Function that relates the circular functions and hyperbolic functions without using complex numbers
Graph of the Gudermannian function
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]
gd
x
=
∫
0
x
1
cosh
t
d
t
.
{\displaystyle \operatorname {gd} x=\int _{0}^{x}{\frac {1}{\cosh t}}\,dt.}
Properties [ edit ]
Alternative definitions [ edit ]
gd
x
=
arcsin
(
tanh
x
)
=
arctan
(
sinh
x
)
=
arccsc
(
coth
x
)
=
sgn
(
x
)
⋅
arccos
(
sech
x
)
=
sgn
(
x
)
⋅
arcsec
(
cosh
x
)
=
2
arctan
[
tanh
(
1
2
x
)
]
=
2
arctan
(
e
x
)
−
1
2
π
.
{\displaystyle {\begin{aligned}\operatorname {gd} x&=\arcsin \left(\tanh x\right)=\arctan(\sinh x)=\operatorname {arccsc} (\coth x)\\&=\operatorname {sgn} (x)\cdot \arccos \left(\operatorname {sech} x\right)=\operatorname {sgn} (x)\cdot \operatorname {arcsec} (\cosh x)\\&=2\arctan \left[\tanh \left({\tfrac {1}{2}}x\right)\right]\\&=2\arctan(e^{x})-{\tfrac {1}{2}}\pi .\end{aligned}}}
Some identities [ edit ]
sin
(
gd
x
)
=
tanh
x
;
csc
(
gd
x
)
=
coth
x
;
cos
(
gd
x
)
=
sech
x
;
sec
(
gd
x
)
=
cosh
x
;
tan
(
gd
x
)
=
sinh
x
;
cot
(
gd
x
)
=
csch
x
;
tan
(
1
2
gd
x
)
=
tanh
(
1
2
x
)
.
{\displaystyle {\begin{aligned}\sin(\operatorname {gd} x)=\tanh x;\quad &\csc(\operatorname {gd} x)=\coth x;\\\cos(\operatorname {gd} x)=\operatorname {sech} x;\quad &\sec(\operatorname {gd} x)=\cosh x;\\\tan(\operatorname {gd} x)=\sinh x;\quad &\cot(\operatorname {gd} x)=\operatorname {csch} x;\\\tan \left({\tfrac {1}{2}}\operatorname {gd} x\right)=\tanh \left({\tfrac {1}{2}}x\right).\end{aligned}}}
Inverse [ edit ]
Graph of the inverse Gudermannian function
gd
−
1
x
=
∫
0
x
1
cos
t
d
t
−
π
/
2
<
x
<
π
/
2
=
ln
|
1
+
sin
x
cos
x
|
=
1
2
ln
|
1
+
sin
x
1
−
sin
x
|
=
ln
|
1
+
tan
x
2
1
−
tan
x
2
|
=
ln
|
tan
x
+
sec
x
|
=
ln
|
tan
(
x
2
+
π
4
)
|
=
artanh
(
sin
x
)
=
arsinh
(
tan
x
)
=
2
arctanh
(
tan
x
2
)
=
arcoth
(
csc
x
)
=
arcsch
(
cot
x
)
=
sgn
(
x
)
arcosh
(
sec
x
)
=
sgn
(
x
)
arsech
(
cos
x
)
=
−
i
gd
(
i
x
)
{\displaystyle {\begin{aligned}\operatorname {gd} ^{-1}x&=\int _{0}^{x}{\frac {1}{\cos t}}\,dt\qquad -\pi /2<x<\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)\\&=2\operatorname {arctanh} \left(\tan {\frac {x}{2}}\right)\\&=\operatorname {arcoth} (\csc x)=\operatorname {arcsch} (\cot x)\\&=\operatorname {sgn} (x)\operatorname {arcosh} (\sec x)=\operatorname {sgn} (x)\operatorname {arsech} (\cos x)\\&=-i\operatorname {gd} (ix)\end{aligned}}}
(See inverse hyperbolic functions .)
Some identities [ edit ]
sinh
(
gd
−
1
x
)
=
tan
x
;
csch
(
gd
−
1
x
)
=
cot
x
;
cosh
(
gd
−
1
x
)
=
sec
x
;
sech
(
gd
−
1
x
)
=
cos
x
;
tanh
(
gd
−
1
x
)
=
sin
x
;
coth
(
gd
−
1
x
)
=
csc
x
.
{\displaystyle {\begin{aligned}\sinh(\operatorname {gd} ^{-1}x)=\tan x;\quad &\operatorname {csch} (\operatorname {gd} ^{-1}x)=\cot x;\\\cosh(\operatorname {gd} ^{-1}x)=\sec x;\quad &\operatorname {sech} (\operatorname {gd} ^{-1}x)=\cos x;\\\tanh(\operatorname {gd} ^{-1}x)=\sin x;\quad &\coth(\operatorname {gd} ^{-1}x)=\csc x.\end{aligned}}}
Derivatives [ edit ]
d
d
x
gd
x
=
sech
x
;
d
d
x
gd
−
1
x
=
sec
x
.
{\displaystyle {\frac {d}{dx}}\operatorname {gd} x=\operatorname {sech} x;\quad {\frac {d}{dx}}\;\operatorname {gd} ^{-1}x=\sec x.}
History [ edit ]
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.[4] 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
S
i
n
{\displaystyle {\mathfrak {Sin}}}
and
C
o
s
{\displaystyle {\mathfrak {Cos}}}
).
The notation gd was introduced by Cayley[5] where he starts by calling gd. u the inverse of the integral of the secant function :
u
=
∫
0
ϕ
sec
t
d
t
=
ln
(
tan
(
1
4
π
+
1
2
ϕ
)
)
{\displaystyle u=\int _{0}^{\phi }\sec t\,dt=\ln \left(\tan \left({\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\phi \right)\right)}
and then derives "the definition" of the transcendent:
gd
u
=
i
−
1
ln
(
tan
(
1
4
π
+
1
2
u
i
)
)
{\displaystyle \operatorname {gd} u=i^{-1}\ln \left(\tan \left({\tfrac {1}{4}}\pi +{\tfrac {1}{2}}ui\right)\right)}
observing immediately that it is a real function of u .
Applications [ edit ]
1
2
π
−
gd
x
{\displaystyle {\tfrac {1}{2}}\pi -\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 .[6]
The Gudermannian also appears in a moving mirror solution of the dynamical Casimir effect .[8]
See also [ edit ]
References [ edit ]
^
Olver, F. W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W., eds. (2010), NIST Handbook of Mathematical Functions , Cambridge University Press. Section 4.23(viii) .
^
CRC Handbook of Mathematical Sciences 5th ed. pp. 323–325
^
Weisstein, Eric W. "Gudermannian" . MathWorld .
^
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
^
John S. Robertson (1997). "Gudermann and the Simple Pendulum". The College Mathematics Journal . 28 (4): 271–276. doi :10.2307/2687148 . JSTOR 2687148 . Review .
^ 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 .