從uni4E09角uni6C42uni548C公式uni5230Fourier級數uni6797uni7426uni711C1. Gauss 之uni555Funi767C:uni95DC於uni9AD8斯 (Carl Friedrich Gauss 17771855), 最uni70BA人所uni6D25uni6D25uni6A02uni9053uni7684故事, 莫uni904E於uni5728uni5C0Funi5B78低年級時老師要uni73ED上uni540Cuni5B78uni6C42從 1 uni5230 100 uni7684uni548C等於uni591Auni5C11。uni9AD8斯很快uni5C31得uni5230uni6B63uni78BA答uni6848 5050。uni9019著uni5BE6令他uni7684老師十分驚訝, uni96A8後uni9AD8斯解uni91CB說, uni9996先從 1 uni5230 100 uni5BEBuni4E00uni6B21, 而後再從 100 uni5230 1 倒uni5BEB回來, 兩者垂uni76F4uni76F8uni52A0, uni53EFuni767Cuni73FEuni6BCFuni9805uni90FD是 101, 共有 100uni9805, 100101 = 10100, 再uni9664以 2 uni5C31是 5050。.+ =uni5716uni4E00誠uni5982歷uni53F2上許uni591Auni540D人uni90FD有許uni591A著uni540Duni7684故事,故事uni7684uni771Funi5BE6性, 歷uni53F2性uni5982何? 是uni5426uni771Funi7684uni767Cuni751F? 我想uni90A3已經uni7121uni95DC緊要, uni91CD要uni7684是uni9019個事件uni672Cuni8EABuni7684意義。從數uni5B78uni7684角度而言, uni9AD8斯uni9019個故事uni544A訴我們思考模型uni7684uni91CD要性, uni9019個uni554Funi984Cuni7684思考模型是uni968E梯之個數, 或者是保齡球之球瓶數, 也uni5C31是uni4E09角形之uni9762積,而uni4E09角形uni9762積uni7684uni6C42uni6CD5是uni5C07uni539Funi4E09角形倒uni8F49兩個uni5408併之後,成uni70BA平行uni56DB邊形。Gn = 1 + 2 + 3 + (n1) + nGn =n + (n1) + (n2) + 2 + 1兩者垂uni76F4uni76F8uni52A02Gn = (n + 1) + (n + 1) + (n + 1)所以Gn = 1 + 2 + 3 + (n1) + n = n(n + 1)2 (1.1)uni5728uni4E09角uni51FD數有uni4E00個著uni540Duni7684公式sin + sin2 + sin3 + sinn = sinn2sinn+12 sin 2 (1.2)1112 數uni5B78傳uni64AD 26uni53773uni671F uni6C1191年9月乍uni770B之下 (1.1), (1.2)uni9019兩個公式uni5BE6uni5728是uni592A像了,uni9996先uni7531等uni6BD4級數uni6C42uni548C公式uni53EF得 + 2 + 3 + n = n(n + 1)2 兩邊uni540C時乘個 sin (英文罪也!)sin( + 2 + 3 + n) = sin n(n + 1)2 再故意uni5C07 sin 分uni914D至uni5404uni9805 (姑且不管乘uni6CD5或uni52A0uni6CD5), 左式uni6B63uni597D是公式 (1.2) uni7684左式, 而uni53F3式uni5247是(1.2) uni7684分uni5B50, uni56E0uni6B64再uni9664以 sin 2, uni6B63uni597D是公式 (1.2)! uni56B4uni683Cuni8B49明uni5982下。uni6CD5uni4E00: 我們uni73FEuni5728uni5C31uni5617試uni5229uni7528uni9AD8斯uni7684方uni6CD5來uni8B49明公式(1.2), 令Sn = sin + sin2 + sin(n1) + sinnSn = sinn + sin(n1) + sin2 + sinuni85C9uni7531uni4E09角公式 (uni5C07 x, y uni5BEB成 x = x+y2 + xy2 , y = x+y2 xy2 是典型uni7684手uni6CD5!)sinx+ siny = sin(x+y2 + xy2 )+ sin(x+y2 xy2 ) = 2sin x+y2 cos xy2 (1.3)兩式垂uni76F4uni76F8uni52A02Sn = 2bracketleftBigsin (1 + n)2 cos (1n)2 + sin (1 + n)2 cos (3n)2 + sin (n + 1)2 cos (n3)2 + sin (n + 1)2 cos (n1)2bracketrightBig= 2sin (n + 1)2bracketleftBigcos (1n)2 + cos (3n)2 + cos (n1)2bracketrightBig兩邊uni540C時乘 sin 2, 且再uni5229uni7528uni4E00uni6B21uni4E09角公式得2Sn sin 2 = sin n + 12 bracketleftBigsin n2 + sin(1 n2) + sin(n2 1) + sin n2bracketrightBig整uni7406得 (uni56E0uni70BA sin() = sin)Sn = sinn2sin 2 sinn + 12 uni6CD5uni4E8C: uni6CD5uni4E00uni9019個uni8B49明方uni6CD5其uni5BE6是uni756Buni86C7uni6DFBuni8DB3, 從最uni958Buni59CB乘 2sin 2 uni5C31uni53EFuni8B49明公式 (1.2)。uni7531uni4E09角公式 2sinxsiny = cos(xy)cos(x + y) uni53EF得2sin 2Sn =nsummationdisplayk=12sin 2 sink =nsummationdisplayk=1cos(2 k)cos(2 + k)=nsummationdisplayk=1cos(k12)cos(k + 12) = cos 2cos(n + 12) = 2sin n + 12 sin n2從uni4E09角uni6C42uni548C公式uni5230 Fourier級數 13故 Sn = sin n2 sin 2sin n+12 。uni4E00個uni5C0Duni4E09角uni51FD數有uni771Funi6B63認uni8B58uni7684uni5B78uni751F, uni59CB終是uni5C07uni6B63uni5F26uni51FD數, uni9918uni5F26uni51FD數等齊uni770B待uni7684。uni5C0D於uni9918uni5F26uni51FD數我們也有uni985E似uni7684公式cos + cos2 + cosn = sinn2sin 2 cosn + 12 (1.4)(1.2), (1.4) 兩式uni76F8uni9664uni53EF得uni53E6uni4E00個uni6F02亮uni7684公式sin + sin2 + sin(n1) + sinncos + cos2 + cos(n1) + cosn = tann + 12 (1.5)uni5C07 (1.2), (1.4) 兩式uni5408併, 再uni85C9uni7531 Euler 公式eiz = cosz + isinz, cosz = 12(eiz + eiz), sinz = 12i(eiz eiz) (1.6)uni53EF以從等uni6BD4級數uni7684角度來uni8B49明 (1.2), (1.4)uni9019兩個公式。uni6CD5uni4E09: 令X = cos + cos2 + cosn, Y = sin + sin2 + sinn (1.7)uni5247uni7531等uni6BD4級數uni6C42uni548C之公式uni53EF得X + iY = ei + ei2 + ei3 + ein = ei ei(n+1)1ei (1.8)分uni5B50分uni6BCDuni540C時乘 ei/2X + iY = ei(n+12) ei/2ei/2 ei/2 =ei(n+1)/2(ein/2 ein/2)ei/2 ei/2=12i(ein/2 ein/2)12i(ei/2 ei