Manuscript IX-E-2@1 · IX-E-2.tex · lemma · e70d9077

Lemma: Abel integral

Common-regulator evaluation › The cosine distribution · lines 703–721 · proof included or linked

For $ε > 0$ and $a \in R$ ,

\int_{0}^{\infty} e^{- εq} \frac{sin ( a q )}{q} d q = arctan \frac{a}{ε} .

Proof. The integral vanishes at $a = 0$ . Differentiation with respect to $a$ is justified by the exponential regulator and gives

\int_{0}^{\infty} e^{- εq} cos (a q) d q = Re \frac{1}{ε - ia} = \frac{ε}{ε ^{2} + a ^{2}} .

Integration from $0$ to $a$ proves the assertion.

Displayed formulas in this passage

unlabeled · line 705
$\int_{0}^{\infty} e^{- εq} \frac{sin ( a q )}{q} d q = arctan \frac{a}{ε} .$
unlabeled · line 715
$\int_{0}^{\infty} e^{- εq} cos (a q) d q = Re \frac{1}{ε - ia} = \frac{ε}{ε ^{2} + a ^{2}} .$