Gudermannian function
The Gudermannian function, denoted \gd(x) or \gamma(x), is an odd special function in mathematics that provides a direct relationship between hyperbolic and trigonometric functions without invoking complex numbers, defined as \gd(x) = \int_0^x \sech t \, dt for -\infty < x < \infty.[1][2] It can also be expressed in closed form as \gd(x) = 2 \arctan(\tanh(x/2)), \gd(x) = \arctan(\sinh x), or \gd(x) = 2 \arctan(e^x) - \pi/2.[1][2] Named after the German mathematician Christoph Gudermann (1798–1852), who explored connections between circular and hyperbolic functions in publications starting in 1830, the function serves as a bridge between these two classes of functions through identities such as \sin(\gd(x)) = \tanh x, \cos(\gd(x)) = \sech x, and \tan(\gd(x)) = \sinh x.[3][1] Its derivative is \gd'(x) = \sech x, and it maps the real line to (-\pi/2, \pi/2), making it strictly increasing and bijective.[1][2] One of its most notable applications is in cartography, where it appears in the inverse Mercator projection to relate latitude \phi to the vertical map coordinate y via \phi = \gd(y), facilitating the conformal representation of the Earth's surface on a cylinder.[1] The inverse Gudermannian function, \gd^{-1}(x) = \int_0^x \sec t \, dt for -\pi/2 < x < \pi/2, satisfies \gd^{-1}(x) = \ln(\sec x + \tan x) and extends these relations symmetrically.[2] Additionally, the function connects to other special functions, such as elliptic integrals and the dilogarithm in its antiderivative, and features a Maclaurin series expansion x - \frac{1}{6}x^3 + \frac{1}{24}x^5 - \frac{61}{5040}x^7 + \cdots.[1][2]Definition and Basic Properties
Definition
The Gudermannian function, often denoted as \mathrm{gd}(x) or \gamma(x), is an odd real-valued function that provides a bridge between hyperbolic and trigonometric measures without invoking complex numbers. Its primary definition is given by the definite integral \mathrm{gd}(x) = \int_0^x \sech(t) \, dt, where \sech(t) = 1 / \cosh(t). This integral form originates from the accumulated arc length or area considerations in hyperbolic geometry, reflecting the function's role in mapping hyperbolic parameters to angular ones.[1] Equivalent closed-form expressions for the Gudermannian function include \mathrm{gd}(x) = \arctan(\sinh x) and \mathrm{gd}(x) = 2 \arctan(e^x) - \frac{\pi}{2}. These representations facilitate direct computation and reveal the function's smooth, monotonically increasing behavior from hyperbolic inputs to bounded angular outputs.[1][4] Geometrically, the Gudermannian function interprets \mathrm{gd}(x) as the circular angle \phi corresponding to a hyperbolic angle \psi = x under a stereographic projection, which equates the areas of associated circular and hyperbolic sectors while preserving angles between the two geometries.[5] The function is defined for all real x \in \mathbb{R}, with its range being the open interval (- \pi/2, \pi/2), ensuring it maps the entire real line bijectively onto this angular domain.[1]Inverse Gudermannian function
The inverse Gudermannian function, denoted \operatorname{gd}^{-1}(x) or \operatorname{arcgd}(x), is the inverse of the Gudermannian function \operatorname{gd}(x), providing a bijection from the open interval (-\pi/2, \pi/2) to \mathbb{R}.[6][2] It arises naturally in contexts bridging trigonometric and hyperbolic functions, such as in the Mercator map projection where it relates latitude \phi to the vertical coordinate y.[7] One standard definition is given by the integral \operatorname{gd}^{-1}(x) = \int_0^x \sec t \, dt, valid for -\pi/2 < x < \pi/2.[2][6] This form emphasizes its role as an area function under the secant curve. Equivalent closed-form expressions include \operatorname{gd}^{-1}(x) = \sinh^{-1}(\tan x) = \tanh^{-1}(\sin x), \operatorname{gd}^{-1}(x) = \frac{1}{2} \ln \left( \frac{1 + \sin x}{1 - \sin x} \right), and \operatorname{gd}^{-1}(x) = \ln (\sec x + \tan x) = \ln \left[ \tan \left( \frac{\pi}{4} + \frac{x}{2} \right) \right]. These identities highlight its connections to inverse hyperbolic functions and logarithmic forms, facilitating computations in hyperbolic geometry and projections.[7][6][2] The derivative of the inverse Gudermannian function is \frac{d}{dx} \operatorname{gd}^{-1}(x) = \sec x, which follows directly from the fundamental theorem of calculus applied to its integral definition.[7][6] In applications, it is particularly valuable for converting between spherical latitudes and Cartesian coordinates in cartography; for instance, in the Mercator projection, y = \operatorname{gd}^{-1}(\phi) yields the northward distance from the equator for a given latitude \phi.[7] The function's Maclaurin series expansion is \operatorname{gd}^{-1}(x) = x + \frac{1}{6} x^3 + \frac{1}{24} x^5 + \frac{61}{5040} x^7 + O(x^9), useful for small-angle approximations in numerical evaluations.[7]Fundamental properties
