I am using a modified gamma distribution (MGD) for a particle size distribution of the form

$$n(D) = N_0 D^\mu e^{-\Lambda D^\gamma}$$

where $N_0$ is the scale factor, $D$ is the particle diameter, $\mu, \Lambda,$ and $\gamma$ control the shape, and the integral is

$$\int_0^\infty n(D)dD = \frac{N_0}{\gamma}\Lambda^{-\frac{\mu+1}{\gamma}}\Gamma(\frac{\mu+1}{\gamma})$$

In order for the $N_0$ term to equal the total particle concentration, I normalize the MGD by its integral as

$$n_{norm}(D) = N_0\frac{D^\mu e^{-\Lambda D^\gamma}}{\frac{1}{\gamma}\Lambda^{-\frac{\mu+1}{\gamma}}\Gamma(\frac{\mu+1}{\gamma})}$$

My question is when I impose size limits on D such that it doesn't go from 0 to $\infty$, but something like 2 to 20, how would this normalization change? Is it even possible to normalize it to keep $N_0$ in terms of total particle concentration?


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