Why are hyperbolic functions defined in terms of exponentials?

neginf

Where do the definitions of hyperbolic functions in terms of exponentials come from ?

neginf

Maybe somebody tried e^i*x=cos x+i*sin x and e^-i*x=cos x-i*sin x with i*x instead of x and got functions that acted a lot like regular trig functions so they named them a lot like trig functions, like cos(i*x)=cosh x.
Could that be where those definitions come from ?
If so,do they come from somewhere else also ?

Hurkyl

AFAIK, the hyperbolic trigonometric functions were originally defined geometrically, analogously to the ordinary (circular) trigonometric functions, just based on the rectangular hyperbola rather than the circle.

neginf

Thank you for that.
I wonder if the definitions in terms of exponentials can be gotten from the geometric definition. That would seem more natural than adding and subtracting trig functions with imaginary arguments.

AlephZero

The most "obvious" formula from the geometry of a hyperbola would be cosh^2 x = sinh^2 x + 1.

Another "obvious" starting point from a rectangular hyperbola is to define log(x) as the integral of 1/x, and exp(x) as the inverse of log(x).

From the point of view of "advanced" math, probably the simplest way to define the trig hyperbolic and exponential functions is using power series of complex variables. Then the relations between them are obvious (and they are all so-called analytic functions defined for all complex arguments, which means they have lots of nice properties), but you then have to prove they have something to do with angles, circles, and hyperbolas. But many of the their uses in "advanced" math and physics don't have much to do with angles and hyperbolas anyway...