366 Chapter^6
y v,z ~
lg
I
0 uFig. 6.10
radius of the circle is zero, or (2) the center of the circle is at the origin.
In the first case fz = 0 and the function is monogenic at z; in the second
case fz = 0 and the function is conjugate monogenic at z.
In the next two sections we investigate how in general p = lf~(z)I and
1/J = Arg f M z) depend on () at a fixed z.
6.16 NONCONFORMAL MAPPINGS. THE MAGNIFICATION
RATIO p AS A FUNCTION OF 6
In what follows we assume that fz-/= 0. From formula (6.10-8), namely,
f~(z) = e-i^6 (Deu + iDev)
we find that
where
p^2 = lf~(z)l^2 = (Deu)^2 + (Dev)^2
= ( Ux cos() -1-Uy sin ())^2 + ( Vx cos() + Vy sin ())^2= E cos^2 () + 2F cos() sin()+ G sin^2 ()
E = ux^2 +vx,^2 G = u^2 y +v^2 y
(6.16-1)Equation ( 6.16-1) represents in polar coordinates (p, ()) a quartic called the
elliptic lemniscate t which is the inverse with respect to the unit circle of
tThe elliptic lemniscate is a special case of a family of curves called Perseus's
spiriques (see [44], Vol. 1, Chap. 3).