积分变换习题解答

1()tf00()()cos()sinftatdbtdωωωωωω+∞+∞=+∫∫1()()cos,1()()sinafbfddωτωτπτωτωττπ+∞−∞+∞−∞==∫∫()()()jj11(cosjsin)cos22tftfededfωτωtddττωτωτωτωτππ+∞+∞+∞+∞−−∞−∞−∞−∞==−∫∫∫∫ω()()()00011+coscos(cosjsin)jsin21+()cos()sinsinsinfdtdftddatdbtdfddtτωττωωτωτωτωτωππωωωωωωτωττωωπ+∞+∞+∞+∞−∞−∞−∞+∞+∞+∞+∞−∞−∞−==+∫∫∫∫∫∫∫∫()sincosftddτωτωτωω+∞−∞∫()coscosftddτωτωτωω+∞−∞∫2()tf()tf()()()ωωωdtbtf∫+∞=0sin()()()02sinbfdωτωτπ+∞=∫τ()tf()()()ωωωdtatfcos0∫+∞=()()()02cosafdωτωτπ+∞=∫τ()tf()()jj12tftfededωτωττωπ+∞+∞−−∞−∞=∫∫()()j1cosjsin2tfdedωτωτωττωπ+∞+∞−∞−∞=−∫∫()j01sinjtfdedωτωττωπ+∞+∞−∞=∫∫()j12jtbedωωω+∞−∞=∫()ωbω()()()01cosjsinsin2jbttdbtdωωωωωω+∞+∞−∞=+=∫∫ω()tf()()jj12tftfededωτωττωπ+∞+∞−−∞−∞=∫∫()()j1cosjsin2tfdedωτωτωττωπ+∞+∞−∞−∞=−∫∫()()j01cos2taedatdωωωωω+∞+∞−∞==∫∫ω()ωaω1232()1,||10,||1tftt≤⎧=⎨⎩()ωa0,||12sincos,||140,||1ttdttπωωπωω+∞⎧⎪⎪⎪==⎨⎪⎪⎪⎩∫()tf()()∫==∞+=ωωπωωπωπωsin201sin2cos02ttdttfa()()∫∫+∞=+∞=ωωωωπωωωdttdatcossin02cos0f()0||12sincos01||122240||ttdftttπωωπππωω+∞⎧1⎪⎪+⎪===⎨⎪⎪⎪⎩∫=1,0()0,Atftτ≤≤⎧=⎨⎩()=ωF¶()()jj0ttftftedtAedtωωτ−−+∞==⎡⎤⎣⎦−∞∫∫jij011jjjteeeAAAτωωτωτωωω−−−−−===−−2123()2221,0,1ttftt⎧−=⎨⎩1()⎩⎨⎧≥=−0,2sin0,0ttettft()0,11,101,010,1ttfttt−∞−⎧⎪−−⎪=⎨⎪⎪+∞⎩1()⎩⎨⎧−=1||,01||,12ttttf()()ii12ttftftedtedωωωπ+∞+∞−−∞−∞=∫∫()12ii1112tttedtedωωωπ+∞−−∞−=−∫∫()12i011costttdted12i2301sin2cos2sinsintttttttedωωωωωωπωωωω+∞−∞⎡⎤⎛⎞=−−+⎢⎥⎜⎟⎝⎠⎣⎦∫]ωωωπ+∞−∞=−∫∫()i32sincos1tedωωωωωπω+∞−∞−=∫304sincoscostdωωωωωπω+∞−=∫2()⎩⎨⎧≥=−0,2sin0,0ttettft()()iiii011sin222tttttftftedtedetedtωωωωedωωππ+∞+∞+∞+∞−−−∞−∞−∞==∫∫∫∫−i2i2ii0122itttttedeeeedtωωωπ−+∞+∞−−−∞−=∫∫()()()i2i2i014ittttteedteωωωdωπ+∞+∞−+−−−+−∞=−∫∫()()()()1i21i2i014i1i21i2ttteeedωωωωπωω+∞−+−−−+⎡⎤⎡⎤⎣⎦⎣⎦+∞−∞⎡⎤=−⎢⎥−+−−−+⎢⎥⎣⎦∫()()i1114i1i21i2tedωωπωω+∞−∞⎡⎤−−=−⎢⎥−+−−−+⎣⎦∫()()22452i1cosisin256ttωωdωωωπωω+∞−∞−−=+−+∫()()2224245cos2sin5sin2cos1i256256ttttddωωωωωωωωωωπωωπωω+∞+∞−∞−∞−+−−=+−+−+∫∫()22405cos2sin2256ttdωωωωωπωω+∞−+=−+∫3()⎪⎩⎪⎨⎧−−=,010,101,1tttf()()()iii011sin2itttftftedtedfttdtωωωedωωωππ+∞+∞+∞+∞−−∞−∞−∞==∫∫∫∫1ii01111siniitttdtededωωcosωωωωππ+∞+∞−∞−∞ω−=⋅=∫∫∫ωωωωπtdsincos120∫+∞−=()tf1,0,10−=t()()20000−++tftf31||()tfteβ−=0β||220cos2ttdeβωπωβωβ+∞−=+∫2()tetftcos||−=()∫+∞−=++0||42cos2cos42tedttπωωωω3()⎩⎨⎧≤=,||,0,||,sinππttttf20sin,||sinsin210,||tttdtππωπωωωπ+∞⎧≤⎪=⎨−⎪⎩∫1¶()=tF()||ittfteedtβω+∞−−−∞=⎡⎤⎣⎦∫=ii002cos22tttteeetdtedtωωββω−+∞+∞−−+=∫∫()()()()()()iiii000iitttteeeedtβωβωβωβωβωβω+∞+−−−−+∞−−−+⎡⎤=+=+⎣⎦−−−+∫∞22112iiββωβωβω=+=−++()tf()()∫+∞∞−=ωωπωdeFtfti21()2212cosisin2ttdβωωωπβω+∞−∞=++∫2202costdβωωπβω+∞=+∫||220cos2ttdeβωπωβωβ+∞−=+∫2()=ωF¶()[]∫∫+∞∞−−−−+∞∞−−−+==dteeeedtteetfttttttωωiii||i||2cos()()()(){}001i11i11i11i10012tttedtedtedtedtωωωt+∞+∞+−−+−+−−−+⎡⎤⎡⎤⎡⎤⎡⎤⎣⎦⎣⎦⎣⎦⎣⎦−∞−∞=+++∫∫∫∫ω=()()()()()()()()001i11i11i11i10021i11i11i11i1tttteeeeωωωωωωω+∞+∞−+−−−+⎡⎤⎡⎤+−−+⎡⎤⎡⎤⎣⎦⎣⎦⎣⎦⎣⎦−∞−∞⎧⎫1⎪⎪+++⎨⎬+−−+−+−−−+⎪⎪⎩⎭ω()()()()1111121i11i11i11i1ωωωω⎡⎤=+++⎢⎥+−−+−−++⎣⎦24244ωω+=+()tf()()ωωωπωωπωωdedeFtftti42i4422121∫∫+∞∞−+∞∞−++==∫+∞++=042cos4421ωωωωπtd()∫+∞−==++0||42cos22cos42tetftdtππωωωω3()=ωF¶()()iisinttftftedttedtπωωπ+∞−−−∞−==⎡⎤⎣⎦∫∫()∫∫−=−=−πππωωω0sinsini2sinicossintdttdtttt=()()[]∫−−+πωω01cos1cosidttt()()⎥⎥⎦⎤⎢⎢⎣⎡−−−++=ωωωωππ11sin11sini00tt()()()()211sin1sin1sin1siniωπωωπωπωωπω−−−−−+−+=21sini2ωωπ−−=()tf()()∫∫+∞∞−+∞∞−⎟⎠⎞⎜⎝⎛−−==ωωωππωωπωωdedeFtftti2i1sini22121()∫∫+∞+∞∞−−=+−−=0221sinsin2sinicos1siniωωωωππωωωωωππdtdtt()∫∞+⎪⎩⎪⎨⎧≤==−02||,0||,sin221sinsinππππωωωωπttttfdt4()tf()=ωFsinωω