CHAPTER IV. 



FUNCTIONS OF A COMPLEX VARIABLE. 



40. Multiplication of Complex Numbers. We have seen 

 in (5) how the two-dimensional complex 

 number a + ib may be represented in 

 the plane by Argand's diagram. From 

 the definition of addition of complex 

 numbers it follows that two complex 

 numbers are added by the parallelo- 

 gram construction, that is the sum of 

 the two complex numbers p = a x + H>i 

 and q = a 2 + ib 2 is represented by the FlG - 20 - 



diagonal of the parallelogram constructed on lines whose lengths 

 are equal to the moduli of p and q, 



and which make angles with the X-axis equal to the arguments of 

 p and q. 



Hence \ P<1\ ^ \ P \ + \q\ 



If we introduce the polar coordinates 



-i & 



we have 



a = r cos (/>, 

 b = r sin 0, 

 p = a + ib = r (cos + i sin 6). 



