Show that symmetry does not depend complex analysis
Answers
Answer:
Step-by-step explanation:
Let C
C
be a circle or a line belonging to C¯¯¯¯
C
¯
and let z2,z3,z4
z
2
,
z
3
,
z
4
. Two points z
z
and z∗
z
∗
are said to be symmetric with respecto to C
C
if (z,z2,z3,z4)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯=(z∗,z2,z3,z4)
(
z
,
z
2
,
z
3
,
z
4
)
¯
=
(
z
∗
,
z
2
,
z
3
,
z
4
)
.
(i)
(
i
)
Prove that the previous definition doesn't depend on the chosen points z2,z3,z4∈C
z
2
,
z
3
,
z
4
∈
C
but of C
C
.
(ii)
(
i
i
)
Prove that for each z∈C¯¯¯¯
z
∈
C
¯
there is a unique symmetric point z∗
z
∗
with respect to C
C
. The function that assigns to each z
z
its correspondent z∗
z
∗
with respect to C
C
is called symmetry with respect to C
C
. Show that for each Möbius transformation T
T
which maps R¯¯¯¯
R
¯
to C
C
, the function
T∘T−1¯¯¯¯¯¯¯¯¯:C¯¯¯¯→C¯¯¯¯
T
∘
T
−
1
¯
:
C
¯
→
C
¯
is the symmetry with respect to C
C
.
My attempt
(i)
(
i
)
Let C
C
be a circle centered at c
c
of radius R
R
Using invariance of cross ratio under Möbius transformations, and using that zi−c=R
z
i
−
c
=
R
for i=2,3,4
i
=
2
,
3
,
4
and zz¯¯¯=|z|2
z
z
¯
=
|
z
|
2
we get
(z,z2,z3,z4)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯=(z−c,z2−c,z3−c,z4−c)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯=(z¯¯¯−c¯¯,z2−c¯¯¯¯¯¯¯¯¯¯¯¯¯,z3−c¯¯¯¯¯¯¯¯¯¯¯¯¯,z4−c¯¯¯¯¯¯¯¯¯¯¯¯¯)=(z¯¯¯−c¯¯,R2z2−c,R2z3−c,R2z4−c)=(R2z¯¯¯−c¯¯,z2−c,z3−c,z4−c)=(R2z¯¯¯−c¯¯+c,z2,z3,z4)
(
z
,
z
2
,
z
3
,
z
4
)
¯
=
(
z
−
c
,
z
2
−
c
,
z
3
−
c
,
z
4
−
c
)
¯
=
(
z
¯
−
c
¯
,
z
2
−
c
¯
,
z
3
−
c
¯
,
z
4
−
c
¯
)
=
(
z
¯
−
c
¯
,
R
2
z
2
−
c
,
R
2
z
3
−
c
,
R
2
z
4
−
c
)
=
(
R
2
z
¯
−
c
¯
,
z
2
−
c
,
z
3
−
c
,
z
4
−
c
)
=
(
R
2
z
¯
−
c
¯
+
c
,
z
2
,
z
3
,
z
4
)
So if C
C
is a circle, from this equation one deduces the dependence only on C
C
.
(ii)
(
i
i
)
If C
C
is a circle, from the formula z∗=R2z¯¯¯−c¯¯+c
z
∗
=
R
2
z
¯
−
c
¯
+
c
it follows the uniqueness of z∗
z
∗
.
I need help to show (i)
(
i
)
and uniqueness of C
C
if C
C
is a line. I also don't know what to do to show that T∘T−1¯¯¯¯¯¯¯¯¯:C¯¯¯¯→C¯¯¯¯
T
∘
T
−
1
¯
:
C
¯
→
C
¯
is the symmetry with respect to