Chap.3 Laplace Transform

Size: px
Start display at page:

Download "Chap.3 Laplace Transform"

Transcription

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

Chapter 12 Introduction To The Laplace Transform

Chapter 12 Introduction To The Laplace Transform Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and

More information

2. The Laplace Transform

2. The Laplace Transform Th aac Tranorm Inroucion Th aac ranorm i a unamna an vry uu oo or uying many nginring robm To in h aac ranorm w conir a comx variab σ, whr σ i h ra ar an i h imaginary ar or ix vau o σ an w viw a a oin

More information

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion

More information

Laplace Transform. National Chiao Tung University Chun-Jen Tsai 10/19/2011

Laplace Transform. National Chiao Tung University Chun-Jen Tsai 10/19/2011 plc Trnorm Nionl Chio Tung Univriy Chun-Jn Ti /9/ Trnorm o Funcion Som opror rnorm uncion ino nohr uncion: d Dirniion: x x, or Dx x dx x Indini Ingrion: x dx c Dini Ingrion: x dx 9 A uncion my hv nicr

More information

Boyce/DiPrima/Meade 11 th ed, Ch 6.1: Definition of Laplace Transform

Boyce/DiPrima/Meade 11 th ed, Ch 6.1: Definition of Laplace Transform Boy/DiPrima/Mad h d, Ch 6.: Diniion o apla Tranorm Elmnary Dirnial Equaion and Boundary Valu Problm, h diion, by William E. Boy, Rihard C. DiPrima, and Doug Mad 7 by John Wily & Son, In. Many praial nginring

More information

PERIODICAL SOLUTION OF SOME DIFFERENTIAL EQUATIONS UDC 517.9(045)=20. Julka Knežević-Miljanović

PERIODICAL SOLUTION OF SOME DIFFERENTIAL EQUATIONS UDC 517.9(045)=20. Julka Knežević-Miljanović FCT UNIVESITTIS Sri: chanic uomaic Conrol and oboic Vol. N o 7 5. 87-9 PEIODICL SOLUTION OF SOE DIFFEENTIL EQUTIONS UDC 57.95= Julka Knžvić-iljanović Facul o ahmaic Univri o lgrad E-mail: knzvic@oincar.ma.bg.ac.u

More information

Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System

Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +

More information

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar

More information

THE LAPLACE TRANSFORM

THE LAPLACE TRANSFORM THE LAPLACE TRANSFORM LEARNING GOALS Diniion Th ranorm map a ncion o im ino a ncion o a complx variabl Two imporan inglariy ncion Th ni p and h ni impl Tranorm pair Baic abl wih commonly d ranorm Propri

More information

LaPlace Transform in Circuit Analysis

LaPlace Transform in Circuit Analysis LaPlac Tranform in Circui Analyi Obciv: Calcula h Laplac ranform of common funcion uing h dfiniion and h Laplac ranform abl Laplac-ranform a circui, including componn wih non-zro iniial condiion. Analyz

More information

Transfer function and the Laplace transformation

Transfer function and the Laplace transformation Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and

More information

Final Exam : Solutions

Final Exam : Solutions Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b

More information

Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule

Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -

More information

Lecture 4: Laplace Transforms

Lecture 4: Laplace Transforms Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions

More information

Poisson process Markov process

Poisson process Markov process E2200 Quuing hory and lraffic 2nd lcur oion proc Markov proc Vikoria Fodor KTH Laboraory for Communicaion nwork, School of Elcrical Enginring 1 Cour oulin Sochaic proc bhind quuing hory L2-L3 oion proc

More information

XV Exponential and Logarithmic Functions

XV Exponential and Logarithmic Functions MATHEMATICS 0-0-RE Dirnial Calculus Marin Huard Winr 08 XV Eponnial and Logarihmic Funcions. Skch h graph o h givn uncions and sa h domain and rang. d) ) ) log. Whn Sarah was born, hr parns placd $000

More information

Spring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review

Spring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw. conx and dircion Imagin a sysm ha varis in im; w migh plo is oupu vs. im. A plo migh imply an quaion, and h quaion is usually an

More information

Why Laplace transforms?

