Master Dissertation
Master Dissertation
Master Dissertation
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Proof. Reduce ρ to be slowly varying<br />
ρ(t) = t ω r(t) = t ω t1<br />
exp η(t) +<br />
t<br />
h(s)<br />
s ds<br />
<br />
, for 0 < t < t1,<br />
where we substitute r by the representation given by Theorem 6.2. Since<br />
h(s) → 0 for s → 0 there exists s0(ɛ) ≤ t1 such that |h(s)| ≤ ɛ for s ≤ s0(ɛ).<br />
So if 0 < t ≤ s0(ɛ), then<br />
<br />
<br />
<br />
<br />
s0(ɛ)<br />
t<br />
h(s)<br />
s ds<br />
<br />
<br />
<br />
≤<br />
s0(ɛ)<br />
t<br />
<br />
<br />
<br />
h(s) <br />
<br />
s ds ≤<br />
s0(ɛ)<br />
t<br />
ɛ<br />
s ds = ɛ ln s0(ɛ) − ɛ ln t<br />
<br />
s0(ɛ)<br />
= ɛ ln .<br />
t<br />
Hence<br />
ρ(t) = t ω s0(ɛ) h(s)<br />
exp η(t) +<br />
t s ds<br />
<br />
≤ t ω <br />
s0(ɛ)<br />
exp η(t) + ɛ ln<br />
t<br />
= t ω−ɛ e η(t) s ɛ 0(ɛ)<br />
≤ Ct ω−ɛ ,<br />
where C = s ɛ 0 (ɛ) sup 0