1550251515-Classical_Complex_Analysis__Gonzalez_

270 Chapter^5

y

x

Fig. 5.~!2

Exercises 5.3

1. Let ai, a 2 , ... , an be n given distinct complex numbers and /31, /32,

... , f3n be any given complex numbers. Prove that there is a unique

polynomial P( z) of degree less than n such that P( a k) = /3 k ( k =

1,2, ... ,n).

If Qk(z) = I]i#(z -aj), show that P(z) is given by the Lagrange

interpolation formula

2. Under the mapping defined by w = z^2 , find:
   (a) The images of the hyperbolas x^2 - y^2 = a and xy = b (a, b
   constants).
   (b) The images of the lines x = c1 and y = c2 ( c1 , c2 con st ants).
   ( c) The image of any straight line not passing through the origin.
   ( d) Also show that the images of the circles lz -rl = r (r > 0) are the
   cardioids p = 2r^2 (1 + cos'lj;), where w = peiiP.


3. Determine the order of the following rational functions and find their
   zeros and poles:

z^4 -16

(a) w-- --z (^5) +1 (b) w = z4+1
z^2 + 9