Why Laplace transforms? MAE4 Linar ircui Why Lalac ranform? Firordr R cc v v v KVL S R inananou for ach Subiu lmn rlaion v S Ordinary diffrnial quaion in rm of caacior volag Lalac ranform Solv Invr LT V u, v Ri, i A R V A _ v

More information

Laplace Transforms recap for ccts

Laplace Transforms recap for ccts Lalac Tranform rca for cc Wha h big ida?. Loo a iniial condiion ron of cc du o caacior volag and inducor currn a im Mh or nodal analyi wih -domain imdanc rianc or admianc conducanc Soluion of ODE drivn

More information

4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b

4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b 4. Th Uniform Disribuion Df n: A c.r.v. has a coninuous uniform disribuion on [a, b] whn is pdf is f x a x b b a Also, b + a b a µ E and V Ex4. Suppos, h lvl of unblivabiliy a any poin in a Transformrs

More information

fiziks Institute for NET/JRF, GATE, IIT JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES MATEMATICAL PHYSICS SOLUTIONS are

fiziks Institute for NET/JRF, GATE, IIT JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES MATEMATICAL PHYSICS SOLUTIONS are MTEMTICL PHYSICS SOLUTIONS GTE- Q. Considr an ani-symmric nsor P ij wih indics i and j running from o 5. Th numbr of indpndn componns of h nsor is 9 6 ns: Soluion: Th numbr of indpndn componns of h nsor

More information

Instructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems

Instructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi

More information

Ma/CS 6a Class 15: Flows and Bipartite Graphs

Ma/CS 6a Class 15: Flows and Bipartite Graphs //206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d

More information

Let s look again at the first order linear differential equation we are attempting to solve, in its standard form:

Let s look again at the first order linear differential equation we are attempting to solve, in its standard form: Th Ingraing Facor Mhod In h prvious xampls of simpl firs ordr ODEs, w found h soluions by algbraically spara h dpndn variabl- and h indpndn variabl- rms, and wri h wo sids of a givn quaion as drivaivs,

More information

REPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if.

REPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if. Tranform Mhod and Calculu of Svral Variabl H7, p Lcurr: Armin Halilovic KTH, Campu Haning E-mail: armin@dkh, wwwdkh/armin REPETITION bfor h am PART, Tranform Mhod Laplac ranform: L Driv h formula : a L[

More information

Elementary Differential Equations and Boundary Value Problems

Elementary Differential Equations and Boundary Value Problems Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ

More information

S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]

S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15] S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:

More information

Institute of Actuaries of India

Institute of Actuaries of India Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6

More information

Midterm exam 2, April 7, 2009 (solutions)

Midterm exam 2, April 7, 2009 (solutions) Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions

More information

CSE 245: Computer Aided Circuit Simulation and Verification

CSE 245: Computer Aided Circuit Simulation and Verification CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy

More information

Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall. Chapr Rviw 0 6. ( a a ln a. This will qual a if an onl if ln a, or a. + k an (ln + c. Thrfor, a an valu of, whr h wo curvs inrsc, h wo angn lins will b prpnicular. 6. (a Sinc h lin passs hrough h origin

More information

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o

More information

CIVL 8/ D Boundary Value Problems - Quadrilateral Elements (Q4) 1/8

CIVL 8/ D Boundary Value Problems - Quadrilateral Elements (Q4) 1/8 CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 1/8 ISOPARAMERIC ELEMENS h inar rianguar mn and h iinar rcanguar mn hav vra imporan diadvanag. 1. Boh mn ar una o accura rprn curvd oundari, and. h provid

More information

Applied Statistics and Probability for Engineers, 6 th edition October 17, 2016

Applied Statistics and Probability for Engineers, 6 th edition October 17, 2016 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 CHATER Scion - -. a d. 679.. b. d. 88 c d d d. 987 d. 98 f d.. Thn, = ln. =. g d.. Thn, = ln.9 =.. -7. a., by symmry. b.. d...6. 7.. c...

