In mathematics, the Gudermannian function relates a hyperbolic angle measure ψ {\textstyle \psi } to a circular angle measure ϕ {\textstyle \phi } called...

} where gd(ψ) is the Gudermannian function. The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones...

{e^{x}-e^{-x}}{e^{x}+e^{-x}}}} Arctangent function f ( x ) = arctan ⁡ x {\displaystyle f(x)=\arctan x} Gudermannian function f ( x ) = gd ⁡ ( x ) = ∫ 0 x d t cosh...

under rotation. The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions that does not involve...

The definite integral of the secant function starting from 0 {\displaystyle 0} is the inverse Gudermannian function, gd − 1 . {\textstyle \operatorname...

trigonometric functions etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function. Indicator function: maps x to either...

{x-c}{a}}\right|^{2b}}}} Hyperbolic secant. This is also the derivative of the Gudermannian function. f ( x ) = sech ⁡ ( x ) = 2 e x + e − x {\displaystyle f(x)=\operatorname...

September 1852) was a German mathematician noted for introducing the Gudermannian function and the concept of uniform convergence, and for being the teacher...

1)=\operatorname {gd} u} where gd {\displaystyle \operatorname {gd} } is the Gudermannian function. In general if neither of p,q is d then pq ⁡ ( u , 1 ) = pq ⁡ (...

(}{\tfrac {1}{2}}\pi +x{\bigr )}}} (the integral of sec, the inverse Gudermannian function), arcsec ⁡ ( 2 e x ) , {\textstyle \operatorname {arcsec} {\bigl...

(trigonometric function), Cosecant – see Trigonometric function atan2 cis—see Euler's formula Cofunction Exsecant Gudermannian function Inverse trigonometric...

{\textstyle \operatorname {gd} ^{-1}} is the inverse Gudermannian function, the integral of the secant function. There are a number of reasons why this particular...

Gaussian function Gudermannian function Half-exponential function Half-life Hyperbolic function Inflation, inflation rate Interest Lambert W function Lifetime...

number of hyperbolic sines. The Gudermannian function gives a direct relationship between the trigonometric functions and the hyperbolic ones that does...

Dirichlet distribution, a probability distribution used in statistics Gudermannian function, used in map-making Georgia Dome, a stadium in Atlanta, Georgia...

\sec x,} log sec ⁡ x , {\textstyle \log \,\sec x,} the inverse Gudermannian function log tan ⁡ 1 2 ( x + 1 2 π ) {\textstyle \log \,\tan {\tfrac {1}{2}}{\bigl...

tilenames". openstreetmap.org. Retrieved 2018-11-14. See Inverse Gudermannian_function: gd − 1 ⁡ x = ln ⁡ | tan ⁡ ( x 2 + π 4 ) | {\displaystyle \operatorname...

⁡ v {\displaystyle \operatorname {gd} v=\arctan \sinh v} is the Gudermannian function. Several other curves can be derived from the hyperbola by inversion...

where sinh, cosh, tanh, sech and csch are hyperbolic functions and gd is the Gudermannian function. János Bolyai discovered a construction which gives...

The expression on the right of the second equation defines the Gudermannian function; i.e., φ = gd(⁠y/R⁠): the direct equation may therefore be written...

)-e\tanh ^{-1}(e\sin \phi )\right]\end{aligned}}} where gd(x) is the Gudermannian function. (See also Mercator projection.) The conformal latitude defines...

{gd} } and gd − 1 {\displaystyle \operatorname {gd} ^{-1}} are the Gudermannian function and its inverse, gd ⁡ ψ = arctan ⁡ ( sinh ⁡ ψ ) , {\displaystyle...

secant function. The cumulative distribution function (cdf) of the standard distribution is a scaled and shifted version of the Gudermannian function, F (...

function Lambert's problem Lambert's theorem on the parabola Lambert's trinomial equation Lambertian function (inverse of the Gudermannian function)...

two numbers. (Also written as hcf.) gd – Gudermannian function. GF – Galois field. GF – generating function. GL – general linear group. G.M. – geometric...

{\displaystyle \textstyle \operatorname {gd} ^{-1}} ⁠ is the inverse Gudermannian function. Ideal points (points at infinity) in the Poincaré half-plane model...

Hypergeometric function 2F1, etc. Leopold Bernhard Gegenbauer: Gegenbauer polynomials Gottlieb polynomial Gould polynomial Christoph Gudermann: Gudermannian function...

{v}{c}}\right)^{2}}}}\end{aligned}}} and showed the relation of rapidity to the Gudermannian function and the angle of parallelism: v c = th ⁡ u = tg ⁡ ψ = sin ⁡ gd ⁡...

the Gudermannian angle of u, and u is the anti-Gudermannian of α. Thus he explains, "the present treatment is in terms of the anti-Gudermannian of the...