Gudermannian function

Gudermannian function
Gudermannian function
General information
General definition
Domain and Range
Domain	(−∞, +∞) a
Codomain	[–π/2, π/2] a
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
Derivative
	a 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[1][2][3]

\operatorname {gd} x=\int _{0}^{x}\operatorname {sech} t\,dt.

The inverse of the Gudermannian function, $\operatorname {gd} ^{-1}x$ , is sometimes known as the Lambertian function and written $\operatorname {lam} x$ .[4]

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\,dt\quad {\text{for}}\;-{\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 {d}{dx}}\operatorname {gd} x=\operatorname {sech} x;\quad {\frac {d}{dx}}\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.[5] Gudermann had published articles in Crelle's Journal that were collected in Theorie der potenzial- oder cyklisch-hyperbolischen Functionen (1833), a book which expounded sinh and cosh to a wide audience (under the guises of ${\mathfrak {Sin}}$ and ${\mathfrak {Cos}}$ ).

The notation $\operatorname {gd}$ was introduced by Cayley[6] where he starts by calling $\operatorname {gd} u$ the inverse of the integral of the secant function:

u=\int _{0}^{\phi }\sec t\,dt=\ln \left(\tan \left({\frac {\pi }{4}}+{\frac {\phi }{2}}\right)\right)

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 in the definition of the transverse Mercator projection.[7]

The Gudermannian appears in a non-periodic solution of the inverted pendulum.[8]

The Gudermannian also appears in a moving mirror solution of the dynamical Casimir effect.[9]

References

Olver, F. W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W., eds. (2010), NIST Handbook of Mathematical Functions, Cambridge University Press
CRC Handbook of Mathematical Sciences 5th ed. pp. 323–325
Weisstein, Eric W. "Gudermannian". MathWorld.
Lee, L.P. (1976). Conformal Projections Based on Elliptic Functions. Supplement No. 1 to Canadian Cartographer, Vol 13. (Designated as Monograph 16)
George F. Becker, C. E. Van Orstrand. Hyperbolic functions. Read Books, 1931. Page xlix. Scanned copy available at archive.org
Cayley, A. (1862). "On the transcendent gd. u". Philosophical Magazine. 4th Series. 24 (158): 19–21. doi:10.1080/14786446208643307.
Osborne, P (2013), The Mercator projections, p74{{citation}}: CS1 maint: postscript (link)
John S. Robertson (1997). "Gudermann and the Simple Pendulum". The College Mathematics Journal. 28 (4): 271–276. doi:10.2307/2687148. JSTOR 2687148.
Good, Michael R. R.; Anderson, Paul R.; Evans, Charles R. (2013). "Time dependence of particle creation from accelerating mirrors". Physical Review D. 88 (2): 025023. arXiv:1303.6756. Bibcode:2013PhRvD..88b5023G. doi:10.1103/PhysRevD.88.025023. S2CID 56113100.

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[1] Olver, F. W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W., eds. (2010), NIST Handbook of Mathematical Functions, Cambridge University Press

[2] CRC Handbook of Mathematical Sciences 5th ed. pp. 323–325

[weinstein-3] Weisstein, Eric W. "Gudermannian". MathWorld.

[lee_exact-4] Lee, L.P. (1976). Conformal Projections Based on Elliptic Functions. Supplement No. 1 to Canadian Cartographer, Vol 13. (Designated as Monograph 16)

[5] George F. Becker, C. E. Van Orstrand. Hyperbolic functions. Read Books, 1931. Page xlix. Scanned copy available at archive.org

[6] Cayley, A. (1862). "On the transcendent gd. u". Philosophical Magazine. 4th Series. 24 (158): 19–21. doi:10.1080/14786446208643307.

[7] Osborne, P (2013), The Mercator projections, p74{{citation}}: CS1 maint: postscript (link)

[8] John S. Robertson (1997). "Gudermann and the Simple Pendulum". The College Mathematics Journal. 28 (4): 271–276. doi:10.2307/2687148. JSTOR 2687148.

[9] Good, Michael R. R.; Anderson, Paul R.; Evans, Charles R. (2013). "Time dependence of particle creation from accelerating mirrors". Physical Review D. 88 (2): 025023. arXiv:1303.6756. Bibcode:2013PhRvD..88b5023G. doi:10.1103/PhysRevD.88.025023. S2CID 56113100.

Gudermannian function

Properties

Alternative definitions

Some identities

Inverse

Some identities

Derivatives

History

Applications

See also

References