More information

5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t

5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t AP CALCULUS FINAL UNIT WORKSHEETS ACCELERATION, VELOCTIY AND POSITION In problms -, drmin h posiion funcion, (), from h givn informaion.. v (), () = 5. v ()5, () = b g. a (), v() =, () = -. a (), v() =

More information

The Laplace Transform

The Laplace Transform Th Lplc Trnform Dfiniion nd propri of Lplc Trnform, picwi coninuou funcion, h Lplc Trnform mhod of olving iniil vlu problm Th mhod of Lplc rnform i ym h rli on lgbr rhr hn clculu-bd mhod o olv linr diffrnil

More information

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35 MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h

More information

Chapter 9 The Laplace Transform

Chapter 9 The Laplace Transform Chapr 9 Th Laplac Tranform 熊红凯特聘教授 hp://min.ju.du.cn 电子工程系上海交通大学 7 Topic 9. DEFINATION OF THE LAPLACE TRANSFORM 9. THE REGION OF CONVERGENCE FOR LAPLACE THANSFORMS 9. PROPERTIES OF THE LAPLACE TRANSFORM

More information

Chapter 3: Fourier Representation of Signals and LTI Systems. Chih-Wei Liu

Chapter 3: Fourier Representation of Signals and LTI Systems. Chih-Wei Liu Chapr 3: Fourir Rprsnaion of Signals and LTI Sysms Chih-Wi Liu Oulin Inroducion Complx Sinusoids and Frquncy Rspons Fourir Rprsnaions for Four Classs of Signals Discr-im Priodic Signals Fourir Sris Coninuous-im

More information

Double Slits in Space and Time

Double Slits in Space and Time Doubl Slis in Sac an Tim Gorg Jons As has bn ror rcnly in h mia, a am l by Grhar Paulus has monsra an inrsing chniqu for ionizing argon aoms by using ulra-shor lasr ulss. Each lasr uls is ffcivly on an

More information

Chapter 13 Laplace Transform Analysis

Chapter 13 Laplace Transform Analysis Chapr aplac Tranorm naly Chapr : Ouln aplac ranorm aplac Tranorm -doman phaor analy: x X σ m co ω φ x X X m φ x aplac ranorm: [ o ] d o d < aplac Tranorm Thr condon Unlaral on-dd aplac ranorm: aplac ranorm

More information

Math 3301 Homework Set 6 Solutions 10 Points. = +. The guess for the particular P ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos 2 t : 4D= 2

Math 3301 Homework Set 6 Solutions 10 Points. = +. The guess for the particular P ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos 2 t : 4D= 2 Mah 0 Homwork S 6 Soluions 0 oins. ( ps) I ll lav i o you o vrify ha y os sin = +. Th guss for h pariular soluion and is drivaivs is blow. Noi ha w ndd o add s ono h las wo rms sin hos ar xaly h omplimnary

More information

AN INTRODUCTION TO FOURIER ANALYSIS PROF. VEDAT TAVSANOĞLU

AN INTRODUCTION TO FOURIER ANALYSIS PROF. VEDAT TAVSANOĞLU A IRODUCIO O FOURIER AALYSIS PROF. VEDA AVSAOĞLU 994 A IRODUCIO O FOURIER AALYSIS ABLE OF COES. HE FOURIER SERIES ---------------------------------------------------------------------3.. Priodic Funcions-----------------------------------------------------------------------3..

More information

Summary: Solving a Homogeneous System of Two Linear First Order Equations in Two Unknowns

Summary: Solving a Homogeneous System of Two Linear First Order Equations in Two Unknowns Summary: Solvng a Homognous Sysm of Two Lnar Frs Ordr Equaons n Two Unknowns Gvn: A Frs fnd h wo gnvalus, r, and hr rspcv corrspondng gnvcors, k, of h coffcn mar A Dpndng on h gnvalus and gnvcors, h gnral

More information

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm

More information

Heat flow in composite rods an old problem reconsidered

Heat flow in composite rods an old problem reconsidered Ha flow in copoi ro an ol probl rconir. Kranjc a Dparn of Phyic an chnology Faculy of Eucaion Univriy of jubljana Karljva ploca 6 jubljana Slovnia an J. Prnlj Faculy of Civil an Goic Enginring Univriy

More information

(1) Then we could wave our hands over this and it would become:

(1) Then we could wave our hands over this and it would become: MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and

More information

B) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of

B) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of Sampl Final 00 1. Suppos z = (, y), ( a, b ) = 0, y ( a, b ) = 0, ( a, b ) = 1, ( a, b ) = 1, and y ( a, b ) =. Thn (a, b) is h s is inconclusiv a saddl poin a rlaiv minimum a rlaiv maimum. * (Classiy

