During a presentation for a PhD engineering topic I had shown the $\operatorname{atan2}(y, x)$ function within a slide as part of an equation. I received a remark by a professor that $\operatorname{atan2}(y, x)$ is not official and that I should not use it. Then I sent him the definition of $\operatorname{atan2}(y, x)$ as:
and he said that I should've used that one instead of $\operatorname{atan2}(y, x).$
I'd like to ask if there is a correct mathematical approach towards using this function within papers and scientific presentations so I can avoid any further remarks. Using the complete mathematical definition seems rather cumbersome, is there any simplified approach? What would be the correct approach in such situations? In the end I'd like to ask if there is any guideline book that contains the official math notation which can only be used in science papers or presentations.
Best Answer
I would be fine with seeing $\operatorname{atan2}(y, x)$ in a mathematics paper and just assume that the author prefers that notation over alternatives. But a brief explanation for readers not familiar with the notation wouldn't hurt, eg "$\operatorname{atan2}(y, x)$ is the polar angle of the point $(x, y) \ne (0, 0)$ taking values in $(-\pi, \pi]$" or "$\operatorname{atan2}(y, x) = \arg(x + iy)$" or one of the explicit formulas. If the exact range of values (looks like $(-\pi, \pi]$ in your formula) is important then I would recommend including it explicitly in any case.
I think the general rule of thumb is that any notation that is unambiguous and explicitly defined is ok, anything else depends on the field and you should use your (or your advisor's or ...) field-specific experience and judgment. In particular a guideline of the type you suggest that tries to cover all of mathematics would not be very useful.