基本的な関数のフーリエ変換
基本的な関数のフーリエ変換
基本的な関数のフーリエ変換は次のようになる。
2乗絶対可積分関数
\[ \begin{array}{|c|c|c|c|c|} \hline & f\left(x\right) & \mathcal{F}_{1,x}\left[f\left(x\right)\right]\left(\xi\right) & \mathcal{F}_{2,x}\left[f\left(x\right)\right]\left(k\right) & \mathcal{F}_{3,x}\left[f\left(x\right)\right]\left(\nu\right)\\ & & =\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi i\xi x}dx & =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)e^{-ikx}dx & =\int_{-\infty}^{\infty}f\left(x\right)e^{-i\nu x}dx\\ \hline (1) & e^{-\left|a\right|x}H\left(x\right) & \frac{1}{\left|a\right|+2\pi i\xi} & \frac{1}{\sqrt{2\pi}\left(\left|a\right|+ik\right)} & \frac{1}{\left|a\right|+i\nu}\\ \hline (2) & e^{-\left|a\right|\left|x\right|} & \frac{2\left|a\right|}{\left|a\right|^{2}+4\pi^{2}\xi^{2}} & \sqrt{\frac{2}{\pi}}\frac{\left|a\right|}{\left|a\right|^{2}+k^{2}} & \frac{2\left|a\right|}{\left|a\right|^{2}+\nu^{2}}\\ \hline (3) & e^{-\alpha x^{2}},\Re\left(\alpha\right)>0 & e^{-\frac{\left(\pi\xi\right)^{2}}{\alpha}}\sqrt{\frac{\pi}{\alpha}} & e^{-\frac{k^{2}}{4\alpha}}\frac{1}{\sqrt{2\alpha}} & e^{-\frac{\nu^{2}}{4\alpha}}\sqrt{\frac{\pi}{\alpha}}\\ \hline (4) & \cosh^{-1}\left(ax\right) & \frac{\pi}{\left|a\right|}\cosh^{-1}\left(\frac{\xi}{a}\pi^{2}\right) & \frac{1}{\left|a\right|}\sqrt{\frac{\pi}{2}}\cosh^{-1}\left(\frac{k}{2a}\pi\right) & \frac{\pi}{\left|a\right|}\cosh^{-1}\left(\frac{\nu}{2a}\pi\right)\\ \hline (5) & \mathrm{rect}\left(ax\right) & \frac{1}{\left|a\right|}\mathrm{sinc}\left(\frac{\xi}{a}\pi\right) & \frac{1}{\left|a\right|\sqrt{2\pi}}\mathrm{sinc}\left(\frac{\xi}{2a}\right) & \frac{1}{\left|a\right|}\mathrm{sinc}\left(\frac{\nu}{2a}\right)\\ \hline (6) & \mathrm{sinc}\left(ax\right) & \frac{\pi}{\left|a\right|}\mathrm{rect}\left(\frac{\xi}{a}\pi\right) & \frac{1}{\left|a\right|}\sqrt{\frac{\pi}{2}}\mathrm{rect}\left(\frac{\xi}{2a}\right) & \frac{\pi}{\left|a\right|}\mathrm{rect}\left(\frac{\nu}{2a}\right)\\ \hline (7) & \mathrm{sinc}^{2}\left(ax\right) & \frac{\pi}{\left|a\right|}\mathrm{tri}\left(\frac{\xi}{a}\pi\right) & \frac{1}{\left|a\right|}\sqrt{\frac{\pi}{2}}\mathrm{tri}\left(\frac{\xi}{2a}\right) & \frac{\pi}{\left|a\right|}\mathrm{tri}\left(\frac{\nu}{2a}\right)\\ \hline (8) & \mathrm{tri}\left(ax\right) & \frac{1}{\left|a\right|}\mathrm{sinc}^{2}\left(\frac{\xi}{a}\pi\right) & \frac{1}{\left|a\right|\sqrt{2\pi}}\mathrm{sinc}^{2}\left(\frac{\xi}{2a}\right) & \frac{1}{\left|a\right|}\mathrm{sinc}^{2}\left(\frac{\nu}{2a}\right) \\\hline \end{array} \]
超函数
\[ \begin{array}{|c|c|c|c|c|} \hline & f\left(x\right) & \mathcal{F}_{1,x}\left[f\left(x\right)\right]\left(\xi\right) & \mathcal{F}_{2,x}\left[f\left(x\right)\right]\left(k\right) & \mathcal{F}_{3,x}\left[f\left(x\right)\right]\left(\nu\right)\\ & & =\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi i\xi x}dx & =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)e^{-ikx}dx & =\int_{-\infty}^{\infty}f\left(x\right)e^{-i\nu x}dx\\ \hline (9) & 1 & \delta\left(\xi\right) & \sqrt{2\pi}\delta\left(k\right) & 2\pi\delta\left(\nu\right)\\ \hline (10) & \delta\left(x\right) & 1 & \frac{1}{\sqrt{2\pi}} & 1\\ \hline (11) & e^{iax} & \delta\left(\xi-\frac{a}{2\pi}\right) & \sqrt{2\pi}\delta\left(k-a\right) & 2\pi\delta\left(\nu-a\right)\\ \hline (12) & \cos\left(ax\right) & \frac{\delta\left(\xi-\frac{a}{2\pi}\right)+\delta\left(\xi+\frac{a}{2\pi}\right)}{2} & \sqrt{2\pi}\frac{\delta\left(k-a\right)+\delta\left(k+a\right)}{2} & \pi\left(\delta\left(\nu-a\right)+\delta\left(\nu+a\right)\right)\\ \hline (13) & \sin\left(ax\right) & \frac{\delta\left(\xi+\frac{a}{2\pi}\right)-\delta\left(\xi-\frac{a}{2\pi}\right)}{2}i & \sqrt{2\pi}\frac{\delta\left(k+a\right)-\delta\left(k-a\right)}{2}i & \pi i\left(\delta\left(\nu+a\right)-\delta\left(\nu-a\right)\right)\\ \hline (14) & e^{iax^{2}} & \sqrt{\frac{\pi}{\left|a\right|}}e^{i\left(-\frac{\pi^{2}\xi^{2}}{a}+\sgn\left(a\right)\frac{\pi}{4}\right)} & \frac{1}{\sqrt{2\left|a\right|}}e^{i\left(-\frac{\xi^{2}}{4a}+\sgn\left(a\right)\frac{\pi}{4}\right)} & \sqrt{\frac{\pi}{\left|a\right|}}e^{i\left(\frac{\nu^{2}}{4a}+\sgn\left(a\right)\frac{\pi}{4}\right)}\\ \hline (15) & \cos\left(ax^{2}\right) & \sqrt{\frac{\pi}{\left|a\right|}}\cos\left(\frac{\pi^{2}\xi^{2}}{a}-\sgn\left(a\right)\frac{\pi}{4}\right) & \frac{1}{\sqrt{2\left|a\right|}}\cos\left(\frac{k^{2}}{4a}-\sgn\left(a\right)\frac{\pi}{4}\right) & \sqrt{\frac{\pi}{\left|a\right|}}\cos\left(\frac{\nu^{2}}{4a}-\sgn\left(a\right)\frac{\pi}{4}\right)\\ \hline (16) & \sin\left(ax^{2}\right) & -\sqrt{\frac{\pi}{\left|a\right|}}\sin\left(\frac{\pi^{2}\xi^{2}}{a}-\sgn\left(a\right)\frac{\pi}{4}\right) & -\frac{1}{\sqrt{2\left|a\right|}}\sin\left(\frac{k^{2}}{4\left|a\right|}-\sgn\left(a\right)\frac{\pi}{4}\right) & -\sqrt{\frac{\pi}{\left|a\right|}}\sin\left(\frac{\nu^{2}}{4\left|a\right|}-\sgn\left(a\right)\frac{\pi}{4}\right)\\ \hline (17) & x^{n} & \left(\frac{i}{2\pi}\right)^{n}\delta^{\left(n\right)}\left(\xi\right) & \sqrt{2\pi}i^{n}\delta^{\left(n\right)}\left(k\right) & 2\pi i^{n}\delta^{\left(n\right)}\left(\nu\right)\\ \hline (18) & \frac{1}{x} & -i\pi\sgn\left(\xi\right) & -i\sqrt{\frac{\pi}{2}}\sgn\left(k\right) & -i\pi\sgn\left(\nu\right)\\ \hline (19 & \frac{1}{x^{n}} & \left(-i\pi\right)^{n}\frac{\left(2\xi\right)^{n-1}}{\left(n-1\right)!}\sgn\left(\xi\right) & \sqrt{\frac{\pi}{2}}\left(-i\right)^{n}\frac{k^{n-1}}{\left(n-1\right)!}\sgn\left(k\right) & \pi\left(-i\right)^{n}\frac{\nu^{n-1}}{\left(n-1\right)!}\sgn\left(\nu\right)\\ \hline (20) & \frac{1}{\sqrt{\left|x\right|}} & \frac{1}{\sqrt{\left|\xi\right|}} & \frac{1}{\sqrt{\left|k\right|}} & \frac{\sqrt{2\pi}}{\sqrt{\left|\nu\right|}}\\ \hline (21) & \frac{1}{x\pm\left|a\right|i} & \mp2\pi iH\left(\pm\xi\right)e^{-2\pi\left|\xi\right|\left|a\right|} & \mp\sqrt{2\pi}iH\left(\pm k\right)e^{-\left|k\right|\left|a\right|} & \mp2\pi iH\left(\pm\nu\right)e^{-\left|\nu\right|\left|a\right|}\\ \hline (22) & H_{c}\left(x\right) & \frac{1}{2}\left(\frac{1}{i\pi\xi}+\delta\left(\xi\right)\right) & \sqrt{\frac{\pi}{2}}\left(\frac{1}{i\pi k}+\delta\left(k\right)\right) & \pi\left(\frac{1}{i\pi\nu}+\delta\left(\nu\right)\right)\\ \hline (23 & \sgn\left(x\right) & \frac{1}{i\pi\xi} & \sqrt{\frac{2}{\pi}}\cdot\frac{1}{ik} & \frac{2}{i\nu}\\ \hline (24) & \mathrm{comb}_{T}\left(x\right) & \frac{1}{T}\mathrm{comb}_{\frac{1}{T}}\left(x\right) & \frac{\sqrt{2\pi}}{T}\mathrm{comb}_{\frac{2\pi}{T}}\left(k\right) & \frac{2\pi}{T}\mathrm{comb}_{\frac{2\pi}{T}}\left(\nu\right) \\\hline \end{array} \]
基本的な関数のフーリエ変換は次のようになる。
2乗絶対可積分関数
\[ \begin{array}{|c|c|c|c|c|} \hline & f\left(x\right) & \mathcal{F}_{1,x}\left[f\left(x\right)\right]\left(\xi\right) & \mathcal{F}_{2,x}\left[f\left(x\right)\right]\left(k\right) & \mathcal{F}_{3,x}\left[f\left(x\right)\right]\left(\nu\right)\\ & & =\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi i\xi x}dx & =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)e^{-ikx}dx & =\int_{-\infty}^{\infty}f\left(x\right)e^{-i\nu x}dx\\ \hline (1) & e^{-\left|a\right|x}H\left(x\right) & \frac{1}{\left|a\right|+2\pi i\xi} & \frac{1}{\sqrt{2\pi}\left(\left|a\right|+ik\right)} & \frac{1}{\left|a\right|+i\nu}\\ \hline (2) & e^{-\left|a\right|\left|x\right|} & \frac{2\left|a\right|}{\left|a\right|^{2}+4\pi^{2}\xi^{2}} & \sqrt{\frac{2}{\pi}}\frac{\left|a\right|}{\left|a\right|^{2}+k^{2}} & \frac{2\left|a\right|}{\left|a\right|^{2}+\nu^{2}}\\ \hline (3) & e^{-\alpha x^{2}},\Re\left(\alpha\right)>0 & e^{-\frac{\left(\pi\xi\right)^{2}}{\alpha}}\sqrt{\frac{\pi}{\alpha}} & e^{-\frac{k^{2}}{4\alpha}}\frac{1}{\sqrt{2\alpha}} & e^{-\frac{\nu^{2}}{4\alpha}}\sqrt{\frac{\pi}{\alpha}}\\ \hline (4) & \cosh^{-1}\left(ax\right) & \frac{\pi}{\left|a\right|}\cosh^{-1}\left(\frac{\xi}{a}\pi^{2}\right) & \frac{1}{\left|a\right|}\sqrt{\frac{\pi}{2}}\cosh^{-1}\left(\frac{k}{2a}\pi\right) & \frac{\pi}{\left|a\right|}\cosh^{-1}\left(\frac{\nu}{2a}\pi\right)\\ \hline (5) & \mathrm{rect}\left(ax\right) & \frac{1}{\left|a\right|}\mathrm{sinc}\left(\frac{\xi}{a}\pi\right) & \frac{1}{\left|a\right|\sqrt{2\pi}}\mathrm{sinc}\left(\frac{\xi}{2a}\right) & \frac{1}{\left|a\right|}\mathrm{sinc}\left(\frac{\nu}{2a}\right)\\ \hline (6) & \mathrm{sinc}\left(ax\right) & \frac{\pi}{\left|a\right|}\mathrm{rect}\left(\frac{\xi}{a}\pi\right) & \frac{1}{\left|a\right|}\sqrt{\frac{\pi}{2}}\mathrm{rect}\left(\frac{\xi}{2a}\right) & \frac{\pi}{\left|a\right|}\mathrm{rect}\left(\frac{\nu}{2a}\right)\\ \hline (7) & \mathrm{sinc}^{2}\left(ax\right) & \frac{\pi}{\left|a\right|}\mathrm{tri}\left(\frac{\xi}{a}\pi\right) & \frac{1}{\left|a\right|}\sqrt{\frac{\pi}{2}}\mathrm{tri}\left(\frac{\xi}{2a}\right) & \frac{\pi}{\left|a\right|}\mathrm{tri}\left(\frac{\nu}{2a}\right)\\ \hline (8) & \mathrm{tri}\left(ax\right) & \frac{1}{\left|a\right|}\mathrm{sinc}^{2}\left(\frac{\xi}{a}\pi\right) & \frac{1}{\left|a\right|\sqrt{2\pi}}\mathrm{sinc}^{2}\left(\frac{\xi}{2a}\right) & \frac{1}{\left|a\right|}\mathrm{sinc}^{2}\left(\frac{\nu}{2a}\right) \\\hline \end{array} \]
超函数