More information

H is equal to the surface current J S

H is equal to the surface current J S Chapr 6 Rflcion and Transmission of Wavs 6.1 Boundary Condiions A h boundary of wo diffrn mdium, lcromagnic fild hav o saisfy physical condiion, which is drmind by Maxwll s quaion. This is h boundary condiion

More information

14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions

14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions 4.0 rincipl of Macroconomic Fall 005 Quiz 3 Soluion Shor Quion (30/00 poin la a whhr h following amn ar TRUE or FALSE wih a hor xplanaion (3 or 4 lin. Each quion coun 5/00 poin.. An incra in ax oday alway

More information

EXERCISE - 01 CHECK YOUR GRASP

EXERCISE - 01 CHECK YOUR GRASP DIFFERENTIAL EQUATION EXERCISE - CHECK YOUR GRASP 7. m hn D() m m, D () m m. hn givn D () m m D D D + m m m m m m + m m m m + ( m ) (m ) (m ) (m + ) m,, Hnc numbr of valus of mn will b. n ( ) + c sinc

More information

INTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS

INTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS adjoint...6 block diagram...4 clod loop ytm... 5, 0 E()...6 (t)...6 rror tady tat tracking...6 tracking...6...6 gloary... 0 impul function...3 input...5 invr Laplac tranform, INTRODUCTION TO AUTOMATIC

More information

= x. I (x,y ) Example: Translation. Operations depend on pixel s Coordinates. Context free. Independent of pixel values. I(x,y) Forward mapping:

= x. I (x,y ) Example: Translation. Operations depend on pixel s Coordinates. Context free. Independent of pixel values. I(x,y) Forward mapping: Gomric Transormaion Oraions dnd on il s Coordinas. Con r. Indndn o il valus. (, ) ' (, ) ' I (, ) I ' ( (, ), ( ) ), (,) (, ) I(,) I (, ) Eaml: Translaion (, ) (, ) (, ) I(, ) I ' Forward Maing Forward

More information

Phys463.nb Conductivity. Another equivalent definition of the Fermi velocity is

Phys463.nb Conductivity. Another equivalent definition of the Fermi velocity is 39 Anohr quival dfiniion of h Fri vlociy is pf vf (6.4) If h rgy is a quadraic funcion of k H k L, hs wo dfiniions ar idical. If is NOT a quadraic funcion of k (which could happ as will b discussd in h

More information

whereby we can express the phase by any one of the formulas cos ( 3 whereby we can express the phase by any one of the formulas

whereby we can express the phase by any one of the formulas cos ( 3 whereby we can express the phase by any one of the formulas Third In-Class Exam Soluions Mah 6, Profssor David Lvrmor Tusday, 5 April 07 [0] Th vrical displacmn of an unforcd mass on a spring is givn by h 5 3 cos 3 sin a [] Is his sysm undampd, undr dampd, criically

More information

On the Speed of Heat Wave. Mihály Makai

On the Speed of Heat Wave. Mihály Makai On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor 1.

More information

Boyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues

Boyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm

More information

symmetric/hermitian matrices, and similarity transformations

symmetric/hermitian matrices, and similarity transformations Linar lgbra for Wirlss Communicaions Lcur: 6 Diffrnial quaions, Grschgorin's s circl horm, symmric/hrmiian marics, and similariy ransformaions Ov Edfors Dparmn of Elcrical and Informaion Tchnology Lund

More information

Math 266, Practice Midterm Exam 2

Math 266, Practice Midterm Exam 2 Mh 66, Prcic Midrm Exm Nm: Ground Rul. Clculor i NOT llowd.. Show your work for vry problm unl ohrwi d (pril crdi r vilbl). 3. You my u on 4-by-6 indx crd, boh id. 4. Th bl of Lplc rnform i vilbl h l pg.

