2.5 • THE RECIPROCAL TR.ANSFOR.MA'I'ION W = ~ 91- Limits involving oo. The function f (z) is said to have the limit L as z approaches
oo, and we write lim f (z) = Liff for every g > 0 there exists an R > 0 such that
•-oo
f (z) E D, (L) (i.e., If (z) - LI < g) whenever lzl > R. Likewise, :r-.zo lim f (z) = oo
iff for every R > 0 there exists o > 0 such that If (z)I > R whenever z E Ds (zo)
(i.e., 0 < lz - zol < o). Use this definition to
(a) show that lim! = O.
.i-oo :r
(b) show that z-0 lim! .z: = oo.
Show that the reciprocal transformation w = ~ maps the vertical strip given by
0 < x < ~ onto the region in the right half-plane Re ( w) > 0 that lies outside the
disk D 1 (1) = {w: lw - l l < l }.
Find the image of the disk D! ( -¥) = { z : I z + ¥I < n under f (z) = ~.
Show that the reciprocal transformation maps the disk lz - ll < 2 onto the region
that lies exterior to the circle { w : jw +! I = ~ }.
F ind the image of t he half-plane y > ~ -x under the mapping w = ~-
Show that the half-plane y < x - ~ is mapped onto the disk lw - 1 - ii < v'2 by
the reciprocal transformation.
Find the image of the quadrant x > 1, y > 1 under t he mapping w = ~·
Show that the transformation w = ~ maps the disk lz -i i < 1 onto the lower
half-plane Im(w) < -1.
Show that t he transformation w =^2 -;• = - 1 +~maps the disk lz -11 < 1 onto
t he right half-plane Re (w) > 0.
Show that the parabola 2x = 1 - y^2 is mapped onto the cardioid p = 1 +cos</> by
the reciprocal transformation.
Use the definition in Exercise 9 to prove that lim ¥-'t = 1.
•-oo
Show that z = x + iy is mapped onto the point ( ~•+~'+P ~·+~'+i> ~;::~r:,) on
t he Riemann sphere.
Explain how the quantities +oo, - oo, and oo differ. How are they similar?