Many emerging light-emitting diode (LED) applications require short-distance illumination. For that, we derive an exact formula of the illumination profile produced by Lambertian LEDs, in which all parameters are fundamental constants. Moreover, we propose a general, simple, but accurate mathematical model of the irradiance pattern, valid for Lambertian and non-Lambertian LEDs at short distances. This model provides the irradiance profile as a function of both the LED-target distance and the coordinates of every point on the irradiated surface. The model is formulated in terms of simple sigmoid functions, in which LED size and light flux are fundamental parameters. The accuracy of the irradiance pattern model is tested both with theory and with experimental measurements.