Manuscript I-B-VP@3 · I-B-VP-v3.tex · theorem · 93c04eb4

Theorem: Small-radius asymptotics with corrected remainders

Scale-dependent diagonal nonnegative extensions · lines 2239–2317 · proof included or linked

I-B-VP@3#thm:small

As $ε ↓ 0$ , $Φ_{ν} \to 1$ . For $0 < ν < 1/2$ ,

Φ_{ν} = 1 - A_{ν} ε^{2 ν} + R_{ν} (ε),

where

A_{ν} = \frac{c _{0}}{ν} \frac{π Γ ( 1 - 2 ν ) Γ ( 1 - ν )}{4Γ ( 3/2 - ν )}

and

R_{ν} (ε) = ⎩ ⎨ ⎧ O (ε^{4 ν}), O (ε^{4/3} lo g (1/ ε)), O (ε^{2 - 2 ν}), 0 < ν < 1/3, ν = 1/3, 1/3 < ν < 1/2.

At $ν = 1/2$ ,

Φ_{1/2} (ε) = 1 - 2 ε e^{2 ε} E_{1} (2 ε) = 1 - 2 ε lo g (1/ ε) + O (ε) .

For $1/2 < ν < 1$ ,

Φ_{ν} = 1 - B_{ν} ε^{2 - 2 ν} + O (ε^{2 ν}),

where

B_{ν} = \frac{c _{0}}{ν} \frac{4 ^{ν} Γ ( ν ) ^{2}}{4} \frac{π}{2 ν sin ( π ( 1 - ν ) / ν )} .

The appropriately rescaled current and stress converge to their $E_{0}$ values.

Proof. The bound by one in the coefficient and the integrability of $x K_{ν}^{2}$ give dominated convergence. For $ν < 1/2$ , expand once:

\frac{1}{1 + ( ε / x ) ^{2 ν}} = 1 - (ε / x)^{2 ν} + \frac{( ε / x ) ^{4 ν}}{1 + ( ε / x ) ^{2 ν}} .

The first correction is the convergent Mellin integral defining $A_{ν}$ . In the last term use $K_{ν} (x)^{2} = O (x^{- 2 ν})$ near zero and split at $x = ε, 1$ . The remaining power integral is governed by $x^{1 - 6 ν}$ , giving the three remainder regimes.

For $ν > 1/2$ , subtract the leading small- $x$ expression

K_{ν} (x)^{2} = a_{ν} x^{- 2 ν} + O (1), a_{ν} = 2^{2 ν - 2} Γ (ν)^{2} .

The leading deficit is obtained by $x = ε y$ and

\int_{0}^{\infty} \frac{y ^{1 - 2 ν}}{1 + y ^{2 ν}} d y = \frac{π}{2 ν sin ( π ( 1 - ν ) / ν )} .

After this subtraction the remainder divided by $ε^{2 ν}$ is integrable both at zero and infinity, which proves the stated bound.

For $ν = 1/2$ , use $K_{1/2} (x)^{2} = π e^{- 2 x} / (2 x)$ and integrate directly. The stress limits follow by the same domination applied to the derivative integrands.

Displayed formulas in this passage

unlabeled · line 2243
$Φ_{ν} = 1 - A_{ν} ε^{2 ν} + R_{ν} (ε),$
unlabeled · line 2247
$A_{ν} = \frac{c _{0}}{ν} \frac{π Γ ( 1 - 2 ν ) Γ ( 1 - ν )}{4Γ ( 3/2 - ν )}$
unlabeled · line 2253
$R_{ν} (ε) = ⎩ ⎨ ⎧ O (ε^{4 ν}), O (ε^{4/3} lo g (1/ ε)), O (ε^{2 - 2 ν}), 0 < ν < 1/3, ν = 1/3, 1/3 < ν < 1/2.$
unlabeled · line 2262
$Φ_{1/2} (ε) = 1 - 2 ε e^{2 ε} E_{1} (2 ε) = 1 - 2 ε lo g (1/ ε) + O (ε) .$
unlabeled · line 2268
$Φ_{ν} = 1 - B_{ν} ε^{2 - 2 ν} + O (ε^{2 ν}),$
unlabeled · line 2273
$B_{ν} = \frac{c _{0}}{ν} \frac{4 ^{ν} Γ ( ν ) ^{2}}{4} \frac{π}{2 ν sin ( π ( 1 - ν ) / ν )} .$
unlabeled · line 2286
$\frac{1}{1 + ( ε / x ) ^{2 ν}} = 1 - (ε / x)^{2 ν} + \frac{( ε / x ) ^{4 ν}}{1 + ( ε / x ) ^{2 ν}} .$
unlabeled · line 2300
$K_{ν} (x)^{2} = a_{ν} x^{- 2 ν} + O (1), a_{ν} = 2^{2 ν - 2} Γ (ν)^{2} .$
unlabeled · line 2305
$\int_{0}^{\infty} \frac{y ^{1 - 2 ν}}{1 + y ^{2 ν}} d y = \frac{π}{2 ν sin ( π ( 1 - ν ) / ν )} .$

Theorem: Small-radius asymptotics with corrected remainders

Displayed formulas in this passage

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