88 



FUNCTIONS OF A COMPLEX VARIABLE. [INT. IV. 



FIG. 23. 



45. Integral of a Function of a Complex Variable. 



Since the complex variable has two degrees of freedom, its integral 

 is not of so simple a nature as that of a single real variable. 

 Suppose the variable z moves from a point A to a point B along 

 any continuous path. The definite integral of f(z) =u + iv along 

 this path will be defined as the line integral 



rB rB 



F = f(z) dz=\ (u + iv) (dx + idy) 



J A J A 



[ B . [ B 



= I udx vdy + i I vdx + udy. 



J A J A 



Now in virtue of the equations (A) both the integrals above 

 are independent of the path, so that F is a function of z. It is 

 evidently -ty + i<f> of the last section. This is on the supposition 

 that the functions u, v are uniform and continuous in the whole 

 region considered. If this is the case the function w u + iv is 

 called holomorphic. 



If w becomes discontinuous in the region considered it ceases 

 to be true that the integral is the same over two paths AB 

 between which lies a point of discontinuity of the function w. 



