MATEMATICĂ 3 PROBLEME DE REFLECŢIE
|
|
- Blaze Sullivan
- 6 years ago
- Views:
Transcription
1 Recapitulare din liceu MATEMATIĂ 3 ANALIZĂ OMPLEXĂ PROBLEME DE REFLEŢIE. Scrieţi numerele următoare sub forma a + bi, unde a, b R: a) 3i + i ; b) i + i ;. Reolvaţi în ecuaţiile: ( + i)( i) c) ( + i)(4 3i) ; (4 + 5i) + i3 d). ( + i) a) + 7 6i = ; b) + = i + 3i. 3. Descrieţi mulţimea punctelor din planul complex, care verifică relaţiile: a) Re(( + i) ) = ; b) = i ; c) = ; d) Re( ) = 3 i ; e) i = ; f) =Re(). 4. a) Dacă =, arătaţi că 3 4. b) Găsiţi o majorare convenabilă pentru 3 + +, dacă. 5. Găsiţi un număr complex ce verifică ecuaţiile: a) = + i; b) + + i = Pentru orice două numere complexe şi, arătaţi că: + + = ( + ). 7. Scrieţi următoarele numere complexe în formă polară: a) 3i; b) 5 5i; c) 3i; d). 3 + i ( 8. alculaţi + ) i. 9. Arătaţi că + cos θ + cos θ cos nθ = + sin ( ) n + θ sin θ, unde < θ < π.. Reolvaţi ecuaţia: = ( + i) 5.. Repreentaţi grafic soluţiile următoarelor relaţii:
2 a) + 3i = ; b) = 4; c) Re( )=; d) <Re( ) < 4; e) i < ; f) 3 + 4i 5. Funcţii complexe. Evaluaţi funcţia f() = ln + iarg() în punctele: a) ; b) 4i; c) + i. 3. Aflaţi părţile reale şi imaginare (u şi v) ale funcţiilor: a) f() = 3 + 6; b) f() = + ; c) f() = + ; d) f() = e +i. 4. Găsiţi domeniul şi imaginea funcţiei f() = Arătaţi că funcţia f() = ei + e i este periodică de perioadă π. 6. Găsiţi imaginea S a mulţimii S, prin funcţia complexă w = f(), dacă: (a) f() =, S este dreapta y = x; (b) f() = 3, S este semiplanul cu Im() > ; (c) f() = 3, S este banda verticală infinită Re() < 3; (d) f() = i + 4, S este banda oriontală infinită < Im() <. Limită şi continuitate 7. alculaţi: (a) lim +i ( + i); (b) lim i e. 8. alculaţi: (a) lim ( ); i (b) lim 5 + ; i (c) (d) lim e iπ/4 lim +i ( + + ; ) ; 4 (e) lim i + i ; ( + i) (f) lim +i ( + i) ; (g) lim + i ( + i) ; (h) lim i + i ; (i) lim i + ; (j) lim i i. 9. Arătaţi că f este continuă în punctul indicat: (a) f() = i + 3 i, = i; (b) f() = Re() + i, = e iπ/4 ;
3 3, = (c) f() = i, 3 = + i 3., = Derivabilitate. Olomorfie. Folosind proprietăţile derivării, calculaţi f () pentru funcţiile: a) f() = ( i) 5 +i 4 3 +i 6 ; b) f() = ( 6 )( + 5i); c) f() = i 3 + i ; d) f() = ( 4 i + )., =. Arătaţi că funcţia f() = x y 3 x + y + + y 3 ix3 x + y, renţiabilă în =. nu este dife-. Utiliaţi regula lui Guillaume de L Hôpital pentru a calcula limitele: 7 + i a) lim i 4 + ; b) lim +i + ; c) lim + i ; d) lim i Arătaţi că următoarele funcţii nu sunt analitice: a) f() =Re(); b) f() = ; c) f() = ; x d) f() = x + y + i y x + y. 4. Arătaţi că următoarele funcţii sunt analitice pe un domeniu adecvat, folosind ecuaţiile auchy -Riemann şi continuitatea derivatelor parţiale ale funcţiilor u şi v: (a) f() = e x cos y ie x sin y; (b) f() = 4x + 5x 4y i(8xy + 5y ); x (c) f() = (x ) + y i y (x ) + y ; (d) f() = cos θ i sin θ r r. 5. Aflaţi numerele a, b, c, d, pentru care următoarele funcţii sunt analitice: (a) f() = 3x