74 7C.] COMPLEX VARIABLES. 73 



positive direction (i.e. from x to y) through an angle 0, where 

 r *J(a? + Z&amp;gt; 2 ), and 6 is the least positive value of arc tan - . With 



8 



respect to the expression a + ib, r is called the modulus, and 6 

 the amplitude. 



The meanings of subtraction and division of vectors follow at 

 once from the considerations that they are the operations inverse 

 to those of addition and multiplication, respectively. 



With these conventions, the addition, multiplication, &c., of 

 vectors are performed according to the same laws of operation as 

 in common algebra. 



76. For shortness we denote the complex quantities &amp;lt;/&amp;gt; + iijr, 

 and x -f- iy by the letters w, and z, respectively. These symbols 

 not being required at present in their former meanings may 

 without inconvenience have these new ones assigned to them. Then 

 w being any function of z t according to the definition of Art. 74, 

 we have corresponding to any point P of the plane xy (which we 

 may call the plane of the variable z) one or more definite values 

 of w. Let us choose any one of these, and denote it by a point P 

 of which &amp;lt;/&amp;gt;, ty are the rectangular co-ordinates in a second plane 

 (the plane of the function w). If P trace out any curve in the 

 plane of z } P will trace out a corresponding curve in the plane 

 of w. By mapping out the positions of P corresponding to the 

 various points P of the plane xy, we may exhibit graphically all 

 the properties of the function w. 



Let now Q be a point infinitely near to P, and let Q be the 

 corresponding point infinitely near to P . We may denote PQ 

 by dz, P Q by dw. The vector P Q may be obtained from the 



vector PQ by multiplying it by the differential coefficient -T- , 



whose value is by definition dependent only on the position of P, 

 and not on the direction of the element dz (PQ). Now the effect 



of multiplying any vector by the complex quantity -y- is to 



increase its length in some definite ratio, and to turn it in the 

 positive direction through some definite angle. Hence, in the 



