Juan Hewlett

2022-01-15

What is $\mathrm{arctan}\left({z}_{1}\right)±\mathrm{arctan}\left({z}_{2}\right)$ with ${z}_{1},{z}_{2}\in \mathbb{C}$
On wikipedia, there is the following identity:
$\mathrm{arctan}\left(u\right)±\mathrm{arctan}\left(v\right)=\mathrm{arctan}\left(\frac{u±v}{1\mp uv}\right)$
However when I try some $u,v\in \mathbb{C}$ to check, the formula does not hold. Is there an equivalent formula for complex numbers?

Terry Ray

Expert

The problem is that arctan is a multivalued function. If you want a specific function, you need to specify which branch you are using. For example, the principal branch has real part in $\left(-\frac{\pi }{2},\frac{\pi }{2}\right]$. Other branches will differ from that one by an integer multiple of $\pi$. So the correct results are
$\mathrm{arctan}\left(u\right)±\mathrm{arctan}\left(v\right)=\mathrm{arctan}\left(\frac{u±v}{1\mp uv}\right)+n\pi$
where n is an integer. If you are using the principal branch, it is the integer needed to put the real part of the arctan on the right in the interval $\left(-\frac{\pi }{2},\frac{\pi }{2}\right]$

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