More information

1973 AP Calculus BC: Section I

1973 AP Calculus BC: Section I 97 AP Calculus BC: Scio I 9 Mius No Calculaor No: I his amiaio, l dos h aural logarihm of (ha is, logarihm o h bas ).. If f ( ) =, h f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, h h s of valus for which f

More information

2008 AP Calculus BC Multiple Choice Exam

2008 AP Calculus BC Multiple Choice Exam 008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl

More information

3(8 ) (8 x x ) 3x x (8 )

3(8 ) (8 x x ) 3x x (8 ) Scion - CHATER -. a d.. b. d.86 c d 8 d d.9997 f g 6. d. d. Thn, = ln. =. =.. d Thn, = ln.9 =.7 8 -. a d.6 6 6 6 6 8 8 8 b 9 d 6 6 6 8 c d.8 6 6 6 6 8 8 7 7 d 6 d.6 6 6 6 6 6 6 8 u u u u du.9 6 6 6 6 6

More information

1 Finite Automata and Regular Expressions

1 Finite Automata and Regular Expressions 1 Fini Auom nd Rgulr Exprion Moivion: Givn prn (rgulr xprion) for ring rching, w migh wn o convr i ino drminiic fini uomon or nondrminiic fini uomon o mk ring rching mor fficin; drminiic uomon only h o

More information

1973 AP Calculus AB: Section I

1973 AP Calculus AB: Section I 97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=

More information

Lecture 1: Growth and decay of current in RL circuit. Growth of current in LR Circuit. D.K.Pandey

Lecture 1: Growth and decay of current in RL circuit. Growth of current in LR Circuit. D.K.Pandey cur : Growh and dcay of currn in circui Growh of currn in Circui us considr an inducor of slf inducanc is conncd o a DC sourc of.m.f. E hrough a rsisr of rsisanc and a ky K in sris. Whn h ky K is swichd

More information

Lecture 26: Leapers and Creepers

Lecture 26: Leapers and Creepers Lcur 6: Lapr and Crpr Scrib: Grain Jon (and Marin Z. Bazan) Dparmn of Economic, MIT May, 005 Inroducion Thi lcur conidr h analyi of h non-parabl CTRW in which h diribuion of p iz and im bwn p ar dpndn.

More information

6.8 Laplace Transform: General Formulas

6.8 Laplace Transform: General Formulas 48 HAP. 6 Laplace Tranform 6.8 Laplace Tranform: General Formula Formula Name, ommen Sec. F() l{ f ()} e f () d f () l {F()} Definiion of Tranform Invere Tranform 6. l{af () bg()} al{f ()} bl{g()} Lineariy

More information

AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )

AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 ) AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc

More information

Coherence and interactions in diffusive systems. Lecture 4. Diffusion + e-e interations

Coherence and interactions in diffusive systems. Lecture 4. Diffusion + e-e interations Cohrnc and inracions in diffusiv sysms G. Monambaux cur 4 iffusion + - inraions nsiy of sas anomaly phasing du o lcron-lcron inracions - inracion andau Frmi liquid picur iffusion slows down lcrons ( )

More information

Introduction to SLE Lecture Notes

Introduction to SLE Lecture Notes Inroducion o SLE Lecure Noe May 13, 16 - The goal of hi ecion i o find a ufficien condiion of λ for he hull K o be generaed by a imple cure. I urn ou if λ 1 < 4 hen K i generaed by a imple curve. We will

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!

More information

CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees

CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 259] B-Trs Th AVL r and rd-black r allowd som variaion in h lnghs of h diffrn roo-o-laf pahs. An alrnaiv ida is o mak sur ha all roo-o-laf pahs

More information

Microscopic Flow Characteristics Time Headway - Distribution

Microscopic Flow Characteristics Time Headway - Distribution CE57: Traffic Flow Thory Spring 20 Wk 2 Modling Hadway Disribuion Microscopic Flow Characrisics Tim Hadway - Disribuion Tim Hadway Dfiniion Tim Hadway vrsus Gap Ahmd Abdl-Rahim Civil Enginring Dparmn,

More information

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn

More information

Wave Equation (2 Week)

Wave Equation (2 Week) Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris

More information

Partial Fraction Expansion

Partial Fraction Expansion Paial Facion Expanion Whn ying o find h inv Laplac anfom o inv z anfom i i hlpfl o b abl o bak a complicad aio of wo polynomial ino fom ha a on h Laplac Tanfom o z anfom abl. W will illa h ing Laplac anfom.

More information

Charging of capacitor through inductor and resistor

Charging of capacitor through inductor and resistor cur 4&: R circui harging of capacior hrough inducor and rsisor us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R, an inducor of inducanc and a y K in sris.

