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ºñÁÖ±âÇÔ¼ö´Â ¿¬¼ÓÀûÀÎ Æļö¸¦ °¡Áø Á¶È­ÇÔ¼öÀÇ ÇÕÀ¸·Î ³ªÅ¸³¾ ¼ö ÀÖ´Ù.

¾Õ¿¡¼­ÀÇ ÁÖ±â $L$ÀÎ ÁÖ±âÇÔ¼öÀÇ $L$À» °è¼Ó Áõ°¡½ÃÄÑ ¹«ÇÑ ´ë·Î µÇ°Ô Çϸé Áֱ⼺ÀÌ ¾ø¾îÁø ÀÓÀÇÀÇ ÇÔ¼ö°¡ µÇ¾î À̰͵µ Á¶È­ÆÄÀÇ ÇÕ¼ºÀ¸·Î ³ªÅ¸³¾ ¼ö ÀÖÀ» °ÍÀ̶ó´Â »ý°¢À» ½±°Ô ÇÒ ¼ö ÀÖ´Ù. À̶§ $k = 2\pi/L$ °ªÀº ¹«ÇÑÈ÷ ÁÙ¾îµé¾î 0À¸·Î Á¢±ÙÇÑ´Ù. ±×¸®°í $nk$°ªÀº ¿¬¼ÓÀûÀÎ °ªÀ¸·Î µÇ¾î Ǫ¸®¿¡ ÇÕ¼ºÀº Ǫ¸®¿¡ ÀûºÐ(Fourier integral)À¸·Î ¹Ù²î°Ô µÈ´Ù. \[ F(x) = \frac{1}{\pi} \left[ \int_0^\infty A(k) \cos kx dk + \int_0^\infty B(k) \sin kx dk \right] \] \[ A(k) = \int_{-\infty}^\infty F(x) \cos kx dx \] \[ B(k) = \int_{-\infty}^\infty F(x) \sin kx dx \] Áï ÇÔ¼ö ÀÓÀÇÀÇ 1Â÷¿ø ÇÔ¼ö $F(x)$¸¦ ´ëÇÑ ÄÚ»çÀÎÇÔ¼ö¿Í »çÀÎÇÔ¼öÀÇ ¼ººÐÀÇ ÇÕÀ¸·Î ³ªÅ¸³¾ ¼ö ÀÖ°í, À̸¦ Ǫ¸®¿¡ º¯È¯(Fourier transform)À̶ó ÇÑ´Ù. ¸¸ÀÏ ÇÔ¼ö°¡ ¿ìÇÔ¼öÀ̸é $A(k)$¸¸, ±âÇÔ¼öÀ̸é $B(k)$¸¸ ³ªÅ¸³­´Ù. ÀÌ·¯ÇÑ ¿ìÇÔ¼öÀÇ º¯È¯À» Ǫ¸®¿¡ ÄÚ»çÀκ¯È¯(Fourier cosine transform), ±âÇÔ¼öÀÇ º¯È¯À» Ǫ¸®¿¡ »çÀκ¯È¯(Fourier sine transform)À̶ó°í ºÐº°Çؼ­ ÀÏÄ´´Ù.

ÀÌÁ¦ À̵é ÄÚ»çÀÎÇÔ¼ö¿Í »çÀÎÇÔ¼ö·Î ³ªÅ¸³½ º¯È¯À» ÅëÇÕÇϱâ À§ÇØ º¹¼ÒÁö¼öÇÔ¼ö¸¦ ´ÙÀ½°ú °°ÀÌ µµÀÔÇÏÀÚ. ¸¶Áö¸· µÎ ½ÄÀ» ù° ½Ä¿¡ ´Ù½Ã ´ëÀÔÇϸé, \[ F(x) = \frac{1}{\pi} \int_0^\infty \cos kx \int_{-\infty}^\infty F(x') \cos kx' dx' dk + \frac{1}{\pi} \int_0^\infty \sin kx \int_{-\infty}^\infty F(x') \sin kx' dx' dk \] ¿©±â¼­ ÄÚ»çÀÎ ÇÕÀÇ ¹ýÄ¢À» ÀÌ¿ëÇÏ¿© Á¤¸®Çϸé, \[ F(x) = \frac{1}{2\pi}\int_{-\infty}^\infty \left[ \int_{-\infty}^\infty F(x') \cos k(x'-x) dx' \right] dk \] ¿©±â¼­ $k$¿¡ ´ëÇÑ ÀûºÐÀ» $-\infty$·Î È®ÀåÇϸ鼭 $\frac{1}{2}$·Î Çß´Â µ¥ ÀÌ´Â Áß°ýÈ£ ¼ÓÀÇ $k$¿¡ ´ëÇÑ ÇÔ¼ö°¡ ¿ìÇÔ¼öÀ̱⠶§¹®ÀÌ´Ù. ¸¸ÀÏ ÀÌ ÀûºÐ¿¡¼­ $\cos k(x'-x)$ ´ë½Å¿¡ $\sin k(x'-x)$À¸·Î ´ëÄ¡ÇÑ´Ù¸é Áß°ýÈ£ ¼ÓÀÇ ÀûºÐÀÌ $k$¿¡ ´ëÇØ ±âÇÔ¼ö°¡ µÇ±â ¶§¹®¿¡ ±× °á°ú´Â 0 ÀÌ µÈ´Ù. µû¶ó¼­ $\cos k(x'-x)$Ç×À» $\exp ik(x'-x)$³ª $\exp ik(-x'+x)$·Î ¹Ù²Ù¾îµµ ¹«¹æÇÏ´Ù. ¿©±â¼­´Â ÈÄÀÚ¸¦ ÅÃÇϸé, \[ F(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \left[ \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty F(x') e^{-i kx'} dx' \right] e^{i kx} dk \] µû¶ó¼­ ÀÌ ½ÄÀÇ Áß°ýÈ£ ¼ÓÀÇ $k$¿¡ ´ëÇÑ ÇÔ¼ö¸¦ $f(k)$¶ó ÇÑ´Ù¸é ´ÙÀ½°ú °°Àº °ü°è¸¦ ¾òÀ» ¼ö ÀÖ´Ù. \[ F(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(k) e^{i kx} dk \] \[ f(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty F(x) e^{-i kx} dx \] ¿©±â¼­ ÆÄÇüÇÔ¼ö $F(x)$¿¡ ´ëÇÑ ¼ººÐÇÔ¼ö $f(k)$·Î º¯È¯ÇÏ´Â °ÍÀ» Ǫ¸®¿¡ º¯È¯(Fourier transformation)À̶ó ÇÏ°í, $f(k)$¿¡ ´ëÇØ $F(x)$·Î º¯È¯ÇÏ´Â °ÍÀ» ¿ªÇª¸®¿¡ º¯È¯(inverse Fourier transformation)À̶ó ÇÑ´Ù.


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