复变与积分函数

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1、{cÁK‰Y1lÙFá“C†XCFuþ°ŒÆêÆXXCFu{cÁK‰Y08-09Á1.F[f(t)]=F(w)§KF[f(t)cosw0t]=:XCFu{cÁK‰Y08-09Á1.F[f(t)]=F(w)§KF[f(t)cosw0t]=:)Z+1F[f(t)coswt]=f(t)coswtejwtdt001Z+1ejw0tejw0t=f(t)ejwtdt12Z+11j(ww0)t=f(t)edt21Z+11j(w+w0)t+f(t)edt211=[F(ww0)+F(w+w0)]:2XCFu{cÁK‰Y08-09Á1.F[f(t)]=F(w)§K

2、F[f(t)coswt]=1[F(ww)+F(w+w)]:0200)Z+1F[f(t)coswt]=f(t)coswtejwtdt001Z+1ejw0tejw0t=f(t)ejwtdt12Z+11j(ww0)t=f(t)edt21Z+11j(w+w0)t+f(t)edt211=[F(ww0)+F(w+w0)]:2XCFu{cÁK‰Y2.sintjtjf(t)=0jtj>£1¤¦f(t)Fá“C†"£2¤Ñf(t)Fá“È©úª"£3¤OŽÈ©Z+1sinwsinwtdw:01w2XCFu{cÁK‰Y)(1)Z+1F(w)=F[f

3、(t)]=f(t)ejwtdt1Z+=sintejwtdtZ+jtjteejwt=edt2jZ+ej(1w)tZ+ej(1+w)t=dt+dt2j2j2jsinw=:w21XCFu{cÁK‰Y(2)Z+111jwtf(t)=F[f(w)]=F(w)edw21Z+112jsinw=(coswt+jsinwt)dw21w21Z+112jsinwcoswt=dw21w21Z+112sinwsinwt+dw211w2Z+12sinwsinwt=dw:01w2(3)d(2)ŒµZ+1sinwsinw

4、tsintjtjdw=201w20jtj>XCFu{cÁK‰Y08¨09S1.u(t)•ü¼ê§KF[u(t)]=:XCFu{cÁK‰Y08¨09S1.u(t)•ü¼ê§KF[u(t)]=:)„‘P194~8.8"XCFu{cÁK‰Y08¨09S1.u(t)•ü¼ê§KF[u(t)]=1+(w):jw)„‘P194~8.8"XCFu{cÁK‰Y~8.8Áyµü¼êu(t)F¼C†•1+(w):jwyµF(w)=1+(w)§f(t)=F1[f(w)]§KjwZ1+11f(t)=+(w)ejwtdw21jwZ+1

5、Z+11jwt11jwt=(w)edw+edw2121jwZ+111sinwt=+dw;20wR+1sinxd~8.2•dx=§¤±k0x21;t>0;f(t)=u(t)=0;t<0:XCFu{cÁK‰Y2.e2tt0f(t)=0t<0£1¤¦f(t)Fá“C†"£2¤OŽÈ©Z+12cosw+wsinw22dw:4+w20XCFu{cÁK‰Y)(1)Z+1F(w)=F[f(t)]=f(t)ejwtdt1Z+1=e2tejwtdt01=:2+jwXCFu{cÁK‰Y(2)Z+111jwtf(t)=F[F(w)]=F(w)edw21Z+1

6、11jwt=edw212+jwZ+112coswt+wsinwt=dw:04+w2t=1µ2Z+12cosw+wsinw22dw=:04+w2eXCFu{cÁK‰Y09¨10Á1.F¼C†F[(t2)]=:XCFu{cÁK‰Y09¨10Á1.F¼C†F[(t2)]=:)Z+1F[(t2)]=(t2)ejwtdt1jwt=et=2=e2wj:XCFu{cÁK‰Y09¨10Á1.F¼C†F[(t2)]=e2wj:)Z+1F[(t2)]=(t2)ejwtdt1jwt=et=2=e2wj:XCFu{cÁK‰Y2.e2t

7、;t0f(t)=0;t<0£1¤¦f(t)Fá“C†¶£2¤OŽÈ©Z+12cosw+wsinw22dw:4+w20XCFu{cÁK‰Y)(1)Z+1F(w)=F[f(t)]=f(t)ejwtdt1Z+1=e2tejwtdt01=:2+jwXCFu{cÁK‰Y(2)Z+111jwtf(t)=F[F(w)]=F(w)edw21Z+111jwt=edw212+jwZ+112coswt+wsinwt=dw:04+w2t=1µ2Z+12cosw+wsinw22dw=04+w2eXCFu{cÁK