Why Whittaker functions are useful?

Math question please helpp!!!!? Why inverse hyperbolic functions are called area-functions?

• Here is my inspiration : http://answers.yahoo.com/question/index?qid=20090424111332AAEvrKo Inverse trigonometric functions are called arc-functions. They are denoted as sin^-1(x) or arcsin(x). Inverse hyperbolic functions are called area-functions. They are denoted as sinh^-1(x) or arsinh(x). Note that character "c" is missing in arsinh(x), it is always missing, because it is not arc, it is area, read area sine hyperbolic.

• Answer:

  Well, I'm sure you already know the answer. But in case some others do not, here is a brief explanation: x^2 + y^2 = 1 unit circle x^2 - y^2 = 1 unit hyperbola. Unit circle parametrization: (x, y) = (cos(t), sin(t)) Unit hyperbola parameterization: (x, y) = (cosh(t), sinh(t)) In BOTH the unit circle and the unit hyperbola, the area, swept out by the radius vector, that is the area enclosed by the x-axis, the radius vector to (x,y) and the curve (circle or hyperbola), is equal to t/2, so t = cos^-1 (x) = 2*area, t = sin^-1 (y) = 2*area, t = cosh^-1 (x) = 2*area, t = sinh^-1 (y) = 2*area. Using areas as an interpretation of the parameter t unifies the circular and hyperbolic functions. The interpretation of t as arclength is available for the circular funtions (sine and cosine) so we have t = arccos(x), t = arcsin(y). But this interpretation of t is not available for the hyperbolic functions, so we have t = arcosh(x) = arsinh(y) = 2A, where A is the enclosed area of the sector of the hyperbola, described above.