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(a) w = zz, z = 2 + i
(b) w = (z - 1)^2 .Z, z = i
(c) w = zRez, z = 1 - iChapter 6*8. Let w = f(z) = u(x,y) + iv(x,y) E 'D(A).
(a) Assuming that J '=I 0 at a point z E A, show that
d z= -fz d w--fz d-w
J J
(b) Prove thatmax lf~(z)I = lfzl + lfzl
and
min lf~(z)I = llfzl -lf-zll
( c) Prove thatwhere the prime denotes derivation with respect to 0.
( d) If () = arg dz, ()' = arg dw, prove that
d()' J J
d() = (Dou)^2 + (Dov)^2
lf~(z)l^2( e) If J '=I 0, show that
Id() 'I lfzl + lfzl max IJ~(z)I
max d() = llfzl -lf-zll = min ln(z)I
*9. With the same notations as in problem 8, prove:(a) If the origin 0 is exterior to the Kasner circle of f at z, then J > 0,
and OT = J^112 , where OT is the length of a tangent from 0 to
the graph of the Kasner circle.(b) If the Kasner circle passes through 0, then J = 0.
( c) If the origin lies in the interior of the Kasner circle, then J < 0
and OM = (-J)^112 , where OM is half the length of the chord
perpendicular to the diameter of the circle that contains 0.
Thus, in all three c~ses the value of the Jacobian J represents the
"power" of the origin 0 with respect to the circle.- Let
az+bz
w=---
cz+dz
(1)· where a, b, c, dare complex constants such that ad-be '=I 0. If z = x+iy
and w = u +iv, show that J(u,v/x,y) = 0.