Pulse lengtheners are circuits which lengthen a series of pulses without changing the relative pulse amplitudes. An ideal "box-car" pulse lengthener produces flat-topped pulses prolonged throughout the interval between pulses. It has been shown by Ming-Chen Wang and G. E. Uhlenbeck that if a pulse with repetition frequency f0, periodically modulated by a signal with fundamental frequency fm, is put through the ideal pulse lengthener, the expression for the output pulse contains terms with frequencies |sf0+qfm| where s= 0, 1, 2,··· and q = ···,-2,-1,0,1,2,···. The amplitudes of terms with frequencies sf0(s=1, 2,···), however, are zero. The present paper contains a derivation of the result in more detail, and without the restriction to the ideal case. In the present derivation, the pulse form is taken initially to be quite general. Both a formula and graphs are then presented for the amplitudes associated with the output frequencies of the pulse lengthener, when the output pulse height decays exponentially with the time constant α, and lasts throughout the fraction β of the interval between pulses. This reduces further to the ideal case when α=0 and β=1. The frequencies sf0(s=1, 2,···) are ordinarily present, except in the ideal case. An example of a pulse-lengthening circuit is given.