Gudermannian function

Gudermannian function
General information
General definition	$\operatorname {gd} =\int _{0}^{x}\operatorname {sech} t\,\mathrm {d} t$
Domain, Codomain and Image
Domain	(−∞, +∞) ‡
Codomain	[–π/2, π/2] ‡
Basic features
Parity	odd
Specific values
At zero	0
Value at +∞	+π/2
Value at −∞	–π/2
Specific features
Asymptote	±π/2
Root	0
Fixed point	0
Inverse	$f^{-1}(y)=\int _{0}^{y}\sec t\,\mathrm {d} t$
Derivative	$f'(x)=\operatorname {sech} x$
‡ For real numbers.

The Gudermannian function, named after Christoph Gudermann (1798–1852), relates the circular functions and hyperbolic functions without explicitly using complex numbers.

It is defined for all $\,x\,$ by

$\operatorname {gd} x=\int _{0}^{x}\operatorname {sech} t\,\mathrm {d} t.$

The inverse of the Gudermannian function, $\;\operatorname {gd} ^{-1}x\;,$ is sometimes known as the Lambertian function and written $\;\operatorname {lam} x\;.$

Properties

Alternative definitions

${\begin{aligned}\operatorname {gd} x&=\arcsin \left(\tanh x\right)=\arctan(\sinh x)=\operatorname {arccsc} (\coth x)\\[6pt]&=\operatorname {sgn}(x)\arccos \left(\operatorname {sech} x\right)=\operatorname {sgn}(x)\operatorname {arcsec} (\cosh x)\\[6pt]&=2\arctan \left(\tanh {\frac {x}{2}}\right)\\[6pt]&=2\arctan \left(e^{x}\right)-{\frac {\pi }{2}}=-i\ln \left(\operatorname {sech} x+i\tanh x\right)\\[6pt]&=-{\frac {i}{2}}\ln \left({\frac {1+i\sinh x}{1-i\sinh x}}\right)=-i\ln \left({\frac {1+i\sinh x}{\cosh x}}\right)\\[6pt]&=-i\ln \left({\frac {1+i\tanh {\frac {x}{2}}}{1-i\tanh {\frac {x}{2}}}}\right)=-i\ln \left(i\tanh \left({\frac {x}{2}}-{\frac {i\pi }{4}}\right)\right)~.\end{aligned}}$

Some identities

${\begin{aligned}\sin(\operatorname {gd} x)&=\tanh x\;;&\csc(\operatorname {gd} x)&=\coth x\;;\\[6pt]\cos(\operatorname {gd} x)&=\operatorname {sech} x\;;&\sec(\operatorname {gd} x)&=\cosh x\;;\\[6pt]\tan(\operatorname {gd} x)&=\sinh x\;;&\cot(\operatorname {gd} x)&=\operatorname {csch} x\;;\\[6pt]\tan \left({\frac {\operatorname {gd} x}{2}}\right)&=\tanh {\frac {x}{2}}~.\end{aligned}}$

Inverse

Graph of the inverse Gudermannian function

${\begin{aligned}\operatorname {gd} ^{-1}x&=\int _{0}^{x}\sec t\,\mathrm {d} t\qquad {\text{ for }}\quad -{\frac {\pi }{2}}<x<{\frac {\pi }{2}}\\[8pt]&=\ln \left|{\frac {1+\sin x}{\cos x}}\right|={\frac {1}{2}}\ln \left|{\frac {1+\sin x}{1-\sin x}}\right|=\ln \left|{\frac {1+\tan {\frac {x}{2}}}{1-\tan {\frac {x}{2}}}}\right|\\[8pt]&=\ln \left|\tan x+\sec x\right|=\ln \left|\tan \left({\frac {x}{2}}+{\frac {\pi }{4}}\right)\right|\\[8pt]&=\operatorname {artanh} (\sin x)=\operatorname {arsinh} (\tan x)\\[6pt]&=2\operatorname {artanh} \left(\tan {\frac {x}{2}}\right)\\[6pt]&=\operatorname {arcoth} (\csc x)=\operatorname {arcsch} (\cot x)\\[6pt]&=\operatorname {sgn} (x)\operatorname {arcosh} (\sec x)=\operatorname {sgn} (x)\operatorname {arsech} (\cos x)\\[6pt]&=-i\operatorname {gd} (ix)~.\end{aligned}}$

Some identities

${\begin{aligned}\sinh \left(\operatorname {gd} ^{-1}x\right)&=\tan x\;;&\operatorname {csch} \left(\operatorname {gd} ^{-1}x\right)&=\cot x\;;\\[6pt]\cosh \left(\operatorname {gd} ^{-1}x\right)&=\sec x\;;&\operatorname {sech} \left(\operatorname {gd} ^{-1}x\right)&=\cos x\;;\\[6pt]\tanh \left(\operatorname {gd} ^{-1}x\right)&=\sin x\;;&\coth \left(\operatorname {gd} ^{-1}x\right)&=\csc x~.\end{aligned}}$

Derivatives

${\frac {\operatorname {d} }{\operatorname {d} x}}\operatorname {gd} x=\operatorname {sech} x\;;\quad {\frac {\operatorname {d} }{\operatorname {d} x}}\operatorname {gd} ^{-1}x=\sec x~.$

History

The function was introduced by Johann Heinrich Lambert in the 1760s at the same time as the hyperbolic functions. He called it the "transcendent angle," and it went by various names until 1862 when Arthur Cayley suggested it be given its current name as a tribute to Gudermann's work in the 1830s on the theory of special functions. Gudermann had published articles in Crelle's Journal that were later collected in a book which expounded $\;\sinh \;$ and $\;\cosh \;$ to a wide audience (although represented by the symbols $\;{\mathfrak {Sin}}\;$ and $\;{\mathfrak {Cos}}\;$).

The notation $\;\operatorname {gd} \;$ was introduced by Cayley where he starts by calling $\;\operatorname {gd} u\;$ the inverse of the integral of the secant function:

$\operatorname {gd} ^{-1}\phi =u=\int _{0}^{\phi }\sec t\,\mathrm {d} t=\ln \left(\tan \left({\frac {\pi }{4}}+{\frac {\phi }{2}}\right)\right)\qquad {\text{ for }}\quad {\bigl |}\phi {\bigr |}<{\frac {\pi }{2}}~,$

and then derives "the definition" of the transcendent:

$\operatorname {gd} u=-i\ln \left(\tan \left({\frac {\pi }{4}}+{\frac {ui}{2}}\right)\right)$

observing immediately that it is a real function of $\;u~.$

Applications

• The angle of parallelism function in hyperbolic geometry is defined by $\;{\tfrac {\pi }{2}}-\operatorname {gd} x~.$

• On a Mercator projection a line of constant latitude is parallel to the equator (on the projection) and is displaced by an amount proportional to the inverse Gudermannian of the latitude.

• The Gudermannian (with a complex argument) may be used to define the transverse Mercator projection.

• The Gudermannian appears in a non-periodic solution of the inverted pendulum.

• The Gudermannian also appears in a moving mirror solution of the dynamical Casimir effect.

See also

• Hyperbolic secant distribution
• Integral of the secant function
• Inverse hyperbolic function
• Mercator projection
• Tangent half-angle formula
• Tractrix
• Trigonometric identity