\[ \begin{array}{|c|c|c|c|c|} \hline & f\left(x\right) & \mathcal{F}_{1,x}\left[f\left(x\right)\right]\left(\xi\right) & \mathcal{F}_{2,x}\left[f\left(x\right)\right]\left(k\right) & \mathcal{F}_{3,x}\left[f\left(x\right)\right]\left(\nu\right)\\ & & =\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi i\xi x}dx & =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)e^{-ikx}dx & =\int_{-\infty}^{\infty}f\left(x\right)e^{-i\nu x}dx\\ \hline (9) & 1 & \delta\left(\xi\right) & \sqrt{2\pi}\delta\left(k\right) & 2\pi\delta\left(\nu\right)\\ \hline (10) & \delta\left(x\right) & 1 & \frac{1}{\sqrt{2\pi}} & 1\\ \hline (11) & e^{iax} & \delta\left(\xi-\frac{a}{2\pi}\right) & \sqrt{2\pi}\delta\left(k-a\right) & 2\pi\delta\left(\nu-a\right)\\ \hline (12) & \cos\left(ax\right) & \frac{\delta\left(\xi-\frac{a}{2\pi}\right)+\delta\left(\xi+\frac{a}{2\pi}\right)}{2} & \sqrt{2\pi}\frac{\delta\left(k-a\right)+\delta\left(k+a\right)}{2} & \pi\left(\delta\left(\nu-a\right)+\delta\left(\nu+a\right)\right)\\ \hline (13) & \sin\left(ax\right) & \frac{\delta\left(\xi+\frac{a}{2\pi}\right)-\delta\left(\xi-\frac{a}{2\pi}\right)}{2}i & \sqrt{2\pi}\frac{\delta\left(k+a\right)-\delta\left(k-a\right)}{2}i & \pi i\left(\delta\left(\nu+a\right)-\delta\left(\nu-a\right)\right)\\ \hline (14) & e^{iax^{2}} & \sqrt{\frac{\pi}{\left|a\right|}}e^{i\left(-\frac{\pi^{2}\xi^{2}}{a}+\sgn\left(a\right)\frac{\pi}{4}\right)} & \frac{1}{\sqrt{2\left|a\right|}}e^{i\left(-\frac{\xi^{2}}{4a}+\sgn\left(a\right)\frac{\pi}{4}\right)} & \sqrt{\frac{\pi}{\left|a\right|}}e^{i\left(\frac{\nu^{2}}{4a}+\sgn\left(a\right)\frac{\pi}{4}\right)}\\ \hline (15) & \cos\left(ax^{2}\right) & \sqrt{\frac{\pi}{\left|a\right|}}\cos\left(\frac{\pi^{2}\xi^{2}}{a}-\sgn\left(a\right)\frac{\pi}{4}\right) & \frac{1}{\sqrt{2\left|a\right|}}\cos\left(\frac{k^{2}}{4a}-\sgn\left(a\right)\frac{\pi}{4}\right) & \sqrt{\frac{\pi}{\left|a\right|}}\cos\left(\frac{\nu^{2}}{4a}-\sgn\left(a\right)\frac{\pi}{4}\right)\\ \hline (16) & \sin\left(ax^{2}\right) & -\sqrt{\frac{\pi}{\left|a\right|}}\sin\left(\frac{\pi^{2}\xi^{2}}{a}-\sgn\left(a\right)\frac{\pi}{4}\right) & -\frac{1}{\sqrt{2\left|a\right|}}\sin\left(\frac{k^{2}}{4\left|a\right|}-\sgn\left(a\right)\frac{\pi}{4}\right) & -\sqrt{\frac{\pi}{\left|a\right|}}\sin\left(\frac{\nu^{2}}{4\left|a\right|}-\sgn\left(a\right)\frac{\pi}{4}\right)\\ \hline (17) & x^{n} & \left(\frac{i}{2\pi}\right)^{n}\delta^{\left(n\right)}\left(\xi\right) & \sqrt{2\pi}i^{n}\delta^{\left(n\right)}\left(k\right) & 2\pi i^{n}\delta^{\left(n\right)}\left(\nu\right)\\ \hline (18) & \frac{1}{x} & -i\pi\sgn\left(\xi\right) & -i\sqrt{\frac{\pi}{2}}\sgn\left(k\right) & -i\pi\sgn\left(\nu\right)\\ \hline (19 & \frac{1}{x^{n}} & \left(-i\pi\right)^{n}\frac{\left(2\xi\right)^{n-1}}{\left(n-1\right)!}\sgn\left(\xi\right) & \sqrt{\frac{\pi}{2}}\left(-i\right)^{n}\frac{k^{n-1}}{\left(n-1\right)!}\sgn\left(k\right) & \pi\left(-i\right)^{n}\frac{\nu^{n-1}}{\left(n-1\right)!}\sgn\left(\nu\right)\\ \hline (20) & \frac{1}{\sqrt{\left|x\right|}} & \frac{1}{\sqrt{\left|\xi\right|}} & \frac{1}{\sqrt{\left|k\right|}} & \frac{\sqrt{2\pi}}{\sqrt{\left|\nu\right|}}\\ \hline (21) & \frac{1}{x\pm\left|a\right|i} & \mp2\pi iH\left(\pm\xi\right)e^{-2\pi\left|\xi\right|\left|a\right|} & \mp\sqrt{2\pi}iH\left(\pm k\right)e^{-\left|k\right|\left|a\right|} & \mp2\pi iH\left(\pm\nu\right)e^{-\left|\nu\right|\left|a\right|}\\ \hline (22) & H_{c}\left(x\right) & \frac{1}{2}\left(\frac{1}{i\pi\xi}+\delta\left(\xi\right)\right) & \sqrt{\frac{\pi}{2}}\left(\frac{1}{i\pi k}+\delta\left(k\right)\right) & \pi\left(\frac{1}{i\pi\nu}+\delta\left(\nu\right)\right)\\ \hline (23 & \sgn\left(x\right) & \frac{1}{i\pi\xi} & \sqrt{\frac{2}{\pi}}\cdot\frac{1}{ik} & \frac{2}{i\nu}\\ \hline (24) & \mathrm{comb}_{T}\left(x\right) & \frac{1}{T}\mathrm{comb}_{\frac{1}{T}}\left(x\right) & \frac{\sqrt{2\pi}}{T}\mathrm{comb}_{\frac{2\pi}{T}}\left(k\right) & \frac{2\pi}{T}\mathrm{comb}_{\frac{2\pi}{T}}\left(\nu\right) \\\hline \end{array} \]
(1)
\begin{align*} \mathcal{F}_{1,x}\left[e^{-\left|a\right|x}H\left(x\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}e^{-\left|a\right|x}H\left(x\right)e^{-2\pi i\xi x}dx\\ & =\int_{0}^{\infty}e^{-\left|a\right|x}e^{-2\pi i\xi x}dx\\ & =\int_{0}^{\infty}e^{-2\pi i\left(\xi+\frac{\left|a\right|}{2\pi i}\right)x}dx\\ & =\frac{1}{-2\pi i\left(\xi+\frac{\left|a\right|}{2\pi i}\right)}\left[e^{-2\pi i\left(\xi+\frac{\left|a\right|}{2\pi i}\right)x}\right]_{0}^{\infty}\\ & =\frac{1}{2\pi i\left(\xi+\frac{\left|a\right|}{2\pi i}\right)}\\ & =\frac{1}{\left|a\right|+2\pi i\xi} \end{align*}(2)
