Does this function belong to an interesting class of functions?

[PhilDSP]

Hello and thanks for your consideration,

I'd like some insight into the function $f(\phi) = \frac {1 - \phi}{\phi - 1}$

Does this apply to any known modeling situations? Is it recognized as belonging to a more general class of functions that may have interesting or unique characteristics? Or can the function be transformed into a function that does?

 

[Mute]

Your function, as written, is just equal to -1, except when $\phi = 1$, where there is a discontinuity because the denominator vanishes there.

If you want an example of a function that has a similar form but isn't trivially some constant and has some applications, see Mobius transformation. (But note that the Mobius transformation is usually used with complex numbers. I don't know if it is used much in real number applications).

 

[PhilDSP]

Thanks, an association with the Mobius transformation does yield many interesting things to think about, especially since $\phi$ can be complex in the situation where the function popped up.

We could argue that the value becomes 1 when $\phi = 1$ couldn't we? This almost sounds like a spinning sphere where the axis must be aligned parallel to a force acting on the sphere, but which can suddenly undergo a spin flip.

 

[Stephen Tashi]

We could argue that the value becomes 1 when $\phi = 1$ couldn't we?

No, you can't define a function one way and then argue that it has a different definition. You can, however, define a function that is 1 when $\phi = 1$ and equal to -1 elsewhere. You can argue that this definition applies to a certain practical situation. That would be an argument about physics.

 

[HallsofIvy]

We could argue that the value becomes 1 when $\phi = 1$ couldn't we?

No. "f(x)= -1" and "g(x)= -1 if $x\ne 1$, and is not defined at $x= 1$" are two different functions.