- #1
- 24,775
- 792
This function $$\frac{e^x - e^{-x}}{2}$$ is called the hyperbolic sine. I'll refer to it as "hypersine" for short. You could say it "splits the difference" between the rising exponential function e^x and the exponential function run backwards, e^-x which slopes downwards---you take the difference between upwards and downwards sloping exponentials and divide by two.
It's a nice function to get to know, if you aren't familiar with it already. It turns out that in our universe distances, areas, and volumes expand over time according to powers of hypersine.
Distances grow according to the 2/3 power ##(\frac{e^x - e^{-x}}{2})^{2/3}##
Areas grow according to the 4/3 power ##(\frac{e^x - e^{-x}}{2})^{4/3}##
Volumes grow according to the square of the hypersine ##(\frac{e^x - e^{-x}}{2})^2##
The hypersine has a nice symmetry which the ordinary exponential function ex does not have. If you flip it right to left, over the y-axis, and then flip it top to bottom over the x-axis, you wind up with the original function. It is the blue curve in this picture.
It's a nice function to get to know, if you aren't familiar with it already. It turns out that in our universe distances, areas, and volumes expand over time according to powers of hypersine.
Distances grow according to the 2/3 power ##(\frac{e^x - e^{-x}}{2})^{2/3}##
Areas grow according to the 4/3 power ##(\frac{e^x - e^{-x}}{2})^{4/3}##
Volumes grow according to the square of the hypersine ##(\frac{e^x - e^{-x}}{2})^2##
The hypersine has a nice symmetry which the ordinary exponential function ex does not have. If you flip it right to left, over the y-axis, and then flip it top to bottom over the x-axis, you wind up with the original function. It is the blue curve in this picture.
Last edited: