Jordan Lemma Proof - Gauge-institute.org

Jordan Lemma Proof
H. Vic Dannon
The cord between the endpoints over the interval
π
0 ≤ θ ≤
2
has the slope
and its equation is
cos − cos 0 2
=− ,
− 0 π
π
2
π
2
2
y − 1 = − θ .
π
Since
cosθ
is concave down in
π
0 θ
2
≤ ≤ , the cord between the
endpoints is under the graph of
cosθ . That is,
Therefore,
and
2
cos θ ≥ y = 1 − θ.
π
2
−cos θ ≤ − 1 + θ,
π
=
π
θ=
π
2 2
2
mρ( −cos θ)
− mρ+
mρ θ
e d ≤ e
π
∫ ∫ dθ
θ
ρ θ ρ
θ= 0 θ=
0
θ=
π
2
1 m
= ρ
mρ
∫
e
θ=
0
e
2
π
ρ θ
dθ
8