\begin{align*} \mathcal{F}_{1,x}\left[e^{-\left|a\right|\left|x\right|}\right]\left(\xi\right) & =\mathcal{F}_{1,x}\left[e^{-\left|a\right|x}H_{\frac{1}{2}}\left(x\right)+e^{\left|a\right|x}H_{\frac{1}{2}}\left(-x\right)\right]\left(\xi\right)\\ & =\mathcal{F}_{1,x}\left[e^{-\left|a\right|x}H_{\frac{1}{2}}\left(x\right)\right]\left(\xi\right)+\mathcal{F}_{1,x}\left[e^{\left|a\right|x}H_{\frac{1}{2}}\left(-x\right)\right]\left(\xi\right)\\ & =\mathcal{F}_{1,x}\left[e^{-\left|a\right|x}H_{\frac{1}{2}}\left(x\right)\right]\left(\xi\right)+\mathcal{F}_{1,x}\left[e^{-\left|a\right|x}H_{\frac{1}{2}}\left(x\right)\right]\left(-\xi\right)\\ & =\frac{1}{\left|a\right|+2\pi i\xi}+\frac{1}{\left|a\right|-2\pi i\xi}\\ & =\frac{2\left|a\right|}{\left|a\right|^{2}+4\pi^{2}\xi^{2}} \end{align*}(3)
\begin{align*} \mathcal{F}_{1,x}\left[e^{-\alpha x^{2}}\right]\left(\xi\right) & =\int_{-\infty}^{\infty}e^{-\alpha x^{2}}e^{-2\pi i\xi x}dx\\ & =\int_{-\infty}^{\infty}e^{-\alpha\left(x+\frac{\pi i\xi}{\alpha}\right)^{2}-\frac{\left(\pi\xi\right)^{2}}{\alpha}}dx\\ & =e^{-\frac{\left(\pi\xi\right)^{2}}{\alpha}}\int_{-\infty}^{\infty}e^{-\alpha\left(x+\frac{\pi i\xi}{\alpha}\right)^{2}}dx\\ & =e^{-\frac{\left(\pi\xi\right)^{2}}{\alpha}}\sqrt{\frac{\pi}{\alpha}} \end{align*}(4)
\begin{align*} \mathcal{F}_{1,x}\left[\cosh^{-1}\left(ax\right)\right]\left(\xi\right) & =\frac{1}{\left|a\right|}\mathcal{F}_{1,x}\left[\cosh^{-1}\left(x\right)\right]\left(\frac{\xi}{a}\right)\\ & =\frac{1}{\left|a\right|}\int_{-\infty}^{\infty}\cosh^{-1}\left(x\right)e^{-2\pi i\frac{\xi}{a}x}dx\\ & =\frac{2}{\left|a\right|}\int_{-\infty}^{\infty}\frac{e^{-2\pi i\frac{\xi}{a}x}}{e^{x}+e^{-x}}dx\\ & =\frac{2}{\left|a\right|}\int_{-\infty}^{\infty}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz \end{align*} 被積分関数の特異点は\(e^{z}+e^{-z}=0\)となる\(z\)なので\(e^{2z}=-1=e^{i\pi+2n\pi i},n\in\mathbb{Z}\)となり、\(z=\frac{2n+1}{2}\pi i\)となる。このとき、経路\(C_{1}\)を実軸上を\(-R\rightarrow R\)として、経路\(C_{2}\)を\(y\)軸に変更に\(R\rightarrow R+i\pi\)として、経路\(C_{3}\)を\(x\)軸と平行に\(R+i\pi\rightarrow-R+i\pi\)として、経路\(C_{4}\)を\(y\)軸と平行に\(-R+i\pi\rightarrow-R\)とする。
また\(C=C_{1}+C_{2}+C_{3}+C_{4}\)とすると、
\[ \int_{C}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz=\int_{C_{1}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz+\int_{C_{2}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz+\int_{C_{3}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz+\int_{C_{4}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz \] となり\(R\rightarrow\infty\)とする。
閉曲線\(C\)に含まれる特異点は\(z=\)\(\frac{1}{2}\pi i\)のみとなり1位の極なので、
\begin{align*} \int_{C}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz & =2\pi i\Res_{z}\left(\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}},\frac{1}{2}\pi i\right)\\ & =2\pi i\lim_{z\rightarrow\frac{1}{2}\pi i}\left(z-\frac{1}{2}\pi i\right)\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}\\ & =2\pi i\lim_{z\rightarrow\frac{1}{2}\pi i}\frac{e^{-2\pi i\frac{\xi}{a}z}}{\left(e^{z}+e^{-z}\right)'}\\ & =2\pi i\lim_{z\rightarrow\frac{1}{2}\pi i}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}-e^{-z}}\\ & =2\pi i\frac{e^{\pi^{2}\frac{\xi}{a}}}{i-\left(-i\right)}\\ & =\pi e^{\pi^{2}\frac{\xi}{a}} \end{align*} 経路\(C_{2}\)は、
\begin{align*} \left|\int_{C_{2}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz\right| & =\lim_{R\rightarrow\infty}\left|\int_{R}^{R+i\pi}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz\right|\\ & =\lim_{R\rightarrow\infty}\left|i\int_{0}^{\pi}\frac{e^{-2\pi i\frac{\xi}{a}\left(R+iy\right)}}{e^{R+iy}+e^{-R-iy}}dy\right|\cmt{z=R+iy}\\ & =\lim_{R\rightarrow\infty}\left|ie^{-2\pi i\frac{\xi}{a}R}\int_{0}^{\pi}\frac{e^{2\pi\frac{\xi}{a}y}}{e^{R}e^{iy}+e^{-R}e^{-iy}}dy\right|\\ & =\lim_{R\rightarrow\infty}\left|\int_{0}^{\pi}\frac{e^{2\pi\frac{\xi}{a}y}}{e^{R}e^{iy}+e^{-R}e^{-iy}}dy\right|\\ & \leq\lim_{R\rightarrow\infty}\int_{0}^{\pi}\left|\frac{e^{2\pi\frac{\xi}{a}y}}{e^{R}e^{iy}+e^{-R}e^{-iy}}\right|dy\\ & =0 \end{align*} となるので、
\[ \int_{C_{2}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz=0 \] となる。
同様に経路\(C_{4}\)は、
\[ \int_{C_{4}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz=0 \] となる。
経路\(C_{1}\)については、実軸上なので、
\begin{align*} \int_{C_{1}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz & =\lim_{R\rightarrow\infty}\int_{-R}^{R}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz\\ & =\lim_{R\rightarrow\infty}\int_{-R}^{R}\frac{e^{-2\pi i\frac{\xi}{a}x}}{e^{x}+e^{-x}}dx\\ & =\int_{-\infty}^{\infty}\frac{e^{-2\pi i\frac{\xi}{a}x}}{e^{x}+e^{-x}}dx \end{align*} となり、経路\(C_{3}\)は
