Computing complex argument Fresnel integral is a difficult problem meeting in electromagnetic scattering of lossy dielectric wedges. This paper makes use synthetically of series expansion and asymptotic expansion of complex argument Fresnel integral and the connections of the two expansions are found and analyzed. The computing of Fresnel integral in whole complex plane is so solved perfectly. With this method the computing speed is rapid and its precision is high. In addition, the symmetrical relations and complex zeros of Fresnel integral are studied also. Three-dimensional figure and two-dimension contour lines of Fresnel intergral in the complex plane are given.