Well-conditioned boundary integral formulations for the ... - Njit

where µ = √ 1 + z 2 z
+ ln
1+ √ , we get that there exist constants N 1+z 2 1 > 0 and c 1 > 0 such that
1
2 c 1k −2 ≤ n + 1
k 2 (2kI
n+1/2 ′ (k)K n+1/2(k) − 1) ≤ c 1 k −2 , for all n > N 1 , k > 0. (102)
Using the asymptotic expansions (Formula 9.7.2 and 9.7.3 in [1]) which are valid for ν fixed and
z → ∞
√ ( π
K ν (z) ∼ √ e −z 1 + µ − 1 ( )) µ
2
+ O
2z 8z z 2
(
I ν(z)
′ 1
∼ √ e z 1 − µ + 3 ( )) µ
2
+ O
2πz 8z z 2 , µ = 4ν 2 (103)
we get that for a fixed n ≥ 0 there exists ˜k n such that
n + 1
4k 3 ≤ m + 1
k 2 (2kI ′ n+1/2 (k)K n+1/2(k) − 1) ≤ n + 1
k 3 , for all k ≥ ˜k n . (104)
We conclude that there exists a constant ˜c = max(c 1 , N 1+1
k 2
) where ˜k 2 = max 0≤n≤N1 ˜kn , such that
0 ≤ n + 1
k 2 (2kI
n+1/2 ′ (k)K n+1/2(k) − 1) ≤ ˜c
k 2 , for all k ≥ ˜k 2 , for all n ≥ 0. (105)
Furthermore, if we use the following result established in Lemma 3.1 in [17], namely there exists a
constant ˜C 1 and a number ˆk 2 > 0 such that
1
4 (n2 + k 2 ) −1/2 ≤ I n+3/2 (k)K n+3/2 (k) ≤ ˜C 1 (n 2 + k 2 ) −1/2 for all k ≥ ˆk 2 , for all n ≥ 0
together with the estimate established in equation (107), we obtain if we take into account equation
(100) that there exists a constant C 3 and a number k 2 = max{˜k 2 , ˆk 2 } such that
1
4 (n2 + k 2 ) −1/2 ≤ −S (1)
n (ik) ≤ C 3 (n 2 + k 2 ) −1/2 + C 4 k −2 for all k ≥ k 2 , for all n ≥ 0.
(iii) We get from the asymptotic expansions (101) that there exist constants N 1 > 0 and c 1 > 0
such that
1
4 k−1 ≤ I ′ n+1/2 (k)K n+1/2(k) ≤ k −1 , for all n > N 1 , k > 0. (106)
On the other hand, using the asymptotic expansions (103) we get that for a fixed n ≥ 0 there exists
˜k n such that
1
4 k−1 ≤ I ′ n+1/2 (k)K n+1/2(k) ≤ k −1 , for all k ≥ ˜k n . (107)
If we let k 3 = max 0≤n≤N1 ˜kn we obtain that
|J ′ n+1/2 (ik)H(1) n+1/2 (ik)| = 2 π I′ n+1/2 (k)K n+1/2(k) ≤ 2 π k−1 , for all k ≥ k 3 , 0 ≤ n. (108)
The estimate concerning |J n+1/2 (ik)(H (1)
n+1/2 )′ (ik)| follows form the previous estimate and the Wronskian
identity (formula 9.6.15 in [1]). Finally, if we take ˜k 0 = max{k 1 , k 2 , k 3 }, the result of the
Lemma follows.
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