We studied the growth of viscous fingers as a Laplacian growth by conformal mapping. Viscous fingers grow due to Saffman-Taylor instability in the interface between two fluids, when a less viscous fluid pushes a more viscous fluid. As there was an interest in the rectangular Hele-Shaw cell, we solved the Laplacian equation with appropriate boundary conditions by means of conformal mapping techniques. The results were then visualized on a personal computer. Using these techniques, we studied singular effects of surface tension in the dynamics of the finger competition in the Saffman-Taylor problem with channel geometry. We also studied the motion of the interface between the two fluids in a pressure field. The more viscous fluid moves with a velocity proportional to the gradient of its pressure. In the two-dimensional case the interface can be described by a complex function which is analytic. Applying surface tension in the equations causes the tip-splitting at a longer finger (ahead). In zero order finger perturbation we had equal results for with and without surface tension. But for the first order perturbation, there was a difference. For limited surface tension in solutions for larger fingers (ahead), we observed tip splitting for larger fingers, which is completely in agreement with experimental observations.