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Given $X\sim{\cal N}(\mu_X,\sigma^2_X)$, $Y\sim{\cal N}(\mu_Y,\sigma^2_Y)$ I am looking for the p.d.f. of $\operatorname{atan2}(Y,X)$ where $\operatorname{atan2}()$ is the 4-quadrant arc tangent. Does this distribution have a name? Is there any literature about it?

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A nicer--and very suggestive--alternative expression for this value is $\text{arg}(X+iY) = Im(\log(X+iY))$, the polar angle of the complex number $X+iY$. – whuber♦ Aug 22 '12 at 19:18

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I don't think there is a simple expression for the pdf. If there were, then there would be a simple expression for the usual $\arctan$, and of the ratio between two (noncentered) normal distributions. The latter is studied in papers like Marsaglia (1965, 2006) and Cedilnik et al (2004).

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(+1), allowing for one notable exception: when $\mu_X=\mu_Y=0$, the distribution obviously is uniform :-). – whuber♦ Aug 22 '12 at 21:17
@whuber You probably want to insist on $\sigma_X^2 = \sigma_Y^2$ too in addition to $\mu_X = \mu_Y$. But then, Douglas Zare was talking about noncentered normal distributions... – Dilip Sarwate Aug 22 '12 at 21:49

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