Function of Gudermann

In Mathematical, the function of Gudermann , noted Gd , named in the honor of Christoph Gudermann (1798 - 1852), establishes the link between the circular trigonometry and the hyperbolic trigonometry without utilizing the complex numbers.

Definition

The function of Gudermann is defined on the whole of realities by:

$\ begin {align} {\ rm {Gd}} (T) &= \ int_0^t \ frac {of the} {\ cosh U} \ \$

&= \ arcsin \ left (\ tanh T \ right) = \ mbox {sign} (T). \ arccos \ left (\ mbox {sech} T \ right) \ \ \ &= \ arctan \ left (\ sinh T \ right) = \ mbox {sign} (T). \ mbox {arcsec} \ left (\ cosh T \ right) \ \ &= \ mbox {arccot} \ left (\ mbox {csch} T \ right) = \ mbox {arccsc} \ left (\ coth T \ right) \ \ &=2 \ arctan \ left (\ tanh \ frac {T} {2} \ right) =2 \ arctan e^t- \ frac {\ pi} {2}. \ end {align} \, \!

The Dérivée from the function of Gudermann is the function $1/\ cosh = \ mbox {sech} \!$ .

Reality $\ theta= \ mbox {Gd} (T) \!$ , called sometimes gudermannien of $t \!$ , is connected to this last by the relations:

$\ begin {align} {\ color {white} \ dowry} &= \ tanh T; \ quad \ cos \ theta =1/\ cosh t= \ mbox {sech} T; \ \
\ tan \ theta&= \ sinh T; \ quad \ tan \ frac {\ theta} {2} = \ tanh \ frac {T} {2}. \ end {align} \, \!$

Opposite function

The opposite function of Gudermann is defined on

] - \ pi/2, \ pi/2 by:

\begin{align}

\ mbox {arcgd} (\ theta) &= {\ rm {Gd}} ^ {- 1} (\ theta) = \ int_0^ \ theta \ frac {of the} {\ cos U}, \ \ &= \ mbox {arctanh} (\ sin \ theta) = \ mbox {sign} (\ theta) {.}\ mbox {arccosh} (\ dry \ theta), \ \ &= {} \ ln (\ tan \ theta+ \ dry \ theta) = \ ln \ left (\ tan \ left (\ frac {\ theta} {2} + \ frac {\ pi} {4} \ right) \ right), \ \ &= {} \ frac {1} {2} \ ln \ frac {1+ \ sin \ theta} {1 \ sin \ theta}. \ end {align} \, \!

The Dérivée from the opposite function of Gudermann is the function $1/\ cos = \ mbox {dry} \!$ .

Applications

cordonnées of Mercator of a point of the sphere is defined by $x=longitude \!$ and $y= \ mbox {Gd} ^ {- 1} (latitude) \!$ .

They are thus defined so that the Loxodromie S of the sphere are represented by lines in the

x plan, there \!

change of variable $\ theta= \ mbox {Gd} (T) \!$ makes it possible to transform integrals of circular functions into hyperbolic integrals of functions; for example, $\ int_0^ {\ pi/2} {(\ cos \ theta) ^n D \ theta} = \ int_0^ {+ \ infty} {(\ mbox {sech} T) ^n dt}$ .
This explains why one can choose circular functions or hyperbolic during change of variables in the calculation of integrals:

When one meets

\ sqrt {1-x^2}

, one uses

x= \ cos \ theta \!

x=1/\ cosh T \!

, and one uses also

x= \ sin \ theta \!

x= \ tanh T \!

When one meets $\ sqrt {1+x^2}$ , one uses $x= \ tan \ theta \!$ or $x= \ sinh T \!$ .

Parameterization of a circle or a hyperbolic line.

If one poses

\ begin {boxes} \ begin {align} x&= \ cos \ theta = 1 {\ cosh T} \ \ there & = \ sin \ theta = \ tanh T \ end {align} \ end {boxes}

, one has obviously a parameterization half-circle of ray 1 in the half-plane

x>0 \!

;

\ theta \!

is curvilinear distance in the Euclidean half-plane enters the point

(X, there) \!

and the point

(1,0) \!

, and

t \!

is also a distance, but measured between these two points in the half-plane considered as half-plane of Poincaré for the hyperbolic Geometry.

See too

the projection of Mercator
elliptic functions of Jacobi

References

CRC Handbook off Mathematical Sciences 5th ED. p 323-5.

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