What is arctan⁡(z1)±arctan⁡(z2) with z1,z2∈C On wikipedia, there is the following identity: arctan⁡(u)±arctan⁡(v)=arctan⁡(u±v1∓uv) However when...

Juan Hewlett

Juan Hewlett

Answered

2022-01-15

What is arctan(z1)±arctan(z2) with z1,z2C
On wikipedia, there is the following identity:
arctan(u)±arctan(v)=arctan(u±v1uv)
However when I try some u,vC to check, the formula does not hold. Is there an equivalent formula for complex numbers?

Answer & Explanation

Terry Ray

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 (π2,π2]. Other branches will differ from that one by an integer multiple of π. So the correct results are
arctan(u)±arctan(v)=arctan(u±v1uv)+nπ
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 (π2,π2]

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