Hence, when 0(a;,y) and tli{x,y) have been expressed in terms of X and Y, they may be taken 

 as the potential and the stream function for a new motion described in terms of X and Y. The 

 original boundaries on the s-plane become transformed into boundaries of a different shape on 

 the Z-pIane; and the curves, ^ = constant and i/( = constant, transform into curves for the same 

 constant values of <p and ip on the Z-plane. Thus the known flow described in terms of z is 

 transformed into another type of flow satisfying different boundary conditions. 



An alternative mathematical statement is the following. Let (x,y), i// (x,y) be a known 

 pair of conjugate functions, and let x{X,Y), y{X,Y) be any other pair of conjugate functions in 

 terms of the variables X and Y. Then a new pair of conjugate functions in terms of X and Y 

 can be obtained by substituting in <^(x,y) and i/j {x,y) the expressions for x and y in terms of 

 A" and Y. They may be written (f,[x(X,Y), y(X,Y)], ^[aj(Z,y), y{X,Y)] 



Any boundary that is a streamline on the 2-plane remains a streamline on the Z-plane. 

 Sources and sinks also remain sources and sinks of the same strength; and the circulation 

 around any closed curve retains the same value around the transformed curve. For, the volume 

 of fluid emitted from a line source, per second and per unit length, is represented by the de- 

 crease in ip as the source is encircled once in the positive direction, according to a principle 

 stated in Section 40, whereas the circulation around a closed curve is similarly represented by 

 the decrease in as the curve is traversed in the positive direction, and these changes in 

 and (A are invariant under the transformation. 



27. THE LAURENT SERIES 



Many series of positive powers are limited in -their range of convergence. For example, 



-^ = 1 + 3 +3-^ + . . . . 

 1-3 



converges only within the unit circle defined by | s | = 1, On the other hand, negative powers 

 such as 1/3 or 1/3^ are regular functions of 3 except only at 3 = 0. These observations 

 suggest that series containing both positive and negative powers might be useful. 



In books on functions of a complex variable it is shown that, if /(s) is regular at all 

 points near a given point 3 = c, it can be expanded in a series of the form 



f(2) = . . . b^{2-c)~^ + b^{z-c)~^ + a^ + a^iz-c) +a^{z-c)'^ . . . . 



where the a's and 6's are constants and all positive and negative powers of (z-c) may occur. 

 This is called a Laurent Series. It converges at any z ^ c throughout the interior of a circle 

 drawn about c as center and passing through the singularity nearest to c; if f(z) has no singu- 

 larity except perhaps at c iLself, the series converges for all z = c. 



If the series contains negative powers of unlimited order with nonvanishing coefficients, 

 f{z) has an essential singularity at 3 = c; if the series begins with a term containing a definite 

 negative power, namely, b (s-c)""", f(z) has a pole of order wi at 3 = c; if m = 1, the pole is 



51 



