<DIV class=board>책에 나오는 내용입니다. 참고가 되면 좋겠습니다. <BR><BR>The analogy between the equation x^2 + y^2 = 1, which represents a circle whose radius is 1 and whose center is at the origin of an orthogonal coordinate system, and that of a hyperbola, x^2 - y^2 = l, started to attract interest from mathematicians in the latter part of the 17C. <BR><BR>Since the equations of the circle and hyperbola differ only by a minus sign, and their areas may be expressed in trigonometric and logarithmic functions, respectively, the idea developed that the imaginary unit was involved in a relation between trigonometric functions and the logarithmic functions and the hyperbola by hyperbolic functions- that is, factors of the imaginary unit and trigonometric functions. <BR><BR>Many participated in the development of hyperbolic functions, but the most comprehensive early publications(1768, 1770) were by the German mathematician J.H.Lambert, who introduced the names and notations that we still use. <BR><BR>The appellation hyperbolic functions is due to a comparison between the Pythagorean identities sin^2 θ +cos ^2 θ = 1 for trigonometric, or circular, functions and cosh^2 θ - sinh^2 θ = 1 for hyperbolic functions : <BR>※ While cosθ = x, sinθ=y are the parametric equations of the circle x^2 + y^2 = 1, cosh t = x, sinh t = y, are the parametric equations of one branch of the hyperbola x^2 - y^2 = 1 <BR><BR>※ In the trigonometric case, θ is a radian measure of <BR>∠POQ, which represents twice the area of the sector POQ. In the hyperbolic case, t is not a measure of an angle; yet, with the help of integral calculus, t can be shown to represent twice the area of the hyperbolic sector POQ. <BR><BR>Hyperbolic functions have applications in science and engineering, where they may express gradual absorption or decay, e.g., of light, sound, electricity, or radioactivity. They are of importance for finding integrals.</DIV>
