80 FUNCTIONS OF A COMPLEX VARIABLE. [INT. IV. 



Now since 



2T 4! 6! 



it follows that 



cos cf) + i sin < = e**, 



jp | J? | d**. 



It is easy to show that the modulus of the product of two 

 complex numbers is equal to the product of their moduli, and that 

 the argument is equal to the sum of their arguments. For if 



p = &! + ibi = TI (cos </>! + i sin fa) = r^, 

 q = a 2 + ^2 = ?*2 (cos < 2 4- * sin < 2 ) = r*^'* 3 , 

 then > = r^e* ( * l+ * a) = r^ [cos (0! + < 2 ) + i sin (^ + < 2 )]. 



In like manner for the quotient, substituting the words 

 quotient for product, and difference for sum. A complex number 

 vanishes only when its modulus vanishes, and is considered infinite 

 when its modulus is infinite, whatever its argument. 



41. Function of Complex Variable. A function of the 

 complex variable z = x + iy, if given as an analytic expression 

 containing z, will be a certain function of the two real variables 

 x and y and will contain a real part, which we shall denote by 

 u (x, y), and an imaginary part, which we shall denote by iv (x, y). 

 Hence the study of functions of a complex variable may be made 

 to depend on the study of functions of two real variables. Let 



The representation of variable and function by means of abscissa 

 and ordinate of a curve is not here applicable, for both variable 

 and function have two degrees of freedom. The function may be 

 otherwise represented by means of another plane in which we 

 mark off lengths u and v as the rectangular coordinates of another 

 point representing w on another Argand's diagram. To every 

 point x, y in the first plane will then correspond a point u, v in the 

 second plane. As the point x, y moves, so will the point u, v. As 

 the point x, y representing the variable z, describes any curve, u, v, 



