Gudermann function

The Gudermann function in the real

The Gudermann function , named after Christoph Gudermann (1798–1852), creates a connection between the trigonometric and the hyperbolic functions without using the complex numbers . The Gudermann function is an intermediate function in order to obtain an exponential or a hyperbolic function for an argument by applying it to a circular function. It was first described by the Swiss mathematician Johann Heinrich Lambert around 1760, when he wanted to find a direct dependence of the Euler's number on the circle number in experiments with continued fractions for the tangent . He was unable to specify a non-trivial, analytical form for this intermediate function, which he called the “transcendent angle”, nor could he point out any further benefits, since the relationship between and could not be derived from it. ${\displaystyle x}$ ${\displaystyle \pi }$ ${\displaystyle \pi }$ ${\displaystyle e}$

The Gudermann function

Around 1830, while investigating elliptic integrals , Christoph Gudermann accidentally came across a real, non-trivial relationship between circular and exponential functions, which could also be applied to all trigonometric functions. This enabled Lambert's intermediate function to be presented in an analytical way, but received little attention or recognition (see reception of Christoph Gudermann's work). The term Gudermann function was introduced in 1862 by Arthur Cayley when he referred to Gudermann's preliminary work in his own work on elliptical integrals.

The function is defined for by: ${\displaystyle x\in \mathbb {R} }$

${\displaystyle \operatorname {gd} \,(x):=\int _{0}^{x}{\frac {\mathrm {d} t}{\cosh t}}}$

The integral can be resolved with the substitutions and consequently with the differential : ${\displaystyle \operatorname {gd} (x)=\varphi ,\,e^{t}=s}$ ${\displaystyle \mathrm {d} t=e^{-t}\mathrm {d} s=s^{-1}\mathrm {d} s}$

${\displaystyle {\begin{aligned}\varphi ={\varphi }\,(x)&:=\int _{0}^{x}{\frac {\mathrm {d} t}{\cosh t}}=\int _{0}^{x}{\frac {2\,\mathrm {d} t}{e^{t}+e^{-t}}}=\int _{e^{0}}^{e^{x}}{\frac {2s^{-1}\mathrm {d} s}{s+s^{-1}}}=\int _{1}^{e^{x}}{\frac {2\,\mathrm {d} s}{s^{2}+1}}=2\arctan(s){\bigg |}_{1}^{e^{x}}\\&:=2\arctan({e^{x}})-{\tfrac {\pi }{2}}={\tfrac {\pi }{2}}-2\arctan({e^{-x}})\\\end{aligned}}}$

From this explicit formula it can be seen that the value of the Gudermann function represents an angle and the argument represents a scalar for the exponential function. For all further formulations of the Gudermann function with real numbers, the separation of the circular function with an angle and the exponential function with a scalar is retained. Solving for the exponential function results in an expression for half the angle ${\displaystyle \varphi =\varphi \,(x)=\operatorname {gd} (x)}$ ${\displaystyle x}$

${\displaystyle {e^{x}}:=\tan({\tfrac {\pi }{4}}+{\tfrac {\varphi }{2}})={\frac {\tan {\tfrac {\pi }{4}}+\tan {\tfrac {\varphi }{2}}}{1-\tan {\tfrac {\pi }{4}}\tan {\tfrac {\varphi }{2}}}}:={\frac {1+\tan {\tfrac {\varphi }{2}}}{1-\tan {\tfrac {\varphi }{2}}}}\quad {\text{(Gl. 1)}}}$

and from that one gets a relation to the half argument

${\displaystyle \tan {\tfrac {\varphi }{2}}={\frac {e^{x}-1}{e^{x}+1}}={\frac {e^{\tfrac {x}{2}}-e^{-{\tfrac {x}{2}}}}{e^{\tfrac {x}{2}}+e^{-{\tfrac {x}{2}}}}}:=\tanh {\tfrac {x}{2}}\quad {\text{(Gl. 2)}}}$

In particular, the simple representation in Eq. (2) is the central relationship of the Gudermann function, which shows the relationship between an angle for an angle function and a scalar for an exponential function. It leads to the following alternative definitions of the Gudermann function

${\displaystyle {\begin{aligned}{\varphi }\,(x):=\operatorname {gd} x&=2\arctan \left[\tanh \left({\tfrac {1}{2}}x\right)\right]\\&=\arcsin \left(\tanh x\right)=\arctan(\sinh x)=\operatorname {arccsc} (\coth x)\\&=\operatorname {sgn} (x)\cdot \arccos \left(\operatorname {sech} x\right)=\operatorname {sgn} (x)\cdot \operatorname {arcsec} (\cosh x)\end{aligned}}}$

It corresponds roughly to the relationship that Lambert examined

${\displaystyle \tanh {\tfrac {x}{2}}={\tfrac {1}{\mathrm {i} }}\tan {\tfrac {\mathrm {i} x}{2}}=\tan {\tfrac {\varphi (x)}{2}}.}$

