1550251515-Classical_Complex_Analysis__Gonzalez_

370 Chapter^6

Therefore, the directions 81 and 83 will correspond to maxima (minima)

of p, and the directions 02 and 04 to minima (maxima) of p, depending on

whether F > 0 (or F < 0), respectively. When E = G, F-/; 0 equation

(6.16-1) becomes

p^2 = E + F sin 20

which for E -/; 0, F > 0 is represented by the curve shown in Fig. 6.13

when plotted from the point z as origin with E = 2, F·= 1.

Returning to the general case, let Pl be the maximum value of p, and

let p 2 be its minimum. Using (6.16-8) in (6.16-1) we find that

Pi = %(E + G + D) (6.16-9)

and similarly,

(6.16-10)

Hence it follows that

Pi+ p~ = E+ G

PiP~ = i/4[(E + G)2 - D2] = 12

This shows that Pi and p~ are the roots of the equation

p^4 - (!E + G)p^2 + J^2 = O (6.16-11)

In the special case E = G, F _# 0 equation (6.16-11) reduces to

p^4 - 2Ep^2 + J^2 = O (6.16-12)

If J = 0, we get from (6.16-11) that p 2 = 0 and p1 = v'E + G.

Fig. 6.13