分母に双曲線関数で分子に3角関数の定積分

\(R\rightarrow\infty\)とする。
\(C_{1}\)を実軸上を\(x:-R\rightarrow R\)として、\(C_{2}\)は\(y:0\rightarrow\pi\)で\(y\)軸と平行に\(R+iy\)として、\(C_{3}\)は\(x:-R\rightarrow R\)で\(x\)軸と平行に\(x+i\pi\)として、\(C_{4}\)は\(y:\pi\rightarrow0\)で\(y\)軸と平行に\(R+iy\)とする。
また、閉曲線\(C\)を\(C=C_{1}+C_{2}+C_{3}+C_{4}\)とする。
\begin{align*} \int_{C_{1}}\frac{\cos z}{\cosh z}dz & =\lim_{R\rightarrow\infty}\int_{-R}^{R}\frac{\cos x}{\cosh x}dx\\ & =\int_{-\infty}^{\infty}\frac{\cos x}{\cosh x}dx \end{align*} \begin{align*} \left|\int_{C_{2}}\frac{\cos z}{\cosh z}dz\right| & =\left|\lim_{R\rightarrow\infty}\int_{0}^{\pi}\frac{\cos\left(R+iy\right)}{\cosh\left(R+iy\right)}idy\right|\\ & =\left|\lim_{R\rightarrow\infty}\int_{0}^{\pi}\frac{\cos\left(R\right)\cos\left(iy\right)-\sin\left(R\right)\sin\left(iy\right)}{\cosh\left(R\right)\cosh\left(iy\right)+\sinh\left(R\right)\sinh\left(iy\right)}idy\right|\\ & =\left|\lim_{R\rightarrow\infty}\int_{0}^{\pi}\frac{\cos\left(R\right)\cosh\left(y\right)-i\sin\left(R\right)\sinh\left(y\right)}{\cosh\left(R\right)\cos\left(y\right)+i\sinh\left(R\right)\sin\left(y\right)}idy\right|\\ & \leq\lim_{R\rightarrow\infty}\int_{0}^{\pi}\left|\frac{\cos\left(R\right)\cosh\left(y\right)-i\sin\left(R\right)\sinh\left(y\right)}{\cosh\left(R\right)\cos\left(y\right)+i\sinh\left(R\right)\sin\left(y\right)}i\right|dy\\ & =\lim_{R\rightarrow\infty}\int_{0}^{\pi}\frac{\cos^{2}\left(R\right)\cosh^{2}\left(y\right)+\sin^{2}\left(R\right)\sinh^{2}\left(y\right)}{\cosh^{2}\left(R\right)\cos^{2}\left(y\right)+\sinh^{2}\left(R\right)\sin^{2}\left(y\right)}dy\\ & =\int_{0}^{\pi}0dy\\ & =0 \end{align*} \begin{align*} \int_{C_{3}}\frac{\cos z}{\cosh z}dz & =\lim_{R\rightarrow\infty}\int_{R}^{-R}\frac{\cos\left(x+i\pi\right)}{\cosh\left(x+i\pi\right)}dx\\ & =\lim_{R\rightarrow\infty}\int_{R}^{-R}\frac{\cos\left(x\right)\cosh\left(\pi\right)-i\sin\left(x\right)\sinh\left(\pi\right)}{-\cosh\left(x\right)}dx\\ & =-\cosh\left(\pi\right)\lim_{R\rightarrow\infty}\int_{R}^{-R}\frac{\cos\left(x\right)}{\cosh\left(x\right)}dx\\ & =\cosh\left(\pi\right)\lim_{R\rightarrow\infty}\int_{-R}^{R}\frac{\cos\left(x\right)}{\cosh\left(x\right)}dx\\ & =\cosh\left(\pi\right)\int_{-\infty}^{\infty}\frac{\cos\left(x\right)}{\cosh\left(x\right)}dx \end{align*} \begin{align*} \left|\int_{C_{4}}\frac{\cos z}{\cosh z}dz\right| & =\left|\lim_{R\rightarrow-\infty}\int_{\pi}^{0}\frac{\cos\left(R+iy\right)}{\cosh\left(R+iy\right)}idy\right|\\ & =\left|\lim_{R\rightarrow\infty}\int_{\pi}^{0}\frac{\cos\left(-R+iy\right)}{\cosh\left(-R+iy\right)}idy\right|\\ & =\left|\lim_{R\rightarrow\infty}\int_{\pi}^{0}\frac{\cos\left(R-iy\right)}{\cosh\left(R-iy\right)}idy\right|\\ & =\left|-\lim_{R\rightarrow\infty}\int_{0}^{\pi}\frac{\cos\left(R-i\left(\pi-y\right)\right)}{\cosh\left(R-i\left(\pi-y\right)\right)}idy\right|\cmt{y\rightarrow\pi-y}\\ & =\left|\lim_{R\rightarrow\infty}\int_{0}^{\pi}\frac{\cos\left(R+iy\right)}{\cosh\left(R+iy\right)}idy\right|\\ & =\left|\int_{C_{2}}\frac{\cos z}{\cosh z}dz\right|\\ & =0 \end{align*} 被積分関数の特異点は分母が0になるときで、その\(z\)は
\begin{align*} 0 & =\cosh z\\ & =\cos\left(iz\right) \end{align*} となるので
\[ z=-i\left(\frac{\pi}{2}+\pi k\right),k\in\mathbb{Z} \] となり、閉曲線\(C\)内には\(\frac{\pi}{2}i\)のみとなる。
\begin{align*} \int_{C}\frac{\cos z}{\cosh z}dz & =2\pi i\Res\left(\frac{\cos z}{\cosh z},\frac{\pi}{2}i\right)\\ & =2\pi i\lim_{z\rightarrow i\frac{\pi}{2}}\frac{\cos z}{\cosh z}\left(z-\frac{\pi}{2}i\right)\\ & =2\pi i\lim_{z\rightarrow i\frac{\pi}{2}}\frac{\cos z}{\sinh z}\\ & =2\pi i\frac{\cos\left(i\frac{\pi}{2}\right)}{\sinh\left(i\frac{\pi}{2}\right)}\\ & =2\pi i\frac{\cosh\left(\frac{\pi}{2}\right)}{i\sin\left(\frac{\pi}{2}\right)}\\ & =2\pi\cosh\left(\frac{\pi}{2}\right) \end{align*} これらより、
\[ \int_{C}\frac{\cos z}{\cosh z}dz=\int_{C_{1}}\frac{\cos z}{\cosh z}dz+\int_{C_{2}}\frac{\cos z}{\cosh z}dz+\int_{C_{3}}\frac{\cos z}{\cosh z}dz+\int_{C_{4}}\frac{\cos z}{\cosh z}dz \] は、先ほど計算したものを代入して、
\begin{align*} 2\pi\cosh\left(\frac{\pi}{2}\right) & =\int_{-\infty}^{\infty}\frac{\cos x}{\cosh x}dx+0+\cosh\left(\pi\right)\int_{-\infty}^{\infty}\frac{\cos x}{\cosh x}dx+0\\ & =\left(1+\cosh\left(\pi\right)\right)\int_{-\infty}^{\infty}\frac{\cos x}{\cosh x}dx \end{align*} となるので、
\begin{align*} \int_{-\infty}^{\infty}\frac{\cos x}{\cosh x}dx & =\frac{2\pi\cosh\left(\frac{\pi}{2}\right)}{1+\cosh\left(\pi\right)}\\ & =\frac{2\pi\cosh\left(\frac{\pi}{2}\right)}{2\cosh^{2}\frac{\pi}{2}}\\ & =\pi\cosh^{-1}\frac{\pi}{2} \end{align*} となる。