(5-13)     The equation of a Gaussian function, G(x), with a mean value of zero is:

Now we are just interested in finding the value of x at which G(x) =    ½ G(0).

σ is a constant, for any specific distribution, so we can simplify things by multiplying by σ √(2π)  to get:

This will generate a peak with the same width as the original G(x); only the height is scaled.

①  What is the value of G’(x) at the top of the curve, when x=0 ?

Your answer to ① should now lead you to realise that what we are interested in is the value of x when G’(x) = 0.5. Because of the symmetry of the curve we know that G’(x) will equal 0.5 at +/-x, so the width at half height will simply be given by 2x.

So we have to solve the equation:

②  See if you can solve this equation, to find σx  in terms of x.

③  Hence write down an expression for the standard deviation, σx , in relation to the width at half height, Δx½.

④  Now combine this equation with your answer to Question 5-11, to find the relationship between σx and kmax .