Exponential Fraction Simplification

Question

It has been a while since I played with simplification of functions and my memory is a little spotty. I am currently doing some function fitting to data, so I am trying out many permutations of functions to see their results.

For one of my functions I managed to get the below to simplify rather easily:

$$ F(X) = \frac{e^{KX}}{e^{K}} = e^{K(X-1)} $$

Now I have found that I can potently improve my fit with the following alteration:

$$ F(X) = \frac{e^{KX}-\Delta}{e^{K}-\Delta} $$

Where K and $\Delta$ are constant with respect to X.

I am wondering if there is a simplification of the above formula that I could use. I feel like I have seen something like this done before but I am just struggling to remember how to start.

I am not looking for someone to do all the work for me but if someone could get me started with a useful identify or a pointer of what method to use (eg partial fractions ect.) to get me going in the right direction.

Many thanks.

Answer 1

There is nothing helpful you can do on that fraction, that it the simpler form. The only thing you could do is to add and subtract the factor $e^K$ on the numerator, use associativity, and then take some factors out. Indeed nothing very useful!

Answer 2

I think it's as simple as it's going to get already. If we write $y = e^K,$ then we have $${y^X-\Delta\over y - \Delta}$$ and even in the simple case where $X$ is a positive integer and $\Delta=1,$ we have the sum of a geometric progression. The general case must be messier.