Manuscript I-B-VP-2@1 · I-B-VP-2-v1.tex · lemma · f4bd2b11

Lemma: Closed angular sum

An exact summed kernel and its boundary values › Summation before continuation · lines 472–500 · proof included or linked

I-B-VP-2@1#lem:sum, I-B-VP-2@1#eq:Sgeom, I-B-VP-2@1#eq:Sclosed

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For $a = η / α$ ,

S_{β} (η, δ) = = \frac{e ^{- β a + i β δ}}{1 - e ^{- a + i δ}} + \frac{e ^{- (1 - β) a - i (1 - β) δ}}{1 - e ^{- a - i δ}} \frac{e ^{i β δ} sinh (( 1 - β ) a ) + e ^{- i (1 - β) δ} sinh ( β a )}{cosh a - cos δ} .

The formulas include $β = 0$ . In particular, the meromorphic continuation of $S_{β}$ is odd in $η$ .

Proof. Separate $n \geq 0$ and $n \leq - 1$ and sum the two convergent geometric series. Putting their terms over a common denominator gives (eq:Sclosed). Its numerator is odd in $a$ and its denominator is even.

Displayed formulas in this passage

I-B-VP-2@1#eq:Sgeom · line 475
$S_{β} (η, δ) = = \frac{e ^{- β a + i β δ}}{1 - e ^{- a + i δ}} + \frac{e ^{- (1 - β) a - i (1 - β) δ}}{1 - e ^{- a - i δ}} \frac{e ^{i β δ} sinh (( 1 - β ) a ) + e ^{- i (1 - β) δ} sinh ( β a )}{cosh a - cos δ} .$
I-B-VP-2@1#eq:Sclosed · line 475
$S_{β} (η, δ) = = \frac{e ^{- β a + i β δ}}{1 - e ^{- a + i δ}} + \frac{e ^{- (1 - β) a - i (1 - β) δ}}{1 - e ^{- a - i δ}} \frac{e ^{i β δ} sinh (( 1 - β ) a ) + e ^{- i (1 - β) δ} sinh ( β a )}{cosh a - cos δ} .$

Lemma: Closed angular sum

Displayed formulas in this passage

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