y i(ax + by 3); (b) f() = x + axy + by + i(cx + dxy + y ). Augustin auchy ( ) a fost unul dintre cei mai importanţi matematicieni francei. A fost unul dintre pionierii analiei matematice şi a adus o serie de contribuţii şi în domeniul fiicii. Bernhard Riemann (86-866) a fost un matematician german cu importante contribuţii în analia matematică şi geometria diferenţială, unele dintre ele deschiând drumul ulterior spre teoria relativităţii generaliate. 3
4 6. Arătaţi că următoarele funcţii nu sunt analitice, dar sunt diferenţiabile de-a lungul curbelor indicate: (a) f() = x + y + ixy, axa Ox; (b) f() = 3x y 6ix y, axele de coordonate; (c) f() = x x + y + i(y 5y x), dreapta y = x Dacă f() este o funcţie analitică pe un domeniu D şi f() = c, unde c este o constantă reală, arătaţi că funcţia f() este constantă pe D. 8. Arătaţi că următoarele funcţii sunt armonice pe un domeniu adecvat, găsiţile conjugatele armonice şi funcţiile f() din care provin: a) u(x, y) = xy + x + y, f(i) = + 5i; b) u(x, y) = 4xy 3 4x 3 y + x, f( + i) = 5 + 4i. Funcţii elementare 9. alculaţi derivatele următoarelor funcţii: a) f() = e +i ; c) f() = e i e i ; b) f() = 3e ie 3 + i ; d) f() = ie. 3. Aduceţi funcţiile f la forma f() = u(x, y) + iv(x, y): a) f() = e i ; c) f() = e ; b) f() = e +i ; d) f() = e. 3. Găsiţi imaginea benii infinite < x prin transformarea w = e. 3. Aflaţi valorile logaritmilor complecşi: a) ln( 5); d) ln( 3 + i); b) ln( ei); e) ln( + i); c) ln( + i); f) Ln(6 6i); g) Ln( e ); h) Ln [ ( + 3i) 5] ; i) Ln[( + i) 4 ]. 33. Reolvaţi ecuaţiile: a) e = ie 3 ; b) e + e + = ; c) e =. 34. Determinaţi domeniul în care funcţia f este diferenţiabilă şi calculaţi derivata f : a) f() = 3 e i + iln; Ln( i) c) f() = ; + b) f() = ( + )Ln( + ); d) f() =Ln( + ). 35. Determinaţi valorile principale ale următoarelor puteri complexe: a) ( ) 3i ; c) i i π ; b) 3 i π ; d) ( + i) i. 36. Exprimaţi valorile funcţiilor de mai jos sub forma x + iy: a) sin(4i); b) cos( 4i); c) tan(i); d) csc( + i); e) cosh(πi); f) tanh( + 3i). 37. Reolvaţi ecuaţiile: 4
5 a) sin = i; b) cos = 4; c) cosh = i; d) sinh = e. 38. alculaţi derivatele următoarelor ( ) funcţii f(): a) sin( ); c) tan ; b) cos(ie ); d) sec( + ( i)); e) sin sinh ; f) cosh(i + e i ). 39. Determinaţi valorile: a) cos i; b) sin ; c) tan (i); d) tanh ( + i). 4. Reolvaţi ecuaţiile de gradul al doilea: a) + i = ; c) + 3i = ; b) i + i = ; d) 3 + ( 3i) 3i =. Integrale complexe 4. Evaluaţi următoarele integrale de-a lungul curbei indicate: a) ( ) d, unde este: x = t, y = t +, t ; b) d, unde este: x = t, y = t, t < ; ( c) ( + i) 5 ) 3 + i + 8 d, unde este cercul + i = ; d) sin d, unde este linia poligonală formată din segmentele ce unesc = cu = şi = cu = + i. 4. Găsiţi o majorare convenabilă pentru valorile absolute ale integralelor următoare, de-a lungul curbei indicate: e a) d, unde este cercul = 5; + b) ( + 4) d, unde este segmentul ce uneşte = cu = + i; c) d, unde este sfertul cercului = 4, de la = 4i la = 4. 3 Teorema auchy-goursat Evaluaţi următoarele integrale de-a lungul curbei indicate: ( a) ( 3 + 3i) d, = ; b) + ) d, = ; 4 3 Edouard Goursat ( ) a fost un matematician france, cunoscut în special pentru lucrarea ours d analyse mathématique, care a fixat un standard înalt pentru analia matematică, în special pentru analia complexă. 5