The transition from half to whole angles and arguments is made by inserting Eq. (2) in the addition theorem for the tangent of the double angle:

${\displaystyle \tan \varphi =\tan {\tfrac {2\varphi }{2}}:={\frac {2\tan {\tfrac {\varphi }{2}}}{1-\tan ^{2}{\tfrac {\varphi }{2}}}}:={\frac {2\tanh {\tfrac {x}{2}}}{1-\tanh ^{2}{\tfrac {x}{2}}}}={\frac {2\sinh {\tfrac {x}{2}}\cosh {\tfrac {x}{2}}}{\cosh ^{2}{\tfrac {x}{2}}-\sinh ^{2}{\tfrac {x}{2}}}}=2\sinh {\tfrac {x}{2}}\cosh {\tfrac {x}{2}}:=\sinh {x}\quad {\text{(Gl. 3)}}}$

This equation is another key representation. From this, by applying the relevant relationships for angle and hyperbolic functions, the left and right sides can be expressed independently of one another with other angle and hyperbolic functions. Since these relationships are usually of a simple algebraic structure, they can also be easily solved algebraically and the other side of the equation can then usually also be simplified considerably (see below some examples for whole angles and arguments). Of particular interest are the representations in which the tangent or tangent hyperbolic nerve simply occurs, since their inverse functions can be calculated particularly easily with numerical means. So is

${\displaystyle \sin \varphi :=\tanh {x}\quad {\text{(Gl. 4)}}}$

The most important of the possible alternative representations. But any other resolution of the equations formed in this way is also a possible representation of the Gudermann function. ${\displaystyle \varphi =\varphi (x)=\operatorname {gd} (x)}$

The inverse Gudermann function

The inverse function of the Gudermann function can on the one hand be obtained by solving one of its equations and usually has to be represented by means of a logarithm. However, it is also defined independently of the above equations and its derivation follows in an analogous manner to the derivation of the Gudermann function, but complex calculations are necessary for the intermediate steps. ${\displaystyle x}$

The following applies to: ${\displaystyle -{\tfrac {\pi }{2}}<\varphi <{\tfrac {\pi }{2}}}$

${\displaystyle {\begin{aligned}\operatorname {arcgd} \,(\varphi )&:=\operatorname {gd} ^{-1}(\varphi )=\int _{0}^{\varphi }{\frac {\mathrm {d} t}{\cos t}}=\int _{0}^{\varphi }{\frac {2\mathrm {d} t}{e^{\mathrm {i} t}+e^{-\mathrm {i} t}}}=\ldots \\&=\ln \tan \left({\tfrac {\pi }{4}}+{\tfrac {\varphi }{2}}\right)=\ln {\frac {1+\tan {\tfrac {\varphi }{2}}}{1-\tan {\tfrac {\varphi }{2}}}}=\ln {\frac {\cos {\tfrac {\varphi }{2}}+\sin {\tfrac {\varphi }{2}}}{\cos {\tfrac {\varphi }{2}}-\sin {\tfrac {\varphi }{2}}}}\\&=\ln \cot \left({\tfrac {\pi }{4}}-{\tfrac {\varphi }{2}}\right)\quad {\text{(nach Gl. 1)}}\\&=\ln {\frac {1+\tan {\tfrac {\varphi }{2}}}{1-\tan {\tfrac {\varphi }{2}}}}\\&={2}\operatorname {artanh} (\tan {\tfrac {\varphi }{2}})\quad {\text{(nach Gl. 2)}}\\&=\ln {\frac {\left(\cos {\tfrac {\varphi }{2}}+\sin {\tfrac {\varphi }{2}}\right)^{2}}{\cos ^{2}{\tfrac {\varphi }{2}}-\sin ^{2}{\tfrac {\varphi }{2}}}}=\ln {\frac {\cos ^{2}{\tfrac {\varphi }{2}}+\sin ^{2}{\tfrac {\varphi }{2}}+2\cos {\tfrac {\varphi }{2}}\sin {\tfrac {\varphi }{2}}}{1-\sin ^{2}{\tfrac {\varphi }{2}}-\sin ^{2}{\tfrac {\varphi }{2}}}}=\ln {\frac {1+2\cos {\tfrac {\varphi }{2}}\sin {\tfrac {\varphi }{2}}}{1-2\sin ^{2}{\tfrac {\varphi }{2}}}}=\ln {\frac {1+\sin \varphi }{\cos \varphi }}=\ln(\tan \varphi +\sec \varphi )\\&=\ln {\frac {1+\sin \varphi }{\sqrt {1-\sin ^{2}\varphi }}}=\ln {\sqrt {\frac {1+\sin \varphi }{1-\sin \varphi }}}={\frac {1}{2}}\ln {\frac {1+\sin \varphi }{1-\sin \varphi }}\\&=\operatorname {artanh} (\sin \varphi )\quad {\text{(nach Gl. 4)}}\\&=\operatorname {arsinh} (\tan \varphi )\quad {\text{(nach Gl. 3)}}\\&=\operatorname {arcosh} (\sec \varphi )\\\end{aligned}}}$