The Gudermannian function \operatorname{gd}(x) is defined for all real numbers x \in \mathbb{R} and is an odd function, satisfying \operatorname{gd}(-x) = -\operatorname{gd}(x) for all x.[1] Its range is the open interval (-\pi/2, \pi/2), and it provides a bijection from \mathbb{R} onto this interval.[1] The function is continuous and strictly increasing on \mathbb{R}, with \operatorname{gd}(0) = 0.[1] This monotonicity follows from the integral definition \operatorname{gd}(x) = \int_0^x \operatorname{sech} t \, \mathrm{d}t, as the integrand \operatorname{sech} t > 0 for all real t.[2] As x \to \infty, \operatorname{gd}(x) \to \pi/2, and as x \to -\infty, \operatorname{gd}(x) \to -\pi/2.[1] These horizontal asymptotes reflect the bounded nature of the function despite its unbounded domain. The Gudermannian function is infinitely differentiable, with its Taylor series expansion around zero given by \operatorname{gd}(x) = x - \frac{1}{6}x^3 + \frac{1}{24}x^5 - \frac{61}{5040}x^7 + \cdots, converging for all real x.[1]Relations to Trigonometric and Hyperbolic Functions
Circular-hyperbolic identities
The Gudermannian function \gd(x) provides a bridge between circular trigonometric functions and hyperbolic functions through a set of fundamental identities that express trigonometric functions of \gd(x) directly in terms of hyperbolic functions of x. These relations arise from the integral definition \gd(x) = \int_0^x \sech t \, \mathrm{d}t and its equivalent expressions using inverse trigonometric functions.[2][1] Specifically, the sine identity is \sin(\gd(x)) = \tanh x, which follows from \gd(x) = \arcsin(\tanh x). Similarly, \cos(\gd(x)) = \sech x, derived from \gd(x) = \arccos(\sech x). The tangent relation is \tan(\gd(x)) = \sinh x, consistent with \gd(x) = \arctan(\sinh x). These identities hold for x \in (-\infty, \infty), with \gd(x) \in (-\pi/2, \pi/2).[2][1] Additional identities extend to reciprocal functions: \cot(\gd(x)) = \csch x, \sec(\gd(x)) = \cosh x, and \csc(\gd(x)) = \coth x. These can be obtained by taking reciprocals of the primary identities or using the expressions \gd(x) = \arccsc(\coth x) and \gd(x) = \arcsec(\cosh x). Such relations facilitate conversions between hyperbolic and spherical geometries, notably in cartographic projections.[1] A complementary set of identities involves half-arguments: \tanh(x/2) = \tan(\gd(x)/2). Furthermore, \gd(x) = 2 \arctan(e^x) - \pi/2, linking it to the exponential function and underscoring its role in unifying circular and hyperbolic trigonometry. These identities are essential for applications in differential geometry and special function theory.[2][1]Symmetries and periodicity
The Gudermannian function \operatorname{gd}(x) is an odd function, satisfying \operatorname{gd}(-x) = -\operatorname{gd}(x) for all real x, which implies point symmetry (or rotational symmetry of 180 degrees) about the origin.[1][8] This property follows directly from its integral definition \operatorname{gd}(x) = \int_0^x \operatorname{sech} t \, dt, as the integrand \operatorname{sech} t is even.[2] In the complex domain, the function exhibits additional symmetry relations, such as \operatorname{gd}(ix) = i \operatorname{gd}^{-1}(x), linking it to its inverse and highlighting its role in bridging trigonometric and hyperbolic functions without invoking complex exponentials explicitly.[1] The Gudermannian function is not periodic over the real numbers, as it is strictly increasing and bounded, with \lim_{x \to \infty} \operatorname{gd}(x) = \pi/2 and \lim_{x \to -\infty} \operatorname{gd}(x) = -\pi/2, approaching horizontal asymptotes without repetition.[1][2] No quasi-periodic or higher-order periodic behaviors are observed in its real-valued form, consistent with its monotonic nature derived from the positive derivative \operatorname{gd}'(x) = \operatorname{sech} x > 0.[2]Evaluation Methods
Specific values
The Gudermannian function satisfies \operatorname{gd}(0) = 0, as follows directly from its integral definition or any of the equivalent closed-form expressions. It is an odd function, with \operatorname{gd}(-x) = -\operatorname{gd}(x) for all real x, a property inherited from the odd nature of \sinh x and \arctan y. The function maps the real line to the open interval (-\pi/2, \pi/2), approaching these bounds asymptotically: \lim_{x \to \infty} \operatorname{gd}(x) = \pi/2 and \lim_{x \to -\infty} \operatorname{gd}(x) = -\pi/2. The identity \operatorname{gd}(x) = \arctan(\sinh x) enables exact evaluation at points where \sinh x = \tan \theta for known angles \theta \in (-\pi/2, \pi/2). For example:- At x = \sinh^{-1}(1/\sqrt{3}) = \frac{1}{2} \ln 3, \operatorname{[gd](/page/GD)}(x) = \arctan(1/\sqrt{3}) = \pi/6.
- At x = \sinh^{-1}(1) = \ln(1 + \sqrt{2}), \operatorname{[gd](/page/GD)}(x) = \arctan(1) = \pi/4.
- At x = \sinh^{-1}(\sqrt{3}) = \ln(2 + \sqrt{3}), \operatorname{[gd](/page/GD)}(x) = \arctan(\sqrt{3}) = \pi/3.