\begin{align*} \int_{C_{3}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz & =\lim_{R\rightarrow\infty}\int_{R+i\pi}^{-R+i\pi}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz\\ & =\lim_{R\rightarrow\infty}\int_{R}^{-R}\frac{e^{-2\pi i\frac{\xi}{a}\left(-x+i\pi\right)}}{e^{x+i\pi}+e^{-x-i\pi}}dx\cmt{z=x+i\pi}\\ & =e^{2\pi^{2}\frac{\xi}{a}}\lim_{R\rightarrow\infty}\int_{-R}^{R}\frac{e^{-2\pi i\frac{\xi}{a}x}}{e^{x}+e^{-x}}dx\\ & =e^{2\pi^{2}\frac{\xi}{a}}\int_{-\infty}^{\infty}\frac{e^{-2\pi i\frac{\xi}{a}x}}{e^{x}+e^{-x}}dx \end{align*} となる。
これらより、
\begin{align*} \pi e^{\pi^{2}\frac{\xi}{a}} & =\int_{C}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz\\ & =\int_{C_{1}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz+\int_{C_{2}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz+\int_{C_{3}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz+\int_{C_{4}}\frac{e^{-2\pi i\frac{\xi}{a}z}}{e^{z}+e^{-z}}dz\\ & =\int_{-\infty}^{\infty}\frac{e^{-2\pi i\frac{\xi}{a}x}}{e^{x}+e^{-x}}dx+0+e^{2\pi^{2}\frac{\xi}{a}}\int_{-\infty}^{\infty}\frac{e^{-2\pi i\frac{\xi}{a}x}}{e^{x}+e^{-x}}dx+0\\ & =\left(1+e^{2\pi^{2}\frac{\xi}{a}}\right)\int_{-\infty}^{\infty}\frac{e^{-2\pi i\frac{\xi}{a}x}}{e^{x}+e^{-x}}dx \end{align*} となるので、
\begin{align*} \int_{-\infty}^{\infty}\frac{e^{-2\pi i\frac{\xi}{a}x}}{e^{x}+e^{-x}}dx & =\frac{\pi e^{\pi^{2}\frac{\xi}{a}}}{1+e^{2\pi^{2}\frac{\xi}{a}}}\\ & =\frac{\pi}{e^{-\pi^{2}\frac{\xi}{a}}+e^{\pi^{2}\frac{\xi}{a}}}\\ & =\frac{\pi}{2\cosh\left(\pi^{2}\frac{\xi}{a}\right)} \end{align*} となり、
\begin{align*} \mathcal{F}_{1,x}\left[\cosh^{-1}\left(ax\right)\right]\left(\xi\right) & =\frac{2}{\left|a\right|}\int_{-\infty}^{\infty}\frac{e^{-2\pi i\frac{\xi}{a}x}}{e^{x}+e^{-x}}dx\\ & =\frac{2}{\left|a\right|}\cdot\frac{\pi}{2\cosh\left(\pi^{2}\frac{\xi}{a}\right)}\\ & =\frac{\pi}{\left|a\right|\cosh\left(\pi^{2}\frac{\xi}{a}\right)} \end{align*} となる。
(5)
\begin{align*} \mathcal{F}_{1,x}\left[\mathrm{rect}\left(ax\right)\right]\left(\xi\right) & =\frac{1}{\left|a\right|}\mathcal{F}_{1,x}\left[\mathrm{rect}\left(x\right)\right]\left(\frac{\xi}{a}\right)\\ & =\frac{1}{\left|a\right|}\int_{-\infty}^{\infty}\mathrm{rect}\left(x\right)e^{-2\pi i\frac{\xi}{a}x}dx\\ & =\frac{1}{\left|a\right|}\int_{-\frac{1}{2}}^{\frac{1}{2}}e^{-2\pi i\frac{\xi}{a}x}dx\\ & =\frac{1}{\left|a\right|}\cdot\frac{-1}{2\pi i\frac{\xi}{a}}\left[e^{-2\pi i\frac{\xi}{a}x}\right]_{-\frac{1}{2}}^{\frac{1}{2}}\\ & =\frac{-1}{2\pi i\frac{\xi}{a}\left|a\right|}\left(e^{-\pi i\frac{\xi}{a}}-e^{\pi i\frac{\xi}{a}}\right)\\ & =\frac{\sin\left(\frac{\xi}{a}\pi\right)}{\left|a\right|\frac{\xi}{a}\pi}\\ & =\frac{\mathrm{sinc}\left(\frac{\xi}{a}\pi\right)}{\left|a\right|} \end{align*}(6)
\begin{align*} \mathcal{F}_{1,x}\left[\mathrm{sinc}\left(ax\right)\right]\left(\xi\right) & =\frac{1}{\left|a\right|}\mathcal{F}_{1,x}\left[\mathrm{sinc}\left(x\right)\right]\left(\frac{\xi}{a}\right)\\ & =\frac{1}{\left|a\right|}\mathcal{F}_{1,x}\left[\frac{e^{ix}-e^{-ix}}{2ix}\right]\left(\frac{\xi}{a}\right)\\ & =\frac{1}{2i\left|a\right|}\left\{ \mathcal{F}_{1,x}\left[\frac{e^{ix}}{x}\right]\left(\frac{\xi}{a}\right)-\mathcal{F}_{1,x}\left[\frac{e^{-ix}}{x}\right]\left(\frac{\xi}{a}\right)\right\} \\ & =\frac{1}{2i\left|a\right|}\left\{ \mathcal{F}_{1,x}\left[\frac{1}{x}\right]\left(\frac{\xi}{a}-\frac{1}{2\pi}\right)-\mathcal{F}_{1,x}\left[\frac{1}{x}\right]\left(\frac{\xi}{a}+\frac{1}{2\pi}\right)\right\} \\ & =\frac{1}{2i\left|a\right|}\left\{ -i\pi\sgn\left(\frac{\xi}{a}-\frac{1}{2\pi}\right)-\left(-i\pi\sgn\left(\frac{\xi}{a}+\frac{1}{2\pi}\right)\right)\right\} \\ & =\frac{\pi}{2\left|a\right|}\left\{ \sgn\left(\frac{\xi}{a}+\frac{1}{2\pi}\right)-\sgn\left(\frac{\xi}{a}-\frac{1}{2\pi}\right)\right\} \\ & =\frac{\pi}{2\left|a\right|}2\mathrm{rect}\left(\frac{1}{2}\cdot\frac{\left(\frac{\xi}{a}+\frac{1}{2\pi}\right)+\left(\frac{\xi}{a}-\frac{1}{2\pi}\right)}{\left(\frac{\xi}{a}+\frac{1}{2\pi}\right)-\left(\frac{\xi}{a}-\frac{1}{2\pi}\right)}\right)\\ & =\frac{\pi}{\left|a\right|}\mathrm{rect}\left(\pi\frac{\xi}{a}\right) \end{align*}(7)
