We demonstrate by a rigorous theoretical calculation the transient Surface Plasmon Polariton (SPP) mode, which is launched by a beam impinging onto a sub-wavelength real metallic slit in addition to the conventional long-range SPP. Different from the previous works, we find a direct closed-form solution of the Maxwell's equations by the Sommerfeld branch-cut integrals without approximation. The transient wave is a SPP with a complex-valued envelope, expressed as the exponential integral. Its rapid damping may be asymptotically approximated as - ln(x), 1/ x1/2 and 1/x, respectively, depending on the distance range away from the slit. The transit SPP may be considered as a cylindrical wave, which is radiated from the slit, takes the SPP propagation constant at the interface and propagates with additional drop due to the loss of energy pumped to the SPP.