()tf()()()∫∫+∞∞−+∞∞−+==ωωωωωπωωπωdttdeFtftsinicossin2121i()0sin(1)sin11sin1cos22tttddωωωωωωπωπω+∞+∞−∞++−==∫∫()()∫∫+∞+∞−++=001sin211sin21ωωωπωωωπdtdt*∫+∞=02sinπdxxx0u∫∫∫+∞+∞+∞===0002sinsinsinπωωωωωωdxxxduuudu0u()∫∫+∞+∞−=−−=002sinsinπωωωωωωdudu0=u0sin0,udωωω+∞=∫(*)()⎪⎪⎪⎩⎪⎪⎪⎨⎧==1||,01||,411||,21ttttf50()[()()]F0ωπδωωδωω=++−()tf()()ii0011[()()]22ttftFededωωωωπδωωδωωππ+∞+∞−∞−∞==++−∫∫ω00-ii0cos2tteetωωω+==61,0sgn1,0||ttttt−⎧==⎨⎩|sgn|tdt+∞−∞→+∞∫//,0()000tnntnetfttet−⎧⎪==⎨⎪−⎩sgnlim()nntft→∞=[sgn]lim[()]dfnnFtFft→∞=()nft[]nFω=¶()()0i/i/0ttnttnnitftftedteedteedtωω+∞+∞−−−−∞−∞==−⎡⎤⎣⎦∫∫∫ω−=1111iinnωω−+−=222i1nωω−⎛⎞+⎜⎟⎝⎠[]Fω=¶()222,02ilim[]i10,0nnftFnωωωωωω→∞⎧≠−⎪===⎡⎤⎨⎣⎦⎛⎞⎪=+⎩⎜⎟⎝⎠lim()()nnfxfx→+∞=()nfx[]nFω=¶()nft⎡⎤⎣⎦(1{[,2,n=)n→+∞]}Fω()fx[]Fω=¶[()]lim[]nnfxFω→+∞=71()[()()()()22aafttatattδδδδ=++−+++−]2()=ωF¶()[]tf()()ii12tttaedttaedtωωδδ+∞+∞−−−∞−∞⎡=++−⎢⎣∫∫ii22ttaatedttedtωωδδ+∞+∞−−−∞−∞⎤⎛⎞⎛⎞+++−⎜⎟⎜⎟⎥⎝⎠⎝⎠⎦∫∫iiii2212aaaaeeeeωωωω−−⎛⎞=+++⎜⎟⎝⎠coscos2aaωω=+8()cossinftt=t()=ωF¶()icossintftttedtω+∞−−∞=⎡⎤⎣⎦∫∫∫+∞∞−−−+∞∞−−−==dteeedttettttωωi2i2iii2212sin21()()⎥⎦⎤⎢⎣⎡−=∫∫+∞∞−+−+∞∞−−−dtedtett2i2ii41ωω()([]2222i41−−+−=ωπδωπδ)()([]222i−−+=ωδωδπ)93()sinft=t()=ωF¶()3isintfttedtω+∞−−∞=⎡⎤⎣⎦∫=i1(3sinsin3)4tttedtω+∞−−∞−∫i[3(1)3(1)(3)(3)]4πδωδωδωδω=+−−−++−10()sin(5)3fttπ=+()()ii1sin(5)(sin53cos5)32ttFftedttedtttedtωωitωπω+∞+∞+∞−−−∞−∞−∞==+=+∫∫∫−13i[(5)(5)][(5)(5)][(3i)(5)(3i)(5)222ππδωδωπδωδωδωδω=+−−+++−=+++−]−11δ−()()ttδδ=−()fx(,)−∞+∞()()()()(0)tftdtufudufδδ+∞+∞−∞−∞−=−=∫∫δ−()()(0)tftdtfδ+∞−∞=∫()()()()tftdttftdtδδ+∞+∞−∞−∞−=∫∫δ−12¶[]j()()teFϕω=()tϕ¶1[cos()][()()],2tFFϕωω=+−¶1[sin()][()()],2jtFFϕωω=−−()Fω−(F)ω−()()()tjcosjsintetϕϕϕ=+()()()jcosjsintttϕϕϕ−=−e()()()2cos*iit