Find the value of \log \frac{a+bi}{a-bi}

Maximus George

Maximus George

Answered question

2022-01-24

Find the value of loga+biabi

Answer & Explanation

Jordyn Horne

Jordyn Horne

Beginner2022-01-25Added 16 answers

Write a+bi=reiθ; then
a+biabi=reiθreiθ=e2iθ
Thus
loga+biabi=reiθreiθ=loge2iθ=2iθ+2kπi
It's generally wrong to say that θ=arctanba. A simple example is
a+bi=12i32
where the argument is θ=2π3, but
arctanba=arctan(3)=π3
Lily Thornton

Lily Thornton

Beginner2022-01-26Added 19 answers

Let θ=tan1ba
Then tan(2θ)=2tan(θ)1tan2(θ)=
=2ba1b2a2

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