Functions. Limits and Continuity. Arcs and Curves 159length of such an arc is invariant under a change of parameter t = h( 1),provided that h' ('Y) is continuous and that h' ('Y) > 0 on [a', ,8'].
- The curves
11: z = eit, 0 :St :S 27r
12 : z = e^2 trig(t), 0 :::; t :::; 27r
where g(t) = tsin(l/t) fort of 0, and g(O) = O, have the same graph
(the unit circle). Prove that the first curve is rectifiable, whereas the
second is not.- Show that the graph of
z=
A+ Bt + ct^2
1 + t^2-oo :S t :S +oowhe-re A, B, and C are complex constants, is an ellipse (possibly a
degenerated ellipse).
*5. (a) Let 1: z = z(t), a :S t :S ,8. The distance from a point a to 'Y is
defined by
d(a, 1) =inf {d(a, z): z E 1*}
Show that actually
d(a,1) = min{d(a,z): z E 1*}If a of z(t) for all t E [a, ,8], show that there exists a p > 0 such that
lz(t) - al > p for all t E [a, ,8].
(b) If 1* is contained in some open set G, show that
r=d(1*,G')= inf d(z(t),()>0
tE[a,,B]
(EG'- Let 1: z = z( s ), 0 :S s :S L be a simple regular arc par~etrized by
arc length, and let R be a region containing 1. Suppose that zo = z(O)
and z1 = z( L ). Let Zz E 1, with z 2 f:. zo, Zz of z 1 • Show that there
is an arc r of class C^1 with graph in R joining Zo and Z1 which does
not contain z 2 •- Find all subsets of the complex plane which, like the positive real axis or
the unit circle, are (1) subgroups of the multiplication group of complex
numbers and (2) graphs of arcs.
R. Hersh, [Amer. Math. Monthly, 71 (1964), 800] - Let C be a simple closed curve enclosing a region R that is starlike
with star center 0. Show that C is radially symmetric [i.e., such that
r( 8 + 7r) = r( 8)] i:ff every line through 0 divides R into subregions of
equal area.
G. P. Graham and T. A. Porsching, [Amer. Math. Monthly, 73 (1966),
542]