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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.

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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!

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  • $\begingroup$ Thanks for answering, I guess that explains why I was not able to make any headway with this one. $\endgroup$ Jul 27 '18 at 12:52
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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.

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  • $\begingroup$ Thank you for your reply, I marked the other answer as correct just on that basis that on my end this other answer was posted first, although they were equally as valuable. $\endgroup$ Jul 27 '18 at 12:53
  • $\begingroup$ Just to explain how it works, you can upvote as many answers as you please, by clicking on the up arrow. This just means that you consider it valuable. You can only accept one, by checking it. This means it answered your question. You can upvote and accept the same answer. Accepting an answer is encouraged, when appropriate, since people browsing the questions will see that it has been addressed to the OP's satisfaction. $\endgroup$
    – saulspatz
    Jul 27 '18 at 12:59
  • $\begingroup$ I did up-vote both however due to the new joiner limitations unfortunately, it will not show until I get over 15 reputation. Once I gain a bit more reputation this should tick through. $\endgroup$ Jul 27 '18 at 13:09

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