Jordan Lemma Proof - Gauge-institute.org

Jordan Lemma Proof
H. Vic Dannon
In MAPLE, we confirm that
and
θ=
∫
π
2
θ=
0
θ=
∫
π
2
θ=
0
e
e
−2cosθ
−2cosθ
dθ
dθ
=
=
(1/2)*Pi*BesselI(0, 2)-(1/2)*Pi*StruveL(0, 2)
(1/2)*Pi*BesselI(0, 2)-(1/2)*Pi*StruveL(0, 2)
In the notations of Abramowitz, and Stegun
BesselI(0, 2)=Modified Bessel Function I
0(2)
StruveL(0, 2)=Struve Function L (2) .
That is, both integrals equal
π
( 0(2) 0(2)
)
2 I − L .
This confirms Whittaker’s guess of equality.
Nevertheless, it seems easier to complete the Jordan Lemma Proof
by bounding
θ=
∫
π
2
θ=
0
e
−mρcosθ
0
d θ, then by trying to prove the equality.
4. Completed Proof
To complete the Jordan Lemma Proof, we need to bound
ρ
θ=
∫
π
2
θ=
0
e
−ρcos
θ
dθ, similarly to the bounding of
ρ
π
2
θ=
−ρsin
θ
∫ e dθ.
θ=
0
7