# 傅里叶正弦、余弦变换

（重定向自傅立葉正弦變換

## 定义

${\displaystyle 2\int _{-\infty }^{\infty }f(t)\sin(2\pi \omega t)\,dt.}$
${\displaystyle {\sqrt {\frac {2}{\pi }}}\int _{0}^{\infty }f(t)\sin(\omega t)\,dt.}$

${\displaystyle {\hat {f}}^{s}(\omega )=-{\hat {f}}^{s}(-\omega ).}$

${\displaystyle 2\int _{-\infty }^{\infty }f(t)\cos(2\pi \omega t)\,dt.}$

${\displaystyle {\hat {f}}^{c}(\omega )={\hat {f}}^{c}(-\omega ).}$

${\displaystyle 4\int _{0}^{\infty }f(t)\cos(2\pi \omega t)\,dt.}$

${\displaystyle 4\int _{0}^{\infty }f(t)\sin(2\pi \omega t)\,dt.}$

## 傅里叶逆变换

${\displaystyle f(t)=\int _{0}^{\infty }{\hat {f}}^{c}\cos(2\pi \omega t)d\omega +\int _{0}^{\infty }{\hat {f}}^{s}\sin(2\pi \omega t)d\omega ,}$

${\displaystyle {\tfrac {\pi }{2}}\left(f(x+0)+f(x-0)\right)=\int _{0}^{\infty }\int _{-\infty }^{\infty }f(t)\cos(\omega (t-x))dtd\omega ,}$

## 与複指数的关系

{\displaystyle {\begin{aligned}{\hat {f}}(\nu )&=\int _{-\infty }^{\infty }f(t)e^{-2\pi i\nu t}\,dt\\&=\int _{-\infty }^{\infty }f(t)(\cos(2\pi \nu t)-i\,\sin(2\pi \nu t))\,dt&&{\text{Euler's Formula}}\\&=\left(\int _{-\infty }^{\infty }f(t)\cos(2\pi \nu t)\,dt\right)-i\left(\int _{-\infty }^{\infty }f(t)\sin(2\pi \nu t)\,dt\right)\\&={\tfrac {1}{2}}{\hat {f}}^{c}(\nu )-{\tfrac {i}{2}}{\hat {f}}^{s}(\nu )\end{aligned}}}

## 参考

• Whittaker, Edmund, and James Watson, A Course in Modern Analysis, Fourth Edition, Cambridge Univ. Press, 1927, pp. 189, 211
1. ^ Mary L. Boas，在《Mathematical Methods in the Physical Sciences》，第二版，John Wiley & Sons Inc, 1983. ISBN 0-471-04409-1
2. ^ Poincaré, Henri. Theorie analytique de la propagation de chaleur. Paris: G. Carré. 1895: pp. 108ff.