More information

Review Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( )

Review Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( ) Rviw Lcur 5 Firs-ordr circui Th sourc-fr R-C/R-L circui Sp rspons of an RC/RL circui v( ) v( ) [ v( 0) v( )] 0 Th i consan = RC Th final capacior volag v() Th iniial capacior volag v( 0 ) Volag/currn-division

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!

More information

ON FINITE MORSE INDEX SOLUTIONS OF HIGHER ORDER FRACTIONAL LANE-EMDEN EQUATIONS

ON FINITE MORSE INDEX SOLUTIONS OF HIGHER ORDER FRACTIONAL LANE-EMDEN EQUATIONS ON FINITE MORSE INDEX SOLUTIONS OF HIGHER ORDER FRACTIONAL LANE-EMDEN EQUATIONS MOSTAFA FAZLY AND JUNCHENG WEI Abrac. W claify fini Mor indx oluion of h following nonlocal Lan-Emdn quaion u u u for <

More information

A THREE COMPARTMENT MATHEMATICAL MODEL OF LIVER

A THREE COMPARTMENT MATHEMATICAL MODEL OF LIVER A THREE COPARTENT ATHEATICAL ODEL OF LIVER V. An N. Ch. Paabhi Ramacharyulu Faculy of ahmaics, R D collgs, Hanamonda, Warangal, India Dparmn of ahmaics, Naional Insiu of Tchnology, Warangal, India E-ail:

More information

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S.

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S. Rfrc: (i) (ii) (iii) Advcd Egirig Mhmic, K.A. Sroud, Dxr J. Booh Egirig Mhmic, H.K. D Highr Egirig Mhmic, Dr. B.S. Grwl Th mhod of m Thi coi of h followig xm wih h giv coribuio o h ol. () Mid-rm xm : 3%

More information

COHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.

COHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim. MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function

More information

Homework #2: CMPT-379 Distributed on Oct 2; due on Oct 16 Anoop Sarkar

Homework #2: CMPT-379 Distributed on Oct 2; due on Oct 16 Anoop Sarkar Homwork #2: CMPT-379 Disribud on Oc 2 du on Oc 16 Anoop Sarkar anoop@cs.su.ca Rading or his homwork includs Chp 4 o h Dragon book. I ndd, rr o: hp://ldp.org/howto/lx-yacc-howto.hml Only submi answrs or

More information

General Article Application of differential equation in L-R and C-R circuit analysis by classical method. Abstract

General Article Application of differential equation in L-R and C-R circuit analysis by classical method. Abstract Applicaion of Diffrnial... Gnral Aricl Applicaion of diffrnial uaion in - and C- circui analysis by classical mhod. ajndra Prasad gmi curr, Dparmn of Mahmaics, P.N. Campus, Pokhara Email: rajndraprasadrgmi@yahoo.com

More information

FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Linear Systems

FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Linear Systems FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Inroducion and Linar Sysms David Lvrmor Dparmn of Mahmaics Univrsiy of Maryland 9 Dcmbr 0 Bcaus h prsnaion of his marial in lcur will diffr from

More information

[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then

[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then SYSTEM PERFORMANCE Lctur 0: Stady-tat Error Stady-tat Error Lctur 0: Stady-tat Error Dr.alyana Vluvolu Stady-tat rror can b found by applying th final valu thorm and i givn by lim ( t) lim E ( ) t 0 providd

More information

Lecture #6: Continuous-Time Signals

Lecture #6: Continuous-Time Signals EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals Lecure #6: Coninuous-Time Signals. Inroducion In his lecure, we discussed he ollowing opics:. Mahemaical represenaion and ransormaions

More information

Nikesh Bajaj. Fourier Analysis and Synthesis Tool. Guess.? Question??? History. Fourier Series. Fourier. Nikesh Bajaj

Nikesh Bajaj. Fourier Analysis and Synthesis Tool. Guess.? Question??? History. Fourier Series. Fourier. Nikesh Bajaj Guss.? ourir Analysis an Synhsis Tool Qusion??? niksh.473@lpu.co.in Digial Signal Procssing School of Elcronics an Communicaion Lovly Profssional Univrsiy Wha o you man by Transform? Wha is /Transform?

