Juan Hewlett

Answered

2022-01-15

What is $\mathrm{arctan}\left({z}_{1}\right)\pm \mathrm{arctan}\left({z}_{2}\right)$ with $z}_{1},{z}_{2}\in \mathbb{C$

On wikipedia, there is the following identity:

$\mathrm{arctan}\left(u\right)\pm \mathrm{arctan}\left(v\right)=\mathrm{arctan}\left(\frac{u\pm v}{1\mp uv}\right)$

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?

On wikipedia, there is the following identity:

However when I try some

Answer & Explanation

Terry Ray

Expert

2022-01-16Added 50 answers

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 $(-\frac{\pi}{2},\frac{\pi}{2}]$ . Other branches will differ from that one by an integer multiple of $\pi$ . So the correct results are

$\mathrm{arctan}\left(u\right)\pm \mathrm{arctan}\left(v\right)=\mathrm{arctan}\left(\frac{u\pm v}{1\mp uv}\right)+n\pi$

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$(-\frac{\pi}{2},\frac{\pi}{2}]$

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

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