Exponential divided difference ($D_t(\beta ,\gamma )={\int }_0^t{e}^{(t-x)\gamma +x\beta }dx$)

The displayed integral is $(e^{t\beta }-e^{t\gamma })/(\beta -\gamma )$ for distinct exponents and $t{e}^{t\gamma }$ when they coincide. The latter is the continuous limit, so a zero denominator is a removable singularity. This quantity appears in variation-of-constants formula estimates of matrix evolution and avoids omitting an equal-exponent case.