Chap.3 Laplace Transform
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1 Chap. aplac Tranorm Tranorm: An opraion ha ranorm a uncion ino anohr uncion i Dirniaion ranorm: ii x: d dx x x Ingraion ranorm: x: x dx x c Now, conidr a dind ingral k, d,ha ranorm ino a uncion o variabl : i Th ingral i aid o b convrgn i h limi xi: b k, d lim b k, d ii Th ingral i aid o b divrgn i h limi don xi. aplac ranorm: on kind o ingraion ranorm i Diniion [ ] d, whr i a uncion dind or &, ar wo indpndn variabl. Whn h abov ingral convrg, h rul i a uncion o : [ ] d Exampl: α []? [] b b d lim d lim lim b or b b * [ aplac inh ] k k ranorm, o baic uncion coh k k, k [], [ ] n! [ ], n, [ ] n a, a n k in k b a [ k], [ k], co k Edid by Pro. Yung-Jung Hu
2 ii Baic Propri a inar propry α βg α * β * g α βg [ ] [ ] [ ] b Exinc o [ ] [ ] xi whn i rul ingral convrg! Suicin condiion: or h xinc o [ ] Th graph o canno grow ar han h graph o a incra! i aid o b o xponnial ordr. Suicin condiion: * can b a pwi coninuou uncion on h inrval [,. *I i o xponnial ordr, or picn coninuou in h inrval [,, hn [ ] xi!! *An non-picn coninuou could howvr hav [ ] o xi. iii Exponnial ordr Diniion: A uncion i aid o b xponnial ordr c i hr xi conan c, M and T, uch ha c M or all T Exampl: < Exampl: < Exampl: co < Exampl4: n < Edid by Pro. Yung-Jung Hu
3 In ohr word, h graph o on h inrval T, don grow ar han c graph o xponnial uncion M i aid o b o xponnial ordr c. Invr Tranorm i or ingraion ranorm: Thi i kind o invr ranorm or ingraion ii or aplac ranorm: Now, how o ranorm back o?? iii Diniion: I rprn h -ranorm o, i.. [ ] W hn ay i h invr -ranorm o, and can b xprd a ollow: iv πi r r i [ ] d,, r R i inariy propry: [ α βg ] α * [ ] β * [ G ] Exampl: *Invr aplac ranorm o baic uncion n! n a n k in k k k inh k k a cok k coh k k 6 6 co in Edid by Pro. Yung-Jung Hu
4 4 Tranorm o Drivaiv d d d i [ ] d d lim By h xponnial ordr horm: c lim limm lim lim M M, or c, convrg! c M, or < c, divrg! [ ] [ ], or c ii [ ] [ ] c, or c n n n n < n iii [ ] [ ]... iv To olv a dirnial quaion: dy Exampl: y in, y 6 d Tak -ranorm on ach par o D.E.: dy [ ] [ y] [in ], 令 [ y] Y d dy [ ] Y y & [in ] 代回 d 4 Y y Y, y 6 代入 4 6 Y 4 6 A B C Y 4 4 A8, B-, C Y y [ Y ] 8 co in 4 Edid by Pro. Yung-Jung Hu
5 Edid by Pro. Yung-Jung Hu Bhavior o a 並非所有 皆可 invr 回 I i picwi coninuou on, and o xponnial ordr, hn ] [ lim lim I lim, hn can b invr ranormd o.g.: or 無 invr ranorm 6 Tranlaion hiing horm 平移定理 i ir ranlaion horm: Tranlaion on h -axi. I [ ] and a i any ral numbr, hn [ ] a a <proo 已知 [ ] d, hn [ ] d d a a 令 a u, a u d u Exampl: [ ]?? co4 [ ] [ ] 6 6 co4 4 co Exampl:?? 令 B A
6 ii Uni p uncion 單位階梯函數 a Diniion: μ a, or a b Turn-o c: I,, or a y * μ, or <, or c Tranlaion on -axi or : I,, y * μ, or <, or d Compac orm: y * μ, or < In gnral: i g, < a h, a, or can b xprd a g g μ a h μ a *Conidr a gnral uncion dind or. Th picwi dind uncion a * μ a i an nir ranlaion o wih a uni o h righ on -axi. iii Scond ranlaion horm: ranlaion on -axi I [ ], and a a, hn [ a * μ a ] [ ] a [ ] ] a * μ a Exampl: [ * μ ]?? 令 a, μ a μ, a, [ ] By cond ranlaion horm: [ * μ ] 6 Edid by Pro. Yung-Jung Hu
7 *Alrnaiv orm o nd ranlaion horm: a [ * μ a ] [ a] Exampl: [ co * μ π ]?? 令 co, μ a μ π, a π a co π co π co By cond ranlaion horm: [ * μ π ] 7 Edid by Pro. Yung-Jung Hu
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