\begin{align*} \mathcal{F}_{1,x}\left[\mathrm{sinc}^{2}\left(ax\right)\right]\left(\xi\right) & =\frac{1}{\left|a\right|}\mathcal{F}_{1,x}\left[\mathrm{sinc}^{2}\left(x\right)\right]\left(\frac{\xi}{a}\right)\\ & =\frac{1}{\left|a\right|}\mathcal{F}_{1,x}\left[\mathrm{sinc}\left(x\right)\right]\left(\frac{\xi}{a}\right)*_{\frac{\xi}{a}}\mathcal{F}_{1,x}\left[\mathrm{sinc}\left(x\right)\right]\left(\frac{\xi}{a}\right)\\ & =\frac{1}{\left|a\right|}\cdot\pi\mathrm{rect}\left(\pi\frac{\xi}{a}\right)*_{\frac{\xi}{a}}\pi\mathrm{rect}\left(\pi\frac{\xi}{a}\right)\\ & =\frac{\pi^{2}}{\left|a\right|}\int_{-\infty}^{\infty}\mathrm{rect}\left(\pi x\right)\mathrm{rect}\left(\pi\left(\frac{\xi}{a}-x\right)\right)dx\\ & =\frac{\pi}{\left|a\right|}\int_{-\infty}^{\infty}\mathrm{rect}\left(x\right)\mathrm{rect}\left(\pi\frac{\xi}{a}-x\right)dx\cmt{\pi x\rightarrow x}\\ & =\frac{\pi}{\left|a\right|}\mathrm{rect}\left(\pi\frac{\xi}{a}\right)*_{\pi\frac{\xi}{a}}\mathrm{rect}\left(\pi\frac{\xi}{a}\right)\cmt{\because\mathrm{rect}\left(x\right)*_{x}\mathrm{rect}\left(x\right)=\mathrm{tri}\left(x\right)}\\ & =\frac{\pi}{\left|a\right|}\mathrm{tri}\left(\pi\frac{\xi}{a}\right) \end{align*}(8)
\begin{align*} \mathcal{F}_{1,x}\left[\mathrm{tri}\left(ax\right)\right]\left(\xi\right) & =\frac{1}{\left|a\right|}\mathcal{F}_{1,x}\left[\mathrm{tri}\left(x\right)\right]\left(\frac{\xi}{a}\right)\\ & =\frac{1}{\left|a\right|}\mathcal{F}_{1,x}\left[\mathrm{rect}\left(x\right)*\mathrm{rect}\left(x\right)\right]\left(\frac{\xi}{a}\right)\\ & =\frac{1}{\left|a\right|}\mathcal{F}_{1,x}\left[\mathrm{rect}\left(x\right)\right]\left(\frac{\xi}{a}\right)\cdot\mathcal{F}_{1,x}\left[\mathrm{rect}\left(x\right)\right]\left(\frac{\xi}{a}\right)\\ & =\frac{1}{\left|a\right|}\mathcal{F}_{1,x}^{2}\left[\mathrm{rect}\left(x\right)\right]\left(\frac{\xi}{a}\right)\\ & =\frac{1}{\left|a\right|}\mathrm{sinc}^{2}\left(\frac{\xi}{a}\pi\right) \end{align*}(9)
\begin{align*} \mathcal{F}_{1,x}\left[1\right]\left(\xi\right) & =\int_{-\infty}^{\infty}e^{-2\pi i\xi x}dx\\ & =\lim_{n\rightarrow\infty}\int_{-n}^{n}e^{-2\pi i\xi x}dx\\ & =\lim_{n\rightarrow\infty}\frac{1}{2\pi i\xi}\left[e^{2\pi i\xi x}\right]_{-n}^{n}\\ & =\lim_{n\rightarrow\infty}\frac{1}{2\pi i\xi}\left(e^{2\pi i\xi n}-e^{-2\pi i\xi n}\right)\\ & =\lim_{n\rightarrow\infty}\frac{1}{2\pi i\xi}2i\sin\left(2\pi\xi n\right)\\ & =\lim_{n\rightarrow\infty}\frac{1}{\pi\xi}\sin\left(2\pi\xi n\right)\\ & =2\pi\lim_{n\rightarrow\infty}\frac{1}{\pi\cdot2\pi\xi}\sin\left(n\cdot2\pi\xi\right)\\ & =2\pi\delta\left(2\pi\xi\right)\cmt{\because\delta\left(x\right)=\lim_{n\rightarrow\infty}\frac{\sin\left(nx\right)}{\pi x}}\\ & =\delta\left(x\right) \end{align*}(10)
\begin{align*} \mathcal{F}_{1,x}\left[\delta\left(x\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}\delta\left(x\right)e^{-2\pi i\xi x}dx\\ & =1 \end{align*}(11)
\begin{align*} \mathcal{F}_{1,x}\left[e^{iax}\right]\left(\xi\right) & =\mathcal{F}_{1,x}\left[e^{2\pi i\frac{a}{2\pi}x}\right]\left(\xi\right)\\ & =\mathcal{F}_{1,x}\left[1\right]\left(\xi-\frac{a}{2\pi}\right)\\ & =\delta\left(\xi-\frac{a}{2\pi}\right) \end{align*}(12)
\begin{align*} \mathcal{F}_{1,x}\left[\cos\left(ax\right)\right]\left(\xi\right) & =\frac{1}{2}\mathcal{F}_{1,x}\left[e^{iax}+e^{-iax}\right]\left(\xi\right)\\ & =\frac{1}{2}\left(\delta\left(\xi-\frac{a}{2\pi}\right)+\delta\left(\xi+\frac{a}{2\pi}\right)\right) \end{align*}(13)
\begin{align*} \mathcal{F}_{1,x}\left[\sin\left(ax\right)\right]\left(\xi\right) & =\frac{1}{2i}\mathcal{F}_{1,x}\left[e^{iax}-e^{-iax}\right]\left(\xi\right)\\ & =-\frac{i}{2}\left(\delta\left(\xi-\frac{a}{2\pi}\right)-\delta\left(\xi+\frac{a}{2\pi}\right)\right)\\ & =\frac{i}{2}\left(\delta\left(\xi+\frac{a}{2\pi}\right)-\delta\left(\xi-\frac{a}{2\pi}\right)\right) \end{align*}(14)
\begin{align*} \mathcal{F}_{1,x}\left[e^{iax^{2}}\right]\left(\xi\right) & =\int_{-\infty}^{\infty}e^{iax^{2}}e^{-2\pi i\xi x}dx\\ & =\int_{-\infty}^{\infty}e^{iax^{2}-2\pi i\xi x}dx\\ & =\int_{-\infty}^{\infty}e^{ia\left(x-\frac{\pi\xi}{a}\right)^{2}-i\frac{\pi^{2}\xi^{2}}{a}}dx\\ & =e^{-i\frac{\pi^{2}\xi^{2}}{a}}\int_{-\infty}^{\infty}e^{ia\left(x-\frac{\pi\xi}{a}\right)^{2}}dx\\ & =e^{-i\frac{\pi^{2}\xi^{2}}{a}}\int_{-\infty}^{\infty}e^{iax^{2}}dx\cmt{x-\frac{\pi\xi}{a}\rightarrow x}\\ & =2e^{-i\frac{\pi^{2}\xi^{2}}{a}}\int_{0}^{\infty}e^{iax^{2}}dx\\ & =2e^{-i\frac{\pi^{2}\xi^{2}}{a}}\cdot\frac{1}{2}\sqrt{\frac{\pi}{\left|a\right|}}e^{\sgn\left(a\right)\frac{\pi}{4}i}\\ & =\sqrt{\frac{\pi}{\left|a\right|}}e^{-i\frac{\pi^{2}\xi^{2}}{a}}e^{\sgn\left(a\right)i\frac{\pi}{4}}\\ & =\sqrt{\frac{\pi}{\left|a\right|}}e^{i\left(-\frac{\pi^{2}\xi^{2}}{a}+\sgn\left(a\right)\frac{\pi}{4}\right)} \end{align*} または、\begin{align*} \mathcal{F}_{1,x}\left[e^{iax^{2}}\right]\left(\xi\right) & =2e^{-i\frac{\pi^{2}\xi^{2}}{a}}\int_{0}^{\infty}e^{iax^{2}}dx\\ & =2e^{-i\frac{\pi^{2}\xi^{2}}{a}}\sqrt{\frac{\pi}{ai}} \end{align*}
