We have calculated the radiative corrections to the Dalitz plot for ${K_{\mu 3}}^{\pm }$ and ${K_{\mu 3}}^0$ decays, assuming a phenomenological weak $K-\pi$ vertex and using perturbation theory. The answer depends logarithmically on a cutoff. We have evaluated all terms which contribute to order $\alpha$ analytically, without any approximations concerning the smallness of the muon mass or the "real inner bremsstrahlung." Thus, the dependence on the parameter $\xi$ (the ratio of the form factor $\frac{f_{-}}{f_{+}}$) is exact. The radiative corrections to the ${K_{\mu 3}}^0$ Dalitz plot, muon spectrum, and lifetime average around 2% over most of their respective ranges and are not especially sensitive to the cutoff. The radiative corrections to ${K_{\mu 3}}^{\pm }$ decays is a fraction of a percent over most of the Dalitz plot and is sensitive to the cutoff. The radiative correction to the $\Delta I=\frac{1}{2}$ rule prediction for the ratio of the charged and neutral decay rates is approximately 2%. The final-state Coulomb correction accounts for most of this numerical result, the rest being model-dependent noise.

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