Both of the problems of in-plane fracture and out-of-plane transverse flexure of an infinitely large isotropic plate containing a circular arc crack is solved using the complex variable method. A new conformal mapping function is introduced to transform the contour surface of a circular arc crack to a unit circle. Through this mapping function, direct stress integration is performed along the unit circle for various loading conditions. After Cauchy integrals are taken for each term in the boundary equations, the complex stress functions corresponding to the specified loading conditions are determined. Without using the limiting process, the obtained solutions can be reduced to consider the straight crack problem simply by setting a mapping variable equal zero. Moreover, the interaction behavior between the crack tips of a circular arc crack is studied and exhibits different characteristics from those of a pair of collinear cracks. The mapping function is further modified to transform the contour surface of an elliptical crack to a unit circle. The stress intensity factors of an elliptical crack with symmetric geometry are numerically calculated.