CHAPTER II 

 THE USE OF COMPLEX FUNCTIONS IN HYDRODYNAMICS 



In two-dimensional hydrodynamics extensive use is made of functions of a complex 

 variable. In this chapter, therfore, the mode of application of the theory of complex variables 

 in hydrodynamics will be discussed, and a summary will be included of the principal relevant 

 parts of the mathematical theory. 



For convenience of reference, a short table of formulas pertaining to the hyperbolic 

 functions is appended; and some useful series are also listed. 



19. COMPLEX NUMBERS 



The so-called imaginary numbers were invented in order to solve certain algebraic 

 equations, such as a;^ = - 1. A solution of this equation is a; = i, where i is a symbol having 

 the property that ^^ = - 1. In other respects i is assumed to behave like a real number. Ob- 

 viously i^ = ii^ = - i, i^ = {i^)^ = 1, 1/i = - z. The product of i by a real number is called an 

 imaginary number. 



The sum of a real number and an imaginary number is called a complex number; it can 

 be written 



3 = ar + zy 

 where x and y are real numbers. 



The number z* = x - iy is formed from z by changing the sign of the imaginary part and 

 is called the complex conjugate of z. It is often denoted by F. 



Complex numbers are conveniently represented on a plot called the Argand diagram. 

 In this plot the real part x is plotted as abcissa and the imaginary part y with i omitted is 

 plotted as ordinate, as in Figure 18. Either the point {x, y) or the vector drawn from the 

 origin to this point may be regarded as representing the complex number. 



In labeling points and lines on such 

 diagrams, it is convenient sometimes to use 

 special symbols representing geometrical 

 quantities only, and sometimes to use sym- 

 bols that stand for numbers, real or complex. 

 This leads to no difficulty in spite of the 

 logical difference between geometrical mag- 

 nitudes on a plane and complex numbers. 



It is often convenient to express a 

 complex number in terms of the polar co- 

 ordinates 



Figure 18 - Argand diagram. 



33 



