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Lemma A.1 .
Given ν ∈ ℝ \nu\in\mathbb{R} y > 0 y>0
(A.7) I ν ( y ) \displaystyle I_{\nu}(y) = 1 π ∫ 0 π e y cos θ cos ( ν θ ) 𝑑 θ − sin ( ν π ) π ∫ 0 ∞ e − y cosh t − ν t 𝑑 t , \displaystyle=\frac{1}{\pi}\int_{0}^{\pi}e^{y\cos\theta}\cos(\nu\theta)d\theta-\frac{\sin(\nu\pi)}{\pi}\int_{0}^{\infty}e^{-y\cosh t-\nu t}dt,
(A.8) K ν ( y ) \displaystyle K_{\nu}(y) = ∫ 0 ∞ e − y cosh t cosh ( ν t ) 𝑑 t . \displaystyle=\int_{0}^{\infty}e^{-y\cosh t}\cosh(\nu t)dt.