6 c) d, = ; 9 g) d, = ; + 3 ( cosh 3 d) d, = ; h) + ) d, = ; + + sin e) d, = ; ( i) d, = 3; 5) e π f) d, = ; j) d, + i = ; ( + i) 4 + k) + d, (i) =, (ii) =, (iii) 3i = ; 3 + l) 8 + d, (i) 5 =, (ii) i = ; ( 3 m) + ) d, (i) = 5, (ii) i = i ; n) ( i)( 3i) d, i = ; ( ) e o) d, = ; ( p) Re() ) d, unde este triunghiul de vârfuri: =, = + i şi =. Formulele lui auchy 44. Dacă = { = }, calculaţi: e ) i d; 4) e ) i d; 5) e 3) + d; 6) Teorema reiduurilor e 5 ( i) 3 d; e ( + ) d; 4 ( 5) d. 45. Evaluaţi următoarele integrale reale, utiliând teorema reiduurilor: ) ) 3) x + dx; x 4 + dx; x 6 + dx; 4) 5) 6) (x + ) dx; (x + ) 3 dx; (x + )(x + 4) dx; 6
7 7) 8) 9) ) ) ) 3) 4) 5) 6) π π π π π π x + x (x + )(x + 4) dx; x x 4 + dx; x x 4 + 5x + 4 dx; (4x + ) 3 dx; a + cos θ cos 3t 5 4 cos t dt; dθ, a > ; sin θ dθ; ( + cos θ) dθ; + a sin θ + a cos θ dθ, < a < ; dθ, < a < ; 7) 8) 9) ) ) ) 3) π < p < ; 4) 5) 6) π x sin x x + 9 dx; sin x x dx; sin x x + x + dx; cos x dx, a > ; x + a cos mx dx, m > ; a + x cos mx dx, m > ; (a + x ) p cos θ + p dθ, (cos 3 t + sin t) dt; x x + dx; dx. (x + ) n+ Bibliografie [] Mary L. Boas, Mathematical Methods in the Physical Sciences, Wiley, 5. [] Murray Spiegel, Seymour Lipschut, John Schiller, Dennis Spellman, Schaum s Outline of omplex Variables, McGraw-Hill, 9. [3] Dennis G. Zill, Patrick Shanahan, A First ourse in omplex Analysis with Applications, Jones & Bartlett Learning, 8. [4] Alexandru Negrescu Universitatea Politehnica din Bucureşti 7
Teorema Reziduurilor şi Bucuria Integralelor Reale Prezentare de Alexandru Negrescu
Teorema Reiduurilor şi Bucuria Integralelor Reale Preentare de Alexandru Negrescu Integrale cu funcţii raţionale ce depind de sint şi cost u notaţia e it, avem: cost sint i ( + ( dt d i, iar integrarea
More informationComplex Homework Summer 2014
omplex Homework Summer 24 Based on Brown hurchill 7th Edition June 2, 24 ontents hw, omplex Arithmetic, onjugates, Polar Form 2 2 hw2 nth roots, Domains, Functions 2 3 hw3 Images, Transformations 3 4 hw4
More informationON THE QUATERNARY QUADRATIC DIOPHANTINE EQUATIONS (II) NICOLAE BRATU 1 ADINA CRETAN 2
ON THE QUATERNARY QUADRATIC DIOPHANTINE EQUATIONS (II) NICOLAE BRATU 1 ADINA CRETAN ABSTRACT This paper has been updated and completed thanks to suggestions and critics coming from Dr. Mike Hirschhorn,
More informationEE2007 Tutorial 7 Complex Numbers, Complex Functions, Limits and Continuity
EE27 Tutorial 7 omplex Numbers, omplex Functions, Limits and ontinuity Exercise 1. These are elementary exercises designed as a self-test for you to determine if you if have the necessary pre-requisite
