Gudermann function. A definition for the Gudermann function is: gdx = / x. 0 secht dt. The function is also called the Gudermannian function or the hyperbolic. The Gudermannian function, named after Christoph Gudermann ( – ), relates the circular and hyperbolic trigonometric functions. You already asked ten similar questions. Expand 1 cosh ⁡ y in powers of e y, integrate term by term, expand 1 1 + e 2 y in powers of e y.

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Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. Back to Topic List.

Christoph Gudermann topic Christoph Gudermann 25 March — 25 September was a German mathematician noted for introducing the Gudermannian function and the concept of uniform convergence, and for being gudermanniam teacher of Karl Weierstrass, who was greatly influenced by Gudermann’s course on elliptic functions in —, the first such course to be taught in any institute.

Angle Revolvy Brain revolvybrain.

Gudermannian Function

O-K, so what’s this got to do with the antiderivative of the secant? Exponentials Revolvy Brain revolvybrain. If one were to drive a car along a great circle one would hold the steering wheel fixed, but to follow a rhumb line one would have to turn the wheel, turning it more sharply as the poles are approached.

Member feedback about GD: Janet Heine Barnett, “Enter, stage center: I’m surprised that no one has yet put these two together and mentioned the Gudermannian function, especially since a lot of the follow-up posts for the secant antiderivative question dealt with the relevance of this material in the presence of today’s computer algebra systems.

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Math Forum Discussions – The Gudermannian function

Anyway, back to Gudermnanian Katze’s question. Graph of the Gudermannian function. List of special functioon and eponyms topic This is a list of special function eponyms in mathematics, to cover the theory of special functions, the differential equations they satisfy, named differential operators of the theory but not intended to include every mathematical eponym.

Tuchinsky, “An application gudermaannian geography to mathematics: The integral of the secant function of gudermanbian was the subject of one of the “outstanding open problems of the mid-seventeenth century”, solved in by Gudermannan Gregory.


Gudremannian half-angle formula topic In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle.

Geometrically, these are identities involving certain functions of one or more angles. Without trying all that hard, here are some things I googled up that are on-line: Which definition of thr Guddermanian Function are you using? Member feedback about Sigmoid function: Member feedback about Latitude: Algebraic functions Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.

The logistic curve Plot of the error function A sigmoid function is a mathematical function having a characteristic “S”-shaped curve or sigmoid curve. And you are right, then I am saying that I’m sorry because I know that sometimes I am boring. In trigonometry, tangent half-angle formulas relate the tangent of fujction of an angle to trigonometric functions of the entire angle.

They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle. Lines of constant latitude, or parallels, run east—west as circles parallel to the equator.

In a footnote, Hobson gives several other names and some historical notes. The Gudermannian function Posted: Gudermann introduced the concept of uniform convergence in an paper on elliptic functions, but only observed it informally, neither formalizing it nor using it in his proofs. Integral of the secant function topic The integral of the secant function of trigonometry was the subject of one of the “outstanding open problems of the gudefmannian century”, solved in by James Gregory.

Member feedback about Mercator projection: The Gudermannian functionnamed after Christoph Gudermann —relates the circular functions and hyperbolic functions without explicitly using complex numbers. Member feedback about Tangent half-angle formula: The Gudermannian function shows up in a lot of diverse ways in mathematics and the physical sciences.


In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. Although the linear scale is equal in all directions around any point, thus preserving the angles and the shapes of small objects making it a conformal map projectionthe Mercator projection distorts the size of objects as the latitude increases from the Equator to the poles, where the scale becomes infinite.

They fnction distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.

Note that the last one is freely available on-line. Just as the points cos t, sin t form a circle funtion a unit radius, the points cosh t, sinh t form I’m not suggesting that this particular antiderivative is all that important for calculus students to know [1], or even something that calculus students should see derived.


Rhumb line topic Image of a loxodrome, or rhumb line, spiraling towards the North Pole In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north. Can be generated solely by addition, multiplication, and raising to the power of a positive integer. Cartographic projections Guvermannian Brain revolvybrain. This list is limited to abbreviations of two or more letters.

Member feedback about Vladimir Karapetoff: A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are ‘anonymous’, with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations.

The lines from pole to pole are lines of constant longitude, or meridians. The angle depends on the segment length a between the right angle tunction the vertex of the angle of parallelism.

List of trigonometric identities topic Cosines and sines around the unit circle In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. Notation Angles Signs of trigonometric functions in each quadrant. Graph of the inverse Gudermannian function.

Hobson continues, in Section on p.

Introduction The effect of following a rhumb line course on the surface of a globe was first discussed by the Portuguese mathematician Pedro Nunes inin his Treatise in Defense of the Marine Chart, with further fhnction development by Thomas Harriot in the s.

For points on the hyperbola below the x-axis, the area is considered negative see animated version with comparison with the trigonometric circular functions.

Graph of the Gudermannian function The Gudermannian function, named after Christoph Gudermann —relates the circular functions and hyperbolic functions without funcion using complex numbers.