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Series expansion of the integrand around $\varphi=\pi$ and integration gives $$H = 2ika\int_{-\pi/2}^{\pi/2}\cos{(\varphi-\phi)}e^{ika[\cos{\varphi}+\cos{(\varphi-\phi)}]}\ d\varphi$$ $$\qquad\qquad=2ika\left[-2+\frac{1}{12} \left(4 (ka)^2+3 i \pi ka+12\right) (\pi -\phi)^2+{\cal O}(\pi-\phi)^4\right].$$