三角関数と双曲線関数の対数

by nomura · 2020年10月4日

三角関数の対数
対数を多価関数とすると以下が成り立つ。

(1)

\[ \log\sin x=-\log2+\frac{\pi}{2}i-ix-Li_{1}\left(e^{2ix}\right) \]

(2)

\[ \log\cos x=-\log2-ix-Li_{1}\left(-e^{2ix}\right) \]

(3)

\[ \log\tan x=\frac{\pi}{2}i-Li_{1}\left(e^{2ix}\right)+Li_{1}\left(-e^{2ix}\right)+\log1 \]

(1)

\begin{align*} \log\sin x & =\log\frac{e^{ix}-e^{-ix}}{2i}\\ & =\log\frac{ie^{-ix}}{2}+\log\left(1-e^{2ix}\right)\\ & =-\log2+\frac{\pi}{2}i-ix-Li_{1}\left(e^{2ix}\right) \end{align*}

(2)

\begin{align*} \log\cos x & =\log\frac{e^{ix}+e^{-ix}}{2}\\ & =\log\frac{e^{-ix}}{2}+\log\left(1+e^{2ix}\right)\\ & =-\log2-ix-Li_{1}\left(-e^{2ix}\right) \end{align*}

(3)

\begin{align*} \log\tan x & =\log\sin x-\log\cos x\\ & =\frac{\pi}{2}i-Li_{1}\left(e^{2ix}\right)+Li_{1}\left(-e^{2ix}\right)+\log1 \end{align*}

双曲線関数の対数
対数を多価関数とすると以下が成り立つ。

(1)

\[ \log\sinh x=-\log2+x-Li_{1}\left(e^{-2x}\right) \]

(2)

\[ \log\cosh x=-\log2+x-Li_{1}\left(-e^{-2x}\right) \]

(3)

\[ \log\tanh x=-Li_{1}\left(e^{-2x}\right)+Li_{1}\left(-e^{-2x}\right)+\log1 \]

(1)

\begin{align*} \log\sinh x & =\log\left(\frac{1}{i}\sin\left(ix\right)\right)\\ & =-\log i+\log\left(\sin\left(ix\right)\right)\\ & =-\frac{\pi}{2}i-\log2+\frac{\pi}{2}i+x-Li_{1}\left(e^{-2x}\right)\\ & =-\log2+x-Li_{1}\left(e^{-2x}\right) \end{align*}

(2)

\begin{align*} \log\cosh x & =\log\cos\left(ix\right)\\ & =-\log2+x-Li_{1}\left(-e^{-2x}\right) \end{align*}

(3)

\begin{align*} \log\tanh x & =\log\sinh x-\log\cosh x\\ & =-Li_{1}\left(e^{-2x}\right)+Li_{1}\left(-e^{-2x}\right)+\log1 \end{align*}

(3)-2

\begin{align*} \log\tanh x & =\log\left(\frac{1}{i}\tan\left(ix\right)\right)\\ & =-\log i+\log\left(\tan\left(ix\right)\right)\\ & =-\frac{\pi}{2}i+\frac{\pi}{2}i-Li_{1}\left(e^{-2x}\right)+Li_{1}\left(-e^{-2x}\right)+\log1\\ & =-Li_{1}\left(e^{-2x}\right)+Li_{1}\left(-e^{-2x}\right)+\log1 \end{align*}