Кіріспе
Математикалық функция шеңберлік және гиперболалық функцияларды байланыстырады. Математикада Гудерманниан функциясы гиперболалық бұрыш өлшемін –ға, ал шеңберлік бұрыш өлшемін – деп аталатын Гудерманнианға байланыстырады. Гудерманниан функциясы шеңберлік функциялар мен гиперболалық функциялардың арасындағы тығыз байланысты көрсетеді. Оны 1760 жылдары Иоганн Генрих Ламберт енгізді, ал кейіннен Кристоф Гудерманның есімімен аталды, ол 1830 жылы шеңберлік және гиперболалық функциялар арасындағы байланысты сипаттады. Гудерманниан кейде гиперболалық амплитуда деп аталады, бұл Якоби эллипстік амплитудасының параметр болғандағы шектік жағдайы.
Mathematical function relating circular and hyperbolic functions
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted 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 is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted 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 is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
Нақты Гудерманниан функциясы гиперболалық секанттың интегралы ретінде анықталады.
Mathematical function relating circular and hyperbolic functions
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted 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 is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted 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 is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
Нақты кері Гудерманниан функциясы (шеңберлік) секанттың интегралы ретінде анықталуы мүмкін.
Mathematical function relating circular and hyperbolic functions
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted 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 is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted 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 is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
Гиперболалық бұрыш өлшемі антигудерманниан немесе кейде Ламбертиан деп аталады. Геодезия және навигациядағы ендік үшін (кез келген тұрақтымен масштабталған) тарихи түрде (француз тілінде: latitude croissante) меридиандық бөлігі деп аталды. Бұл Меркатор проекциясының тік координатасы. Екі бұрыш өлшемі – және олар ортақ стереографиялық проекция арқылы байланысты, ал бұл сәйкестік күрделі жазықтықта – және анықтамасы ретінде қызмет ете алады:
Mathematical function relating circular and hyperbolic functions
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted 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 is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted 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 is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
TOC
Mathematical function relating circular and hyperbolic functions
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted 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 is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted 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 is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
Жалпылау
Гудерманн функциясы гиперболаның бір тармағындағы нүктелерді жарты шеңбердегі нүктелерге бейнелеу ретінде қарастырылуы мүмкін. Екі қабатты n өлшемді гиперболоидтың бір жағындағы нүктелерді де стереографиялық проекция арқылы n өлшемді жарты шарға бейнелеуге болады. Гиперболалық кеңістіктің жарты шар моделі гиперболалық кеңістікті көрсету үшін мұндай бейнелеуді пайдаланады.
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.