Gudermannian function

The Gudermann function, named after Christoph Gudermann (1798-1852), establishes a connection between the trigonometric and hyperbolic functions without using complex numbers. The Gudermann function is an intermediate function to x to obtain for an argument by applying it to a loop function an exponential or a hyperbolic function. It was first described by the Swiss mathematician Johann Heinrich Lambert in 1760 when the latter was found in experiments with continued fractions for the tangent of a direct dependence of the Euler number of the circle number. He could for this from him, " transcendent angle" called intermediate function not specify a non- trivial analytic form and also reveal no further use, as that would bring the sought relationship between and not infer allowed.

The Gudermann function

In 1830 came Christoph Gudermann in the study of elliptic integrals by chance on a real, non-trivial relationship between the angular and exponential functions, which also could be applied to all trigonometric functions. This Lamberts intermediate function could be represented in analytical form, but found little attention and recognition (see reception of the work of Christopher Gudermann ). The term Gudermann function introduced by Arthur Cayley in 1862, when he moved to Guder 's preparatory work in their own work on elliptic integrals.

The function is defined by:

With the substitutions and, consequently, with the differential to the integral can be resolved:

From this explicit formula it can be seen that the value of Gudermann function represents an angle and the argument a scalar for the exponential function. For all other formulations of Gudermann function with real numbers, the separation of circular function with an angle and exponential function is maintained by a scalar. Solving for the exponential function results in an expression for the half angle

And from this one obtains a relationship with the half argument

In particular, the simple representation of the relationship of the central Gudermann function showing the relationship between an angle of an angular and a scalar function to an exponential function. It corresponds approximately to the context that Lambert has examined

The transition from half to all angles and arguments is accomplished by inserting in the addition theorem for the tangent of the double angle:

This equation is a more central representation. This may result in the left and right side are independent of one another, expressed with other trigonometric and hyperbolic by applying the relevant relations for trigonometric and hyperbolic. Since these relations are generally of simple algebraic structure, they can also be algebraically simply dissolve and the other side of the equation can be then also considerably usually easier (see below some examples of all angles and arguments). Of particular interest are the depictions in which tangent or hyperbolic tangent simply occur because let their inverse functions compute with numerical means particularly easy. is therefore

Of the possible alternative representations of the most important. However, any other resolution of the so-formed according to equations is a possible representation of Gudermann function.

The inverse function Gudermann

The inverse function of Gudermann function can be recovered on the one hand by a resolution whose equations for and must usually be represented by means of logarithms. However, it is also independent of the above equations defined and their derivation follows in a manner analogous to the derivation of the Gudermann function, however complex calculations are needed for the intermediate steps.

For the following applies:

For the numerical evaluation of the inverse Gudermann function, the projector after especially for the middle two-thirds of the definition range is suitable. At the edges of a representation with half angles is preferable because it does not work in shallow areas of the extrema of sine and / or cosine, and therefore have a higher numeric field. For the evaluation of Gudermann function similar considerations apply.

No other relationships

The derivation of the Gudermann function and reversal are those corresponding to the integrand their definition integrals:

Particularly noteworthy is the identity for complex numbers:

The combination of circular and hyperbolic functions is essentially given by:

Practical Application

With the connections shown by circular and hyperbolic functions can be mathematical expressions simplify if necessary.

Because of their simple derivatives of the Gudermann function and its inverse are suitable as a substitute for the integral calculus. To this end Gudermann has used them.

With the Gudermann function or its inverse, the angle of latitude is linked with the north-south component of the Mercator projection. Here, the radius of the earth in particular, the equations

Of importance. Since the local distortion of the Mercator projection depends with the latitude, the relative projection distance from the equator to latitude is the integral of all the distortions about the arc ( arc of the meridian ) from the equator to

For the evaluation may be a representation of the inverse function for Gudermann half angle is preferable.