I have two functions:
A: y(x) = 2 exp(-0.5 (100/sigma)^2)/x
B: y(x) = (2pi sigma) exp(-0.5 (sigma x)^2)
I want to find an algebraic expression relating x to sigma, but can't figure out a way. The pertinent values are between 10<sigma<50, and 0<x<1.
When plotted on a log plot, I noticed that the intersection points fall very closely to the line:
C: y(x) = 60 exp(-53x)
Knowing this, I take the log of equations B and C, solve for x, and come up with a nice approximation for the intersection points, x = 2*53/sigma^2.
FYI, for the quadradic equation, I used the approximation ( for 4AC << B^2 ):
[ -B + sqrt(B^2-4AC) ] /2A = B/A – C/B
My question is, is there an analytic way to get to equation C? Or is there a way to find a similar approximation for the intersection points without using equation C?