(15)
\begin{align*} \mathcal{F}_{1,x}\left[\cos\left(ax^{2}\right)\right]\left(\xi\right) & =\mathcal{F}_{1,x}\left[\frac{1}{2}\left(e^{iax^{2}}+e^{-iax^{2}}\right)\right]\left(\xi\right)\\ & =\frac{1}{2}\left\{ \mathcal{F}_{1,x}\left[e^{iax^{2}}\right]\left(\xi\right)+\mathcal{F}_{1,x}\left[e^{-iax^{2}}\right]\left(\xi\right)\right\} \\ & =\frac{1}{2}\left\{ \sqrt{\frac{\pi}{\left|a\right|}}e^{i\left(-\frac{\pi^{2}\xi^{2}}{a}+\sgn\left(a\right)\frac{\pi}{4}\right)}+\sqrt{\frac{\pi}{\left|a\right|}}e^{-i\left(-\frac{\pi^{2}\xi^{2}}{a}+\sgn\left(a\right)\frac{\pi}{4}\right)}\right\} \\ & =\frac{1}{2}\sqrt{\frac{\pi}{\left|a\right|}}\left\{ e^{-i\left(\frac{\pi^{2}\xi^{2}}{\left|a\right|}-\sgn\left(a\right)\frac{\pi}{4}\right)}+e^{i\left(\frac{\pi^{2}\xi^{2}}{\left|a\right|}-\sgn\left(a\right)\frac{\pi}{4}\right)}\right\} \\ & =\sqrt{\frac{\pi}{\left|a\right|}}\cos\left(\frac{\pi^{2}\xi^{2}}{\left|a\right|}-\sgn\left(a\right)\frac{\pi}{4}\right) \end{align*}(16)
\begin{align*} \mathcal{F}_{1,x}\left[\sin\left(ax^{2}\right)\right]\left(\xi\right) & =\mathcal{F}_{1,x}\left[\frac{1}{2i}\left(e^{iax^{2}}-e^{iax^{2}}\right)\right]\left(\xi\right)\\ & =\frac{1}{2i}\left\{ \mathcal{F}_{1,x}\left[e^{iax^{2}}\right]\left(\xi\right)-\mathcal{F}_{1,x}\left[e^{-iax^{2}}\right]\left(\xi\right)\right\} \\ & =\frac{1}{2i}\left\{ \sqrt{\frac{\pi}{\left|a\right|}}e^{i\left(-\frac{\pi^{2}\xi^{2}}{a}+\sgn\left(a\right)\frac{\pi}{4}\right)}-\sqrt{\frac{\pi}{\left|a\right|}}e^{-i\left(-\frac{\pi^{2}\xi^{2}}{\left|a\right|}+\sgn\left(a\right)\frac{\pi}{4}\right)}\right\} \\ & =\frac{1}{2i}\sqrt{\frac{\pi}{\left|a\right|}}\left\{ e^{-i\left(\frac{\pi^{2}\xi^{2}}{a}-\sgn\left(a\right)\frac{\pi}{4}\right)}-e^{i\left(\frac{\pi^{2}\xi^{2}}{\left|a\right|}-\sgn\left(a\right)\frac{\pi}{4}\right)}\right\} \\ & =-\sqrt{\frac{\pi}{\left|a\right|}}\sin\left(\frac{\pi^{2}\xi^{2}}{a}-\sgn\left(a\right)\frac{\pi}{4}\right) \end{align*}(17)
\begin{align*} \mathcal{F}_{1,x}\left[x^{n}\right]\left(\xi\right) & =\mathcal{F}_{1,x}\left[x^{n}\cdot1\right]\left(\xi\right)\\ & =\left(\frac{i}{2\pi}\right)^{n}\mathcal{F}_{1,x}^{\bullet,\left(n\right)}\left[1\right]\left(\xi\right)\\ & =\left(\frac{i}{2\pi}\right)^{n}\delta^{\left(n\right)}\left(\xi\right) \end{align*}(18)
\begin{align*} \mathcal{F}_{1,x}\left[\frac{1}{x}\right]\left(\xi\right) & =\int_{-\infty}^{\infty}\frac{1}{x}e^{-2\pi i\xi x}dx\\ & =2\pi iH\left(\xi\right)\frac{-1}{2}\Res_{z}\left(\frac{1}{z}e^{-2\pi i\xi z},0\right)+2\pi iH\left(-\xi\right)\frac{1}{2}\Res_{z}\left(\frac{1}{z}e^{-2\pi i\xi z},0\right)\cmt{\because\text{ジョルダンの補題、経路上に特異点}}\\ & =-\pi i\left(H\left(\xi\right)-H\left(-\xi\right)\right)\Res_{z}\left(\frac{1}{z}e^{-2\pi i\xi z},0\right)\\ & =-\pi i\left(H\left(\xi\right)-H\left(-\xi\right)\right)\\ & =-\pi i\sgn\left(\xi\right) \end{align*}(18)-2
\begin{align*} \int_{-\infty}^{\xi}-2\pi ie^{-2\pi i\xi x}d\xi & =\left[\frac{1}{x}e^{-2\pi i\xi x}\right]_{-\infty}^{\xi}\\ & =\frac{1}{x}e^{-2\pi i\xi x}-\frac{1}{x}\left(2\pi^{2}ix\right)\delta\left(2\pi ix\right)\\ & =\frac{1}{x}e^{-2\pi i\xi x}-i\pi\delta\left(x\right) \end{align*} となるので、\begin{align*} \frac{1}{x}e^{-2\pi i\xi x} & =\int_{-\infty}^{\xi}-2\pi ie^{-2\pi i\xi x}d\xi+i\pi\delta\left(x\right)\\ & =-\int_{-\infty}^{\xi}\left(2\pi ie^{-2\pi i\xi x}-i\pi\delta\left(\frac{\xi}{2}\right)\delta\left(x\right)\right)d\xi \end{align*} となる。
これより、
\begin{align*} \mathcal{F}_{1,x}\left[\frac{1}{x}\right]\left(\xi\right) & =\int_{-\infty}^{\infty}\frac{1}{x}e^{-2\pi i\xi x}dx\\ & =-\int_{-\infty}^{\infty}\int_{0}^{\xi}\left(2\pi ie^{-2\pi i\xi x}-i\pi\delta\left(\frac{\xi}{2}\right)\delta\left(x\right)\right)d\xi dx\\ & =-\int_{0}^{\xi}\int_{-\infty}^{\infty}\left(2\pi ie^{-2\pi i\xi x}-i\pi\delta\left(\frac{\xi}{2}\right)\delta\left(x\right)\right)dxd\xi\\ & =-\int_{0}^{\xi}\left(2\pi i\delta\left(\xi\right)-i\pi\delta\left(\frac{\xi}{2}\right)\right)d\xi\\ & =-\left(2\pi iH\left(\xi\right)-i\pi\right)\\ & =-i\pi\left(2H\left(\xi\right)-1\right)\\ & =-i\pi\sgn\left(\xi\right) \end{align*}
(19)
\begin{align*} \mathcal{F}_{1,x}\left[\frac{1}{x^{n}}\right]\left(\xi\right) & =\mathcal{F}_{1,x}\left[\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}\left(\frac{1}{x}\right)^{\left(n-1\right)}\right]\left(\xi\right)\\ & =\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}\mathcal{F}_{1,x}\left[\left(\frac{1}{x}\right)^{\left(n-1\right)}\right]\left(\xi\right)\\ & =\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}\left(2\pi i\xi\right)^{n-1}\mathcal{F}_{1,x}\left[\left(\frac{1}{x}\right)\right]\left(\xi\right)\\ & =\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}\left(2\pi i\xi\right)^{n-1}\left(-i\pi\sgn\left(\xi\right)\right)\\ & =\left(-i\pi\right)^{n}\frac{\left(2\xi\right)^{n-1}}{\left(n-1\right)!}\sgn\left(\xi\right) \end{align*}(20)
