called simple. If f{z) is regular also at c, the negative powers disappear and the series be- 

 comes a Taylor series, converging also at 2 = c. 



The series exists also if f{z) is assumed to be regular merely outside of a given circle 

 centered at c, or between two such circles. Then the series converges at least at all points 

 outside of the given circle, or between the two circles, respectively. 



In any case, if a Laurent Series or Taylor Series representing a function w{z) converges 

 for all large 2, then it can be shown that either values of \w\ exceeding all limits occur when 

 z goes to infinity in certain directions, or else the series contains no positive powers of 3. 



28. COMPLEX INTEGRATION 



An integral with respect to the complex variable 3 = x + iy is defined in the same way 

 as with real variables, but it has some novel properties. 



The indefinite integral of f{z) or //(3) dz is a function F {z) of z whose derivative is 

 /(s), as with real variables. If F {z) is many-valued, care must be taken to select a branch of 

 this function that varies continuously with z. 



In defining the definite integral, it is necessary to specify, in addition to the limits, a 

 definite path of integration connecting them. This may be indicated by adding to the integral 

 sign a symbol designating the path. For example, the integral of /(s) along the path APB in 

 Figure 30 is 



/ 



/■(s) dz = 



lir 



(APB) 



Az-> 



:f{z)^z 





y 



^3 



\ B 



Ay 



/ 



p 



'y^ ^x 



X 







V 



^ 









Figure 30 — Illustrating the definition of 

 a complex integral. 



Here the sum on the right is formed as follows. 

 Choose a large number of points scattered 

 along the curve, and let As stand for the 

 difference in the values of z at any two 

 successive points; thus, in Figure 30, one 

 Az = z^ - 2 the next As = 2^ - 2 , and so 

 on. Multiply each A2 by the value of f(z) at 

 any point on the corresponding segment of the 

 curve; for example, if A2 = 2 - z^, f(z) As 

 may stand for f(z') (z -z^), where 2' is the 

 point shown in Figure 30. All the products 

 thus obtained are to be added, and the limit of 

 this sum is to be taken as the number of points 

 is increased indefinitely in such manner that 

 all of the differences A2 approach zero. 



The value of such an integral is usually 

 a complex number. It can also be written in 



52 



