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Rednas

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I fear that this one is really hard, if not impossible, but an analytic answer would be way more usefull than a numerical one. Who can help me in the right direction?

[itex]\int_0^{arccos(a)} d\phi \frac{cos(\phi)}{(cos(\phi)+\sqrt{cos^2(\phi) - a^2})(x \sqrt{cos^2(\phi)-a^2}+y sin(\phi) + z + x_0 cos(\phi) )}[/itex]

with 0<a<1 and the phi integral only over positive values of the squareroot

Approximations for y=0 and a small are also welcome.

This integral comes from the double integral [itex]\int_0^{\infty} dk\int_0^{2\pi}d\phi \frac{cos(\phi)}{(cos(\phi)+\sqrt{cos^2(\phi) - a^2}} e^{i k(x \sqrt{cos^2(\phi)-a^2}+y sin(\phi) + z + x_0 cos(\phi) )}[/itex]

[itex]\int_0^{arccos(a)} d\phi \frac{cos(\phi)}{(cos(\phi)+\sqrt{cos^2(\phi) - a^2})(x \sqrt{cos^2(\phi)-a^2}+y sin(\phi) + z + x_0 cos(\phi) )}[/itex]

with 0<a<1 and the phi integral only over positive values of the squareroot

Approximations for y=0 and a small are also welcome.

This integral comes from the double integral [itex]\int_0^{\infty} dk\int_0^{2\pi}d\phi \frac{cos(\phi)}{(cos(\phi)+\sqrt{cos^2(\phi) - a^2}} e^{i k(x \sqrt{cos^2(\phi)-a^2}+y sin(\phi) + z + x_0 cos(\phi) )}[/itex]

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