1}\left(x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sec z$: secant function and $x$: real variable

A&S Ref:

4.3.117

Permalink:

http://dlmf.nist.gov/4.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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8: 19.9 Inequalities

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19.9.11 $$\varphi \le F(\varphi ,k)\le \min (\varphi /\mathrm{\Delta },{\mathrm{gd}}^{-1}\left(\varphi \right)),$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Permalink:

http://dlmf.nist.gov/19.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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9: 22.1 Special Notation

… ►The notation $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$ is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). …