Sometimes in experimental studies, I need to fit a peak with an exponentially modified Gaussian (EMG) function. One of the popular formulations of EMG is

$$

y=frac{a_{0}}{2 a_{3}} exp left(frac{a_{2}^{2}}{2 a_{3}^{2}}+frac{a_{1}-x}{a_{3}}right)left(operatorname{erf}left(frac{x-a_{1}}{sqrt{2} a_{2}}-frac{a_{2}}{sqrt{2} a_{3}}right)+frac{a_{3}}{left|a_{3}right|}right)

$$

$$

begin{array}{l}

a_{0}=text { area } \

a_{1}=text { center of the pure Gaussian } \

a_{2}=text { width }(>0) \

a_{3}=text { distortion = time constant }(neq 0)

end{array}

$$

One point which has been bothering me for a while is that certain combinations of width and distortion lead to zeros or meaningless results although those combinations are physically realistic.

Take for example, a peak with an area of 1 centered at 30, its Gaussian standard deviation is 2 and distortion is 0.9, the above formula works very well.

However, if we try to make the peak symmetric, i.e., make the distortion 0.09, the result is zero, although a time constant of 0.09 is very much possible. It is not a software error (tried Excel or Matlab)- another independent scientist also tried it and the same thing happened.

**Alternatively** Numerically, I can use a discrete Fourier transform to convolute a Gaussian and an exponential decay with a time constant of 0.09. I get the desired peak. The result of DFT convolution with width of 2 and time constant of 0.9 match perfectly with the above mentioned formula.

What is the mathematical reason for the above equation to fail at certain combinations of width and distortion? Say $a_{2}$=2 and $a_{3}$=0.09, the result is zero but it works for $a_{3}$=0.9.

Thanks.