cost=1/2(e^jt e^-jt) 求函数f(t)=costsint的傅氏变换
\u590d\u53d8\u51fd\u6570\u9898\uff0c\uff0c\u6c42f(t)=sintcost\u7684\u5085\u91cc\u53f6\u53d8\u6362sintcost=1/2sin2t
F(1/2sin2t)
=\u222b(-\u221e\uff0c+\u221e) 1/2sin2t \u00b7 e^-jwt dt
\u7528\u6b27\u62c9\u516c\u5f0f\u53ef\u5f97\u539f\u5f0f=
1/2\u222b(-\u221e\uff0c+\u221e) j/2( e^-2jt - e^2jt )e^-jwt dt
=j/4\u222b(-\u221e\uff0c+\u221e) e^-j(w+2)t - e^-j(w-2)t dt
\u7528\u03b4\u51fd\u6570\u7684\u5085\u6c0f\u53d8\u6362 \u5f97\u539f\u5f0f=
j/2 \u03c0[\u03b4(w+2)-\u03b4(w-2)]
\u6b27\u62c9\u516c\u5f0f: sin2t=j/2 (e^-2jt - e^2jt)
\u03b4\u51fd\u6570\u7684\u5085\u6c0f\u53d8\u6362:
F(e^jw\u3002t)=\u222b(-\u221e\uff0c+\u221e) e^j(w\u3002-w)t dt =2\u03c0\u03b4(w\u3002-w)
\u6269\u5c55\u8d44\u6599\u5e38\u7528\u5bfc\u6570\u516c\u5f0f\uff1a
1.y=c(c\u4e3a\u5e38\u6570) y'=0
2.y=x^n y'=nx^(n-1)
3.y=a^x y'=a^xlna\uff0cy=e^x y'=e^x
4.y=logax y'=logae/x\uff0cy=lnx y'=1/x
5.y=sinx y'=cosx
6.y=cosx y'=-sinx
7.y=tanx y'=1/cos^2x
8.y=cotx y'=-1/sin^2x
9.y=arcsinx y'=1/\u221a1-x^2
10.y=arccosx y'=-1/\u221a1-x^2
11.y=arctanx y'=1/1+x^2
12.y=arccotx y'=-1/1+x^2
sintcost=1/2sin2t
F(1/2sin2t)
=\u222b(-\u221e\uff0c+\u221e) 1/2sin2t \u00b7 e^-jwt dt
\u7528\u6b27\u62c9\u516c\u5f0f\u53ef\u5f97\u539f\u5f0f=
1/2\u222b(-\u221e\uff0c+\u221e) j/2( e^-2jt - e^2jt )e^-jwt dt
=j/4\u222b(-\u221e\uff0c+\u221e) e^-j(w+2)t - e^-j(w-2)t dt
\u7528\u03b4\u51fd\u6570\u7684\u5085\u6c0f\u53d8\u6362 \u5f97\u539f\u5f0f=
j/2 \u03c0[\u03b4(w+2)-\u03b4(w-2)]
\u6b27\u62c9\u516c\u5f0f: sin2t=j/2 (e^-2jt - e^2jt)
\u03b4\u51fd\u6570\u7684\u5085\u6c0f\u53d8\u6362:
F(e^jw\u3002t)=\u222b(-\u221e\uff0c+\u221e) e^j(w\u3002-w)t dt =2\u03c0\u03b4(w\u3002-w)
\u6269\u5c55\u8d44\u6599\uff1a
\u5085\u6c0f\u53d8\u6362\u7684\u610f\u4e49
\u5085\u91cc\u53f6\u539f\u7406\u8868\u660e\uff1a\u4efb\u4f55\u8fde\u7eed\u6d4b\u91cf\u7684\u65f6\u5e8f\u6216\u4fe1\u53f7\uff0c\u90fd\u53ef\u4ee5\u8868\u793a\u4e3a\u4e0d\u540c\u9891\u7387\u7684\u6b63\u5f26\u6ce2\u4fe1\u53f7\u7684\u65e0\u9650\u53e0\u52a0\u3002
\u800c\u6839\u636e\u8be5\u539f\u7406\u521b\u7acb\u7684\u5085\u91cc\u53f6\u53d8\u6362\u7b97\u6cd5\u5229\u7528\u76f4\u63a5\u6d4b\u91cf\u5230\u7684\u539f\u59cb\u4fe1\u53f7\uff0c\u4ee5\u7d2f\u52a0\u65b9\u5f0f\u6765\u8ba1\u7b97\u8be5\u4fe1\u53f7\u4e2d\u4e0d\u540c\u6b63\u5f26\u6ce2\u4fe1\u53f7\u7684\u9891\u7387\u3001\u632f\u5e45\u548c\u76f8\u4f4d\u3002
\u548c\u5085\u91cc\u53f6\u53d8\u6362\u7b97\u6cd5\u5bf9\u5e94\u7684\u662f\u53cd\u5085\u91cc\u53f6\u53d8\u6362\u7b97\u6cd5\u3002\u8be5\u53cd\u53d8\u6362\u4ece\u672c\u8d28\u4e0a\u8bf4\u4e5f\u662f\u4e00\u79cd\u7d2f\u52a0\u5904\u7406\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5355\u72ec\u6539\u53d8\u7684\u6b63\u5f26\u6ce2\u4fe1\u53f7\u8f6c\u6362\u6210\u4e00\u4e2a\u4fe1\u53f7\u3002
\u56e0\u6b64\uff0c\u53ef\u4ee5\u8bf4\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u5c06\u539f\u6765\u96be\u4ee5\u5904\u7406\u7684\u65f6\u57df\u4fe1\u53f7\u8f6c\u6362\u6210\u4e86\u6613\u4e8e\u5206\u6790\u7684\u9891\u57df\u4fe1\u53f7\uff08\u4fe1\u53f7\u7684\u9891\u8c31\uff09\uff0c\u53ef\u4ee5\u5229\u7528\u4e00\u4e9b\u5de5\u5177\u5bf9\u8fd9\u4e9b\u9891\u57df\u4fe1\u53f7\u8fdb\u884c\u5904\u7406\u3001\u52a0\u5de5\u3002\u6700\u540e\u8fd8\u53ef\u4ee5\u5229\u7528\u5085\u91cc\u53f6\u53cd\u53d8\u6362\u5c06\u8fd9\u4e9b\u9891\u57df\u4fe1\u53f7\u8f6c\u6362\u6210\u65f6\u57df\u4fe1\u53f7\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u5085\u91cc\u53f6\u53d8\u6362
e^(ix) = cost + i sint
e^(- ix) = cost - isint
相加e^(ix) + e^(- ix) = 2cost
得出cost = [ e^(ix) + e^(- ix) ]/2
绛旓細姹傚緱鍑烘潵鐨勶紝鍏堝皢cosx灞曞紑鎴恱鐨勫箓绾ф暟寰楋紝cosx=1-x^2/2锛+x^4/4锛+...+(-1)^n*x^(2n)/(2n)锛+... (1)浠=x^2锛宑os(x^2)=cost=1-t^2/2锛+t^4/4锛+... (2)灏唗=x^2浠ｅ叆鍎(2)寮忎腑锛屽緱 cos(x^2)=1-x^4/2锛+x^8/4锛+...+(-1)^n*x^(4n)/(2n)锛
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