\begin{align*} \mathcal{F}_{1,x}\left[\frac{1}{\sqrt{\left|x\right|}}\right]\left(\xi\right) & =\int_{-\infty}^{\infty}\frac{1}{\sqrt{\left|x\right|}}e^{-2\pi i\xi x}dx\\ & =\int_{-\infty}^{0}\frac{1}{\sqrt{\left|x\right|}}e^{-2\pi i\xi x}dx+\int_{0}^{\infty}\frac{1}{\sqrt{\left|x\right|}}e^{-2\pi i\xi x}dx\\ & =\int_{0}^{\infty}\frac{1}{\sqrt{x}}e^{2\pi i\xi x}dx+\int_{0}^{\infty}\frac{1}{\sqrt{x}}e^{-2\pi i\xi x}dx\\ & =\int_{0}^{\infty}\frac{1}{\sqrt{x}}\left(e^{2\pi i\xi x}+e^{-2\pi i\xi x}\right)dx\\ & =2\int_{0}^{\infty}\frac{\cos\left(2\pi\xi x\right)}{\sqrt{x}}dx\\ & =4\int_{0}^{\infty}\cos\left(2\pi\xi y^{2}\right)dy\cmt{x^{\frac{1}{2}}=y}\\ & =4\int_{0}^{\infty}\cos\left(2\pi\left|\xi\right|y^{2}\right)dy\\ & =\frac{4}{\sqrt{2\pi\left|\xi\right|}}\int_{0}^{\infty}\cos\left(y^{2}\right)dy\cmt{\sqrt{2\pi\left|\xi\right|}y\rightarrow y}\\ & =\frac{4}{\sqrt{2\pi\left|\xi\right|}}\cdot\frac{\sqrt{2\pi}}{4}\\ & =\frac{1}{\sqrt{\left|\xi\right|}} \end{align*}(21)
\begin{align*} \mathcal{F}_{x}\left[\frac{1}{x\pm\left|a\right|i}\right]\left(\xi\right) & =\int_{-\infty}^{\infty}\left(\frac{1}{x\pm\left|a\right|i}\right)e^{-2\pi i\xi x}dx\\ & =\mp2\pi iH\left(\pm\xi\right)\Res\left(\frac{1}{x\pm\left|a\right|i}e^{-2\pi i\xi x},\mp\left|a\right|i\right)\cmt{\because\text{ジョルダンの補題}}\\ & =\mp2\pi iH\left(\pm\xi\right)e^{-2\pi i\xi\left(\mp\left|a\right|i\right)}\\ & =\mp2\pi iH\left(\pm\xi\right)e^{\mp2\pi\xi\left|a\right|}\\ & =\mp2\pi iH\left(\pm\xi\right)e^{-2\pi\left|\xi\right|\left|a\right|}\cmt{\because H\left(\pm x\right)f\left(x\right)=H\left(\pm x\right)f\left(\pm\left|x\right|\right)} \end{align*}(22)
\begin{align*} \mathcal{F}_{1,x}\left[H_{c}\left(x\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}H_{c}\left(x\right)e^{-2\pi i\xi x}dx\\ & =\lim_{n\rightarrow\infty}\int_{0}^{n}e^{-2\pi i\xi x}dx\\ & =\frac{-1}{2\pi i\xi}\lim_{n\rightarrow\infty}\left[e^{-2\pi i\xi x}\right]_{0}^{n}\\ & =\frac{-1}{2\pi i\xi}\lim_{n\rightarrow\infty}\left(e^{-2\pi i\xi n}-1\right)\\ & =\frac{-1}{2\pi i\xi}\lim_{n\rightarrow\infty}\left(-2\pi^{2}i\xi\cdot\frac{e^{-2\pi i\xi n}}{-2\pi i\xi\cdot\pi}-1\right)\\ & =\frac{-1}{2\pi i\xi}\left(-2\pi^{2}\xi i\delta\left(-2\pi i\xi\right)-1\right)\\ & =-i\pi\delta\left(-2\pi i\xi\right)+\frac{1}{2\pi i\xi}\\ & =\frac{1}{2}\delta\left(-2\pi i\xi\right)+\frac{1}{2\pi i\xi}\\ & =\frac{1}{2}\left(\delta\left(\xi\right)+\frac{1}{i\pi\xi}\right) \end{align*}(23)
\begin{align*} \mathcal{F}_{1,x}\left[\sgn\left(x\right)\right]\left(\xi\right) & =\mathcal{F}_{1,x}\left[2H_{\frac{1}{2}}\left(x\right)-1\right]\left(\xi\right)\\ & =2\mathcal{F}_{1,x}\left[H_{\frac{1}{2}}\left(x\right)\right]\left(\xi\right)-\mathcal{F}_{1,x}\left[1\right]\left(\xi\right)\\ & =2\cdot\frac{1}{2}\left(\delta\left(\xi\right)+\frac{1}{i\pi\xi}\right)-\delta\left(x\right)\\ & =\frac{1}{i\pi\xi} \end{align*}(24)
\begin{align*} \mathcal{F}_{x}\left[\mathrm{comb}_{T}\left(x\right)\right]\left(\xi\right) & =\int_{-\infty}^{\infty}\mathrm{comb}_{T}\left(x\right)e^{-2\pi i\xi x}dx\\ & =\int_{-\infty}^{\infty}\sum_{n=-\infty}^{\infty}\delta\left(x-nT\right)e^{-2\pi i\xi x}dx\\ & =\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}\delta\left(x-nT\right)e^{-2\pi i\xi x}dx\\ & =\sum_{n=-\infty}^{\infty}e^{-2\pi i\xi nT}\\ & =\sum_{n=-\infty}^{\infty}e^{2\pi i\xi nT}\\ & =\frac{1}{T}\cdot\frac{1}{T^{-1}}\sum_{n=-\infty}^{\infty}e^{i\frac{2\pi}{T^{-1}}\xi n}\\ & =\frac{1}{T}\mathrm{comb}_{\frac{1}{T}}\left(\xi\right) \end{align*}
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フーリエ変換の定義による違い
\[
\mathcal{F}_{2,x}\left[f\left(x\right)\right]\left(k\right)=\frac{1}{\sqrt{2\pi}}\mathcal{F}_{1,x}\left[f\left(x\right)\right]\left(\frac{k}{2\pi}\right)
\]
偶関数・奇関数のフーリエ変換とフーリエ逆変換
\[
\mathcal{F}_{x}\left[f_{e}\left(x\right)\right]\left(\xi\right)=\mathcal{F}_{x}^{\bullet}\left[f_{e}\left(x\right)\right]\left(\xi\right)
\]
ポアソン和公式
\[
\sum_{n=-\infty}^{\infty}f\left(n\right)=\sum_{\xi=-\infty}^{\infty}\hat{f}\left(\xi\right)
\]
フーリエ変換の性質
\[
\mathcal{F}_{x}\left[f\left(x\right)*g\left(x\right)\right]\left(\xi\right)=\mathcal{F}_{x}\left[f\left(x\right)\right]\left(\xi\right)\mathcal{F}_{x}\left[g\left(x\right)\right]\left(\xi\right)
\]