For the numerical evaluation of the inverse Gudermann function, the representation according to Eq. (4) in particular for the middle two-thirds of the definition range suitable: . A representation with half angles is to be preferred at the edges, because these do not work in the flat areas of the extremes of sine and / or cosine and therefore have a higher numerical sharpness. Similar considerations must be made for the evaluation of the Gudermann function . ${\displaystyle -{\tfrac {\pi }{3}}<\varphi <{\tfrac {\pi }{3}}}$ ${\displaystyle \varphi (x)=\operatorname {gd} (x)}$

More relationships

The derivation of the Gudermann function and its inversion are, according to the integrands, their definition integrals:

${\displaystyle {\begin{aligned}&{\frac {\mathrm {d} }{\mathrm {d} x}}\,\operatorname {gd} (x)=\operatorname {sech} x=\cos(\operatorname {gd} x),\\&{\frac {\mathrm {d} }{\mathrm {d} \varphi }}\,\operatorname {arcgd} \,(\varphi )=\sec \varphi \end{aligned}}}$

Particularly noteworthy is the identity for complex calculations:

${\displaystyle \operatorname {gd} \left({\mathrm {i} }\cdot x\right)={\mathrm {i} }\cdot \operatorname {arcgd} \left(x\right)}$

The connection between circle and hyperbolic functions is essentially given by:

${\displaystyle {\begin{aligned}\sinh(x)&=\tan(\operatorname {gd} \,x)&=&\tan(\varphi )&\quad &{\text{(Gl. 3)}}\\\cosh(x)&=\sec(\operatorname {gd} \,x)&=&\sec(\varphi )\\\tanh(x)&=\sin(\operatorname {gd} \,x)&=&\sin(\varphi )&\quad &{\text{(Gl. 4)}}\\\operatorname {sech} (x)&=\cos(\operatorname {gd} \,x)&=&\cos(\varphi )\\\operatorname {csch} (x)&=\cot(\operatorname {gd} \,x)&=&\cot(\varphi )\\\coth(x)&=\csc(\operatorname {gd} \,x)&=&\csc(\varphi )\\\end{aligned}}}$

Practical use

With the shown connections of circle and hyperbolic functions, mathematical expressions can be simplified if necessary.

Because of its simple derivatives, the Gudermann function and its inverse are suitable as substitutes for integral calculus. Gudermann used it for this purpose.

With the Gudermann function or its inverse, the angle of the geographical latitude is linked with the north-south component of the Mercator projection . The equations in particular are associated with the earth's radius ${\displaystyle \varphi }$ ${\displaystyle y_{N}}$ ${\displaystyle R_{E}}$

${\displaystyle {\begin{aligned}{\text{(Gl. 2)}}&\quad &\tan {\tfrac {\varphi }{2}}&=\tanh {\tfrac {y_{N}}{2R_{E}}}\\{\text{(Gl. 3)}}&\quad &\tan {\varphi }&=\sinh {\tfrac {y_{N}}{R_{E}}}\\{\text{(Gl. 4)}}&\quad &\sin {\varphi }&=\tanh {\tfrac {y_{N}}{R_{E}}}\\\end{aligned}}}$

significant. Since the local distortion of the Mercator projection also depends on the latitude, the relative projection distance from the equator to the latitude is the integral of all distortions over the circular arc ( meridional arc ) from the equator to ${\displaystyle \sec {\varphi }}$ ${\displaystyle {\tfrac {y_{N}}{R_{E}}}}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi }$

${\displaystyle {\frac {y_{N}(\varphi )}{R_{E}}}=\int _{0}^{\varphi }{\frac {\mathrm {d} t}{\cos t}}=\operatorname {artanh} (\sin \varphi ):=\operatorname {arcgd} (\varphi )}$

A representation of the inverse Gudermann function for half angles may be preferable for the evaluation.

See also

The hyperbolic tangent , or the logistic function based on it, shows a sigmoid curve which is very similar to the Gudermann function .

swell

• Eric W. Weisstein : Gudermann function . In: MathWorld (English).

Remarks

1. ↑ The inverse function of the tangent can be developed very efficiently from sine and cosine using the simplified Newton method without divisions, and the area tangent is expressed as a logarithm. This can also be calculated as the inverse of the exponential function using Newton's method or, even more elegantly and efficiently, using the cubically converging Halley method .
2. ↑ Curve comparison Gudermann vs. Hyperbolic tangent - normalized to gd (x) for WolframAlpha