Hyperbolic sector
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it.
The area of a hyperbolic sector in standard position is natural logarithm of b .
When in standard position, a hyperbolic sector determines a hyperbolic triangle, the right triangle with one vertex at the origin, base on the diagonal ray y = x, and third vertex on the hyperbola with the hypotenuse being the segment from the origin to the point (x, y) on the hyperbola.
The length of the base of this triangle is and the altitude is where u is the appropriate hyperbolic angle.
When the length of theses legs is divided by the square root of 2, they can be graphed as the unit hyperbola with hyperbolic cosine and sine coordinates.
The analogy between circular and hyperbolic functions was described by Augustus De Morgan in his Trigonometry and Double Algebra (1849).
[2] William Burnside used such triangles, projecting from a point on the hyperbola xy = 1 onto the main diagonal, in his article "Note on the addition theorem for hyperbolic functions".
[3] It is known that f(x) = xp has an algebraic antiderivative except in the case p = –1 corresponding to the quadrature of the hyperbola.
Whereas quadrature of the parabola had been accomplished by Archimedes in the third century BC (in The Quadrature of the Parabola), the hyperbolic quadrature required the invention in 1647 of a new function: Gregoire de Saint-Vincent addressed the problem of computing the areas bounded by a hyperbola.
[4] Before 1748 and the publication of Introduction to the Analysis of the Infinite, the natural logarithm was known in terms of the area of a hyperbolic sector.
Leonhard Euler changed that when he introduced transcendental functions such as 10x.
Euler identified e as the value of b producing a unit of area (under the hyperbola or in a hyperbolic sector in standard position).
To accommodate the case of negative logarithms and the corresponding negative hyperbolic angles, different hyperbolic sectors are constructed according to whether x is greater or less than one.
