A hyperbola and an ellipse can be described with one equation in which a real value for a given variable yields an ellipse and an imaginary value for the same variable yields a hyperbola. Similarly, e

^{ix}forms a circle in the complex plane, while e

^{x}, as mentioned above, resembles a hyperbola.

Finally, let's look at the area under e

^{x}. If we think of the curve as a hyperbola, that would imply that, in a vague hand-wavy sense, there is a -e

^{-x}somewhere out beyond the vertical asymptote at infinity, forming the other branch of the hyperbola, and having exactly negative of the area under the more visible branch. Well, the integral of e

^{x}from 0 to +inf can be assigned a value in much the same way that a divergent sum can still be assigned a real value. In fact, if you convert the act of integration into a sum of a geometric series(where divergent sums can be easily and coherently defined), and find the limit of the summation as the interval approaches 0(equivalent to shrinking the width of rectangles in an approximation of integration), the integral of e

^{x}from 0 to +inf works out to -1. Since the integral from -inf to 0 is 1, this means the total area under e

^{x}is precisely 0, consistent with the notion of a hypothetical opposite branch of the hyperbola that cancels out the area.

[EDIT] Also of possible relevance: ln(x), the inverse of e

^{x}, is the integral of the right side of 1/x, which is a hyperbola. [/EDIT]

I realize the above analysis isn't exactly rigorous, but that's because my point is simply that this a conjecture/intuition I have about the subject, and I'm open to having all sorts of holes poked in it. What I'm really interested in is this: if e

^{x}can be thought of as the limit of a hyperbola as some property tends toward an extreme, how might we define what that property is, and thus define the family of shapes between hyperbolas and exponential curves? That is, if the exponential curve is the result of stretching(or whatever) a hyperbola until one of its asymptotes vanishes into infinity, what is the equation for a hyperbola that has been finitely "stretched" in the same manner?

And if this is something already well-known, somebody just tell me what it's called so I can know where to look to actually find stuff about it. Thanks.