Limit of the Pearson function for M →∞

The Pearson VII function is defined as

y = (1 + K2 x2)-M. M

We know that a^b = e^{b ln a}. Consequently we can write

( ) 2x2- y = exp - M ln(1+ K M ) ,

and if M →∞ then

x2 x2 ln(1+ K2 M-) ≈ K2 M-,

hence

2 2 y ≈ exp(- K x ),

which is a gaussian function.