Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Gudermannian function
open-in-new
<p>In mathematics, the Gudermannian function relates a <a href="/facts/Hyperbolic_angle/m4IaIh0g">hyperbolic angle</a> measure ψ {\textstyle \psi } to a <a href="/facts/Angle/gFUTEQYq">circular angle</a> measure ϕ {\textstyle \phi } called the <i>gudermannian</i> of ψ {\textstyle \psi } and denoted gd ψ {\textstyle \operatorname {gd} \psi } . The Gudermannian function reveals a close relationship between the <a href="/facts/Circular_function/czej7ur0">circular functions</a> and <a href="/facts/Hyperbolic_function/Y4Cb5S35">hyperbolic functions</a>. It was introduced in the 1760s by <a href="/facts/Johann_Heinrich_Lambert/CG3IHVB9">Johann Heinrich Lambert</a>, and later named for <a href="/facts/Christoph_Gudermann/sJIED65C">Christoph Gudermann</a> who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a <a href="/facts/Limiting_case_(mathematics)/UEehGCom">limiting case</a> of the <a href="/facts/Jacobi_elliptic_functions/BLk6Rxew">Jacobi elliptic amplitude</a> am ( ψ , m ) {\textstyle \operatorname {am} (\psi ,m)} when parameter m = 1. {\textstyle m=1.} </p><p>The <a href="/facts/Real_number/R02gw5Pb">real</a> Gudermannian function is typically defined for − ∞ < ψ < ∞ {\textstyle -\infty <\psi <\infty } to be the integral of the hyperbolic secant </p> ϕ = gd ψ ≡ ∫ 0 ψ sech t d t = arctan ( sinh ψ ) . {\displaystyle \phi =\operatorname {gd} \psi \equiv \int _{0}^{\psi }\operatorname {sech} t\,\mathrm {d} t=\operatorname {arctan} (\sinh \psi ).} <p>The real inverse Gudermannian function can be defined for − 1 2 π < ϕ < 1 2 π {\textstyle -{\tfrac {1}{2}}\pi <\phi <{\tfrac {1}{2}}\pi } as the <a href="/facts/Integral_of_the_secant_function/HC9KYXq1">integral of the (circular) secant</a> </p> ψ = gd − 1 ϕ = ∫ 0 ϕ sec t d t = arsinh ( tan ϕ ) . {\displaystyle \psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\operatorname {sec} t\,\mathrm {d} t=\operatorname {arsinh} (\tan \phi ).} <p>The hyperbolic angle measure ψ = gd − 1 ϕ {\displaystyle \psi =\operatorname {gd} ^{-1}\phi } is called the <i>anti-gudermannian</i> of ϕ {\displaystyle \phi } or sometimes the lambertian of ϕ {\displaystyle \phi } , denoted ψ = lam ϕ . {\displaystyle \psi =\operatorname {lam} \phi .} In the context of <a href="/facts/Geodesy/EJH0Te0m">geodesy</a> and <a href="/facts/Navigation/sGGAJ0AQ">navigation</a> for latitude ϕ {\textstyle \phi } , k gd − 1 ϕ {\displaystyle k\operatorname {gd} ^{-1}\phi } (scaled by arbitrary constant k {\textstyle k} ) was historically called the meridional part of ϕ {\displaystyle \phi } (<a href="/facts/French_(language)/qe1DnaQh">French</a>: <i>latitude croissante</i>). It is the vertical coordinate of the <a href="/facts/Mercator_projection/bYVL5bP9">Mercator projection</a>. </p><p>The two angle measures ϕ {\textstyle \phi } and ψ {\textstyle \psi } are related by a common <a href="/facts/Stereographic_projection/oVyMrWqR">stereographic projection</a> </p> s = tan 1 2 ϕ = tanh 1 2 ψ , {\displaystyle s=\tan {\tfrac {1}{2}}\phi =\tanh {\tfrac {1}{2}}\psi ,} <p>and this identity can serve as an alternative definition for gd {\textstyle \operatorname {gd} } and gd − 1 {\textstyle \operatorname {gd} ^{-1}} valid throughout the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>: </p> gd ψ = 2 arctan ( tanh 1 2 ψ ) , gd − 1 ϕ = 2 artanh ( tan 1 2 ϕ ) . {\displaystyle {\begin{aligned}\operatorname {gd} \psi &={2\arctan }{\bigl (}\tanh {\tfrac {1}{2}}\psi \,{\bigr )},\\[5mu]\operatorname {gd} ^{-1}\phi &={2\operatorname {artanh} }{\bigl (}\tan {\tfrac {1}{2}}\phi \,{\bigr )}.\end{aligned}}}