SOLUTIONS OF HW5N
MINGFENG ZHAO
March 16, 2011
1. [Problem 2, in Page 55] Show that the linear transformation which carries the points z1 and
z2 =6= ∞ into the points w1 6= ∞ and w2 = ∞, respectively, maps the Steiner circles belonging to z1
and z2 into circles about w1 and straight lines through w1 .
2. [Problem 11, in Page 56] Show that there exists a linear transformation which maps four arbitrarily
given points onto the points 1, −1, k, −k, where k depends upon the given points. How many different
solutions are there to this problem, and how are they related?
3. [Problem 16, in Page 56] Map the half-pane Re z ≤
1
2
onto the unit disk |w| ≤ 1 in such a way that
the points z = 0, ∞ correspond to the points w = 0, −1. Divide up the w-plane by the coordinate
axes and the bisectors of the angles between them, draw the corresponding curves in the z-plane, and
investigate how the different sections of the two planes are associated.
4. [Problem 17, in Page 56] Suppose that there are given two circles in the z-plane which have no
common points. Show that the domain bounded by these two circles can be mapped onto an annulus
whose boundary is made up of two concentric circles, and that the annulus is uniquely determined up
to a similarity transformation.
5. [Problem 20, in Page 57] Let K1 and K2 be two circles, one of which lies completely inside the
other. Suppose that there exists a circle k1 with the following properties:
a. It is tangent to K1 and K2 ;
b. If one draws a circle k2 which is tangent to K1 , K2 and k1 , and then a circle k3 which is
tangent to K1 , K2 and k2 , etc.
1
2
MINGFENG ZHAO
c. By induction, a circle kn is eventually ontained which is tangent to K1 , K2 , Kn−1 , and to the
first circle k1 , so that one has a closed chain of mutually tangent circles k1 , k2 , · · · , kn which
are all tangent to the circles K1 and K2 .
Show that under this assumption the chain of circles always close, no matter which of the circles
tangent to K1 and K2 is taken as the initial circle k1 .
6. [Problem 29, in Page 58] Let z1 and z2 be two points in the z-plane. Prove that the chordal distance
between their image points on the Riemann sphere is
CR p
|z1 − z2 |
(1 + |z1 |2 )(1 + |z2 |2 )
,
when the diameter of the Riemann sphere has length R > 0.
7. [Problem 33, in Page 58] Prove that every circle in the z-plane corresponds to a circle on the
Riemann sphere which does not pass through the pole P , and conversely.
8. [Problem 36, in Page 59] Prove that if the fixed points of an elliptic transformation map onto
the opposite ends of a diameter on the Riemann sphere, then this transformation corresponds to a
rotation of the Riemann sphere about the said diameter.
9. [Problem 44, in Page]
Department of Mathematics, University of Connecticut, 196 Auditorium Road, Unit 3009, Storrs, CT
06269-3009
E-mail address: mingfeng.zhao@uconn.edu