# Exponential Fraction Simplification

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.

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!
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.