More information

Logistic equation of Human population growth (generalization to the case of reactive environment).

Logistic equation of Human population growth (generalization to the case of reactive environment). Logisic quaion of Human populaion growh gnralizaion o h cas of raciv nvironmn. Srg V. Ershkov Insiu for Tim aur Exploraions M.V. Lomonosov's Moscow Sa Univrsi Lninski gor - Moscow 999 ussia -mail: srgj-rshkov@andx.ru

More information

On General Solutions of First-Order Nonlinear Matrix and Scalar Ordinary Differential Equations

On General Solutions of First-Order Nonlinear Matrix and Scalar Ordinary Differential Equations saartvlos mcnirbata rovnuli akadmiis moamb 3 #2 29 BULLTN OF TH ORN NTONL DMY OF SNS vol 3 no 2 29 Mahmaics On nral Soluions of Firs-Ordr Nonlinar Mari and Scalar Ordinary Diffrnial uaions uram L Kharaishvili

More information

46. Let y = ln r. Then dy = dr, and so. = [ sin (ln r) cos (ln r)

46. Let y = ln r. Then dy = dr, and so. = [ sin (ln r) cos (ln r) 98 Scion 7.. L w. Thn dw d, so d dw w dw. sin d (sin w)( wdw) w sin w dw L u w dv sin w dw du dw v cos w w sin w dw w cos w + cos w dw w cos w+ sin w+ sin d wsin wdw w cos w+ sin w+ cos + sin +. L w +

More information

Circuit Transients time

Circuit Transients time Circui Tranin A Solp 3/29/0, 9/29/04. Inroducion Tranin: A ranin i a raniion from on a o anohr. If h volag and currn in a circui do no chang wih im, w call ha a "ady a". In fac, a long a h volag and currn

More information

Section 11.6: Directional Derivatives and the Gradient Vector

Section 11.6: Directional Derivatives and the Gradient Vector Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th

More information

Discussion 06 Solutions

Discussion 06 Solutions STAT Discussion Soluions Spring 8. Th wigh of fish in La Paradis follows a normal disribuion wih man of 8. lbs and sandard dviaion of. lbs. a) Wha proporion of fish ar bwn 9 lbs and lbs? æ 9-8. - 8. P

More information

Calculus Revision A2 Level

Calculus Revision A2 Level alculus Rvision A Lvl Tabl of drivativs a n sin cos tan d an sc n cos sin Fro AS * NB sc cos sc cos hain rul othrwis known as th function of a function or coposit rul. d d Eapl (i) (ii) Obtain th drivativ

More information

2. Transfer function. Kanazawa University Microelectronics Research Lab. Akio Kitagawa

2. Transfer function. Kanazawa University Microelectronics Research Lab. Akio Kitagawa . ransfr funion Kanazawa Univrsiy Mirolronis Rsarh Lab. Akio Kiagawa . Wavforms in mix-signal iruis Configuraion of mix-signal sysm x Digial o Analog Analog o Digial Anialiasing Digial moohing Filr Prossor

More information

SUPERCRITICAL BRANCHING DIFFUSIONS IN RANDOM ENVIRONMENT

SUPERCRITICAL BRANCHING DIFFUSIONS IN RANDOM ENVIRONMENT Elc. Comm. in Proa. 6 (), 78 79 ELECTRONIC COMMUNICATIONS in PROBABILITY SUPERCRITICAL BRANCHING DIFFUSIONS IN RANDOM ENVIRONMENT MARTIN HUTZENTHALER ETH Zürich, Dparmn of Mahmaics, Rämisrass, 89 Zürich.

More information

Veer Surendra Sai University of Technology, Burla. S u b j e c t : S i g n a l s a n d S y s t e m s - I S u b j e c t c o d e : B E E

Veer Surendra Sai University of Technology, Burla. S u b j e c t : S i g n a l s a n d S y s t e m s - I S u b j e c t c o d e : B E E Vr Surndra Sai Univriy of Tchnology, Burla Dparmn o f E l c r i c a l & E l c r o n i c E n g g S u b j c : S i g n a l a n d S y m - I S u b j c c o d : B E E - 6 0 5 B r a n c h m r : E E E 5 h m SYLLABUS

More information