三角関数(双曲線関数)の対数とリーマン・ゼータ関数

三角関数の対数とリーマン・ゼータ関数

(1)

\[ \log\left(\sin\left(\pi x\right)\right)=\log\left(\pi x\right)-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}x^{2k} \]

(2)

\[ \log\cos\left(\pi x\right)=-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)\left(2^{2k}-1\right)}{k}x^{2k}+\log1 \]

(3)

\[ \log\tan\left(\pi x\right)=\log\left(\pi x\right)+2\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)\left(2^{2k-1}-1\right)}{k}x^{2k} \]

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(\(\zeta\left(x\right)\)はリーマン・ゼータ関数)

(1)

\begin{align*} \log\left(\sin\left(\pi x\right)\right) & =\log\left(\pi x\right)-\log\frac{\pi x}{\sin\left(\pi x\right)}\\ & =\log\left(\pi x\right)-\log\left(\Gamma(1+x)\Gamma(1-x)\right)\\ & =\log\left(\pi x\right)-\log\Gamma(1+x)-\log\Gamma(1-x)\\ & =\log\left(\pi x\right)-\left(-\gamma x+\sum_{j=2}^{\infty}\frac{(-1)^{j}\zeta\left(j\right)}{j}x^{j}\right)-\left(\gamma x+\sum_{j=2}^{\infty}\frac{(-1)^{j}\zeta\left(j\right)}{j}\left(-x\right)^{j}\right)\slug{mc0bcpgo}\\ & =\log\left(\pi x\right)-\sum_{j=2}^{\infty}\frac{(-1)^{j}\zeta\left(j\right)}{j}\left(1^{j}+(-1)^{j}\right)x^{j}\\ & =\log\left(\pi x\right)-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}x^{2k} \end{align*}

(2)

\begin{align*} \log\cos\left(\pi x\right) & =\log\frac{\sin\left(2\pi x\right)}{2\sin\left(\pi x\right)}\\ & =\log\sin\left(2\pi x\right)-\log2-\log\sin\left(\pi x\right)\\ & =\log\left(2\pi x\right)-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)2^{2k}}{k}x^{2k}-\log2-\left(\log\left(\pi x\right)-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}x^{2k}\right)\\ & =-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)2^{2k}}{k}x^{2k}+\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}x^{2k}\\ & =-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)\left(2^{2k}-1\right)}{k}x^{2k}+\log1 \end{align*}

(3)

\begin{align*} \log\tan\left(\pi x\right) & =\log\frac{\sin\left(\pi x\right)}{\cos\left(\pi x\right)}\\ & =\log\sin\left(\pi x\right)-\log\cos\left(\pi x\right)\\ & =\log\left(\pi x\right)-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}x^{2k}+\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)\left(2^{2k}-1\right)}{k}x^{2k}\\ & =\log\left(\pi x\right)+\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)\left(2^{2k}-2\right)}{k}x^{2k}\\ & =\log\left(\pi x\right)+2\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)\left(2^{2k-1}-1\right)}{k}x^{2k} \end{align*}

(3)-2

\begin{align*} \log\tan\left(\pi x\right) & =\log\frac{2\sin^{2}\left(\pi x\right)}{\sin\left(2\pi x\right)}\\ & =\log2+2\log\sin\left(\pi x\right)-\log\sin\left(2\pi x\right)\\ & =\log2+2\left(\log\left(\pi x\right)-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}x^{2k}\right)-\left(\log\left(2\pi x\right)-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)2^{2k}}{k}x^{2k}\right)\\ & =\log\left(\pi x\right)-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)2}{k}x^{2k}+\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)2^{2k}}{k}x^{2k}\\ & =\log\left(\pi x\right)+2\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)\left(2^{2k-1}-1\right)}{k}x^{2k} \end{align*}

双曲線関数の対数とリーマン・ゼータ関数

(1)

\[ \log\left(\sinh\left(\pi x\right)\right)=\log\left(\pi x\right)-\sum_{k=1}^{\infty}\left(-1\right)^{k}\frac{\zeta\left(2k\right)}{k}x^{2k} \]

(2)

\[ \log\cosh\left(\pi x\right)=-\sum_{k=1}^{\infty}\left(-1\right)^{k}\frac{\zeta\left(2k\right)\left(2^{2k}-1\right)}{k}x^{2k}+\log1 \]

(3)

\[ \log\tanh\left(\pi x\right)=\log\left(\pi x\right)+2\sum_{k=1}^{\infty}\left(-1\right)^{k}\frac{\zeta\left(2k\right)\left(2^{2k-1}-1\right)}{k}x^{2k} \]

-

(\(\zeta\left(x\right)\)はリーマン・ゼータ関数)

(1)

\begin{align*} \log\left(\sinh\left(\pi x\right)\right) & =\log\left(\frac{1}{i}\sin\left(i\pi x\right)\right)\\ & =\log i^{-1}+\log\left(\sin\left(i\pi x\right)\right)\\ & =\log i^{-1}+\log\left(i\pi x\right)-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}\left(ix\right)^{2k}\\ & =\log i^{-1}+\log i+\log\left(\pi x\right)-\sum_{k=1}^{\infty}\left(-1\right)^{k}\frac{\zeta\left(2k\right)}{k}x^{2k}\\ & =\log\left(\pi x\right)-\sum_{k=1}^{\infty}\left(-1\right)^{k}\frac{\zeta\left(2k\right)}{k}x^{2k} \end{align*}

(2)

\begin{align*} \log\cosh\left(\pi x\right) & =\log\cos\left(i\pi x\right)\\ & =-\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)\left(2^{2k}-1\right)}{k}\left(ix\right)^{2k}+\log1\\ & =-\sum_{k=1}^{\infty}\left(-1\right)^{k}\frac{\zeta\left(2k\right)\left(2^{2k}-1\right)}{k}x^{2k}+\log1 \end{align*}

(3)

\begin{align*} \log\tanh\left(\pi x\right) & =\log\left(\frac{1}{i}\tan\left(i\pi x\right)\right)\\ & =\log i^{-1}+\log\tan\left(i\pi x\right)\\ & =\log i^{-1}+\log\left(i\pi x\right)+2\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)\left(2^{2k-1}-1\right)}{k}\left(ix\right)^{2k}\\ & =\log i^{-1}+\log i+\log\left(\pi x\right)+2\sum_{k=1}^{\infty}\left(-1\right)^{k}\frac{\zeta\left(2k\right)\left(2^{2k-1}-1\right)}{k}x^{2k}\\ & =\log\left(\pi x\right)+2\sum_{k=1}^{\infty}\left(-1\right)^{k}\frac{\zeta\left(2k\right)\left(2^{2k-1}-1\right)}{k}x^{2k} \end{align*}

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三角関数(双曲線関数)の対数とリーマン・ゼータ関数

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