@mmcdara 

> Of course such formulae exist.

I don't think so.  At least not that have been derived, so far.

@mmcdara 

When I write "generic," I mean works with values other than 2 and 1 for Sa and Sd.  Unless I am mistaken, equation 10 does not.

In the attached sheet, I have cells for peak (x,y) and half (x, y).  The former works, regardless of supplied values, but the latter only works for 2 and 1.

For the peak, I use =X0 + SQRT(2)*SQRT(((LN(Sa) - LN(Sd))/(Sa - Sd)))

For the two halfs, I use

=X0 + SQRT(2)*SQRT(ABS(LN(1/2 + 1/2*SQRT(1 + 2*EXP(-LN(Sa/Sd)/(Sa - Sd))^2 - 2*EXP(-LN(Sa/Sd)/(Sa - Sd))))))
=X0 + SQRT(2)*SQRT(ABS(LN(1/2 - 1/2*SQRT(1 + 2*EXP(-LN(Sa/Sd)/(Sa - Sd))^2 - 2*EXP(-LN(Sa/Sd)/(Sa - Sd))))))

which are from your equation 10 (the shift with x0 can be set to 0, but is trivial in any event).

Please take a look at DoG.xlsx and let me know what you think I am doing incorrectly.  And apologies if I am being daft.

@mmcdara 

Sorry for the delay.  I was taking my time trying to work through yours since I didn't understand all of it (@tomleslie's I could follow).  I still don't get what the "u" formula is doing.  I have replied to you, now.  And thanks again for the help!

@mmcdara 

Thanks for this really interesting effort.  I apologize that I am too naive in Maple and algebra to follow it perfectly, but I've worked my way though it, and agree with everything, until I get to the "mychange" bit.  However, looking past that, what I gather is that formula 10 is supposed to be a solution.  For values of 2 and 1, it seems to work.  However, I don't think it's generic.  I tried to build it into an Excel sheet (DoG.xlsx) and it doesn't play nice:

Is there some modification that I am not understanding that makes it work for other values?  I think maybe there is no formal solution and a numerical optimization may be all that works.

Please let me know if I am understanding it correctly.

@tomleslie This is great but can it be made generic?  That is, are there formulae that will work with arbitrary values of S_a and S_d?

@Joe Riel

@tomleslie 

First of all, thank you both for your help.  @Joe Riel, I was actually able to get somewhere similar (possibly identical, although the formatting of the expression was different) but I was stumped by the "LambertW" bit.  Looking it up it seems like a function outlining pairs of solutions, of which there would be an infinite number, and I need explicit ones.

In terms of the answer, what I am really hoping to do is work my way through to a solution for Excel.  I have set up a sheet to take some X and Y values and then use the Excel Solver to optimize Ym, X₀, c and d.  Then, I am hoping to have cells that tell me Xs where the funciton crosses 50% maximum.  I should note that the function is only defined, in my case, for for positive Xs, but it may be that it rises again below 0 (and I just don't care) and that's where the third value you are seeing above is.  I am happy to provide some XY data:

For these data, it seems as though

are good fits:

What I am hoping to do is come up with a function I can enter as an Excel formula that will tell me where the point is on the way up and, especially, on the way down, that this curve crosses 50% max.  The data will always have this general form (including positive, real values for X, Y, Ym and, as far as I know, c, and real values for d), but the exact values of Ym, X₀, c and d will vary a bit from fit to fit.