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Wikipedia / Gudermannian function

In mathematics, the Gudermannian function relates a hyperbolic angle to a circular angle called the gudermannian. The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The Gudermannian function and its inverse were used historically to construct tables of hyperbolic functions or to compute hyperbolic functions given only a table of circular functions. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude

The real to be the integral of the hyperbolic secant

as the integral of the (circular) secant

is called the anti-gudermannian or sometimes the lambertian In the context of geodesy and navigation) was historically called the meridional part (French: latitude croissante). It is the vertical coordinate of the Mercator projection.

are related by a common stereographic projection

valid throughout the complex plane:

Circular–hyperbolic identities

We can evaluate the integral of the hyperbolic secant using the stereographic projection (hyperbolic half-tangent) as a change of variables:

For example, the numerically well-behaved formulas

and for complex arguments, care must be taken choosing branches of the inverse functions.)

in terms of the exponential are all Möbius transformations of each-other (specifically, rotations of the Riemann sphere):

, these Möbius transformations can be written in terms of trigonometric functions in several ways,

For example,

Complex values

As a function of a complex variable conformally maps

Analytically continued by reflections For all points in the complex plane, these functions can be correctly written as:

functions to remain invertible with these extended domains, we might consider each to be a multivalued function the principal branch) or consider their domains and codomains as Riemann surfaces.

can be found by:

(In practical implementation, make sure to use the 2-argument arctangent,

can be found by:

Multiplying these together reveals the additional identity

Symmetries

between sine and hyperbolic sine:

The functions are both odd and they commute with complex conjugation. That is, a reflection across the real or imaginary axis in the domain results in the same reflection in the codomain:

The functions are periodic:

This is related to the identity

Specific values

indicates the limit at one end of the infinite strip):

Derivatives

As the Gudermannian and inverse Gudermannian functions can be defined as the antiderivatives of the hyperbolic secant and circular secant functions, respectively, their derivatives are those secant functions:

Argument-addition identities

By combining hyperbolic and circular argument-addition identities,

with the circular–hyperbolic identity,

we have the Gudermannian argument-addition identities:

Further argument-addition identities can be written in terms of other circular functions, but they require greater care in choosing branches in inverse functions. Notably,

which can be used to derive the per-component computation for the complex Gudermannian and inverse Gudermannian.

double-argument identities are

Taylor series

The Taylor series are

are the Euler secant numbers, 1, 0, 1, 0, 5, 0, 61, 0, 1385 ... (sequences , , and in the OEIS). These series were first computed by James Gregory in 1671.

are same as the numerators of the Taylor series for sech and sec, respectively, but shifted by one place.

The reduced unsigned numerators are 1, 1, 1, 61, 277, ... and the reduced denominators are 1, 6, 24, 5040, 72576, ... (sequences and in the OEIS).

History

The function and its inverse are related to the Mercator projection. The vertical coordinate in the Mercator projection is called isometric latitude In terms of latitude on the sphere (expressed in radians) the isometric latitude can be written

(Note: on an ellipsoid of revolution, the relation between geodetic latitude and isometric latitude is slightly more complicated.)

Gerardus Mercator plotted his celebrated map in 1569, but the precise method of construction was not revealed. In 1599, Edward Wright described a method for constructing a Mercator projection numerically from trigonometric tables, but did not produce a closed formula. The closed formula was published in 1668 by James Gregory.

The Gudermannian function per se 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 Christoph Gudermann's work in the 1830s on the theory of special functions. Gudermann had published articles in Crelle's Journal that were later collected in a book).

the Jacobi elliptic amplitude This is the inverse of the integral of the secant function. Using Cayley's notation,

He then derives "the definition of the transcendent",

observing that "although exhibited in an imaginary form, [it] is a real function of

The Gudermannian and its inverse were used to make trigonometric tables column of the trigonometric table.

Generalization

The Gudermannian function can be thought of mapping points on one branch of a hyperbola to points on a semicircle. Points on one sheet of an n-dimensional hyperboloid of two sheets can be likewise mapped onto a n-dimensional hemisphere via stereographic projection. The hemisphere model of hyperbolic space uses such a map to represent hyperbolic space.

Applications

By relating the circular and hyperbolic functions, the Gudermannian function can be used to express one type in terms of the other. It was used historically in constructing tables of hyperbolic functions, or, when provided as a dedicated table, could eliminate the need for separate tables of hyperbolic functions, each of which could be computed by doing a lookup in the Gudermannian table followed by another lookup in a table of circular functions. In some cases the hyperbolic angle corresponding to each circular angle was included as a column of a standard trigonometric table, allowing that single table to be used for both circular and hyperbolic functions.
The angle of parallelism function in hyperbolic geometry is the complement
On a Mercator projection a line of constant latitude is parallel to the equator (on the projection) at a distance proportional to the anti-gudermannian of the latitude.
The Gudermannian function (with a complex argument) may be used to define the transverse Mercator projection.
The Gudermannian function appears in a non-periodic solution of the inverted pendulum.
The Gudermannian function appears in a moving mirror solution of the dynamical Casimir effect.
If an infinite number of infinitely long, equidistant, parallel, coplanar, straight wires are kept at equal potentials with alternating signs, the potential-flux distribution in a cross-sectional plane perpendicular to the wires is the complex Gudermannian function.
The Gudermannian function is a sigmoid function, and as such is sometimes used as an activation function in machine learning.
The (scaled and shifted) Gudermannian function is the cumulative distribution function of the hyperbolic secant distribution.
A function based on the Gudermannian provides a good model for the shape of spiral galaxy arms.