More informationMath 417 Midterm Exam Solutions Friday, July 9, 2010
Math 417 Midterm Exam Solutions Friday, July 9, 010 Solve any 4 of Problems 1 6 and 1 of Problems 7 8. Write your solutions in the booklet provided. If you attempt more than 5 problems, you must clearly
More informationFORMULELE LUI STIRLING, WALLIS, GAUSS ŞI APLICAŢII
DIDACTICA MATHEMATICA, Vol. 34), pp. 53 67 FORMULELE LUI STIRLING, WALLIS, GAUSS ŞI APLICAŢII Eugenia Duca, Emilia Copaciu şi Dorel I. Duca Abstract. In this paper are presented the Wallis, Stirling, Gauss
More informationBarem de notare clasa a V-a
Barem de notare clasa a V-a Problema1. Determinați mulțimile A și B, formate din numere naturale, știind că îndeplinesc simultan condițiile: a) A B,5,6 ; b) B A 0,7 ; c) card AB 3; d) suma elementelor
More informationSoluţii juniori., unde 1, 2
Soluţii juniori Problema 1 Se consideră suma S x1x x3x4... x015 x016 Este posibil să avem S 016? Răspuns: Da., unde 1,,..., 016 3, 3 Termenii sumei sunt de forma 3 3 1, x x x. 3 5 6 sau Cristian Lazăr
More information4.1 Exponential and Logarithmic Functions
. Exponential and Logarithmic Functions Joseph Heavner Honors Complex Analysis Continued) Chapter July, 05 3.) Find the derivative of f ) e i e i. d d e i e i) d d ei ) d d e i ) e i d d i) e i d d i)
More informationMath 312 Fall 2013 Final Exam Solutions (2 + i)(i + 1) = (i 1)(i + 1) = 2i i2 + i. i 2 1
. (a) We have 2 + i i Math 32 Fall 203 Final Exam Solutions (2 + i)(i + ) (i )(i + ) 2i + 2 + i2 + i i 2 3i + 2 2 3 2 i.. (b) Note that + i 2e iπ/4 so that Arg( + i) π/4. This implies 2 log 2 + π 4 i..
More informationEcuatii si inecuatii de gradul al doilea si reductibile la gradul al doilea. Ecuatii de gradul al doilea
Ecuatii si inecuatii de gradul al doilea si reductibile la gradul al doilea Ecuatia de forma Ecuatii de gradul al doilea a + b + c = 0, (1) unde a, b, c R, a 0, - variabila, se numeste ecuatie de gradul
More informationSecond Midterm Exam Name: Practice Problems March 10, 2015
Math 160 1. Treibergs Second Midterm Exam Name: Practice Problems March 10, 015 1. Determine the singular points of the function and state why the function is analytic everywhere else: z 1 fz) = z + 1)z
More informationDifferential Equations: Homework 2
Differential Equations: Homework Alvin Lin January 08 - May 08 Section.3 Exercise The direction field for provided x 0. dx = 4x y is shown. Verify that the straight lines y = ±x are solution curves, y
More informationDivizibilitate în mulțimea numerelor naturale/întregi
Divizibilitate în mulțimea numerelor naturale/întregi Teorema îmărţirii cu rest în mulțimea numerelor naturale Fie a, b, b 0. Atunci există q, r astfel încât a=bq+r, cu 0 r < b. În lus, q şi r sunt unic
More informationMTH3101 Spring 2017 HW Assignment 4: Sec. 26: #6,7; Sec. 33: #5,7; Sec. 38: #8; Sec. 40: #2 The due date for this assignment is 2/23/17.
MTH0 Spring 07 HW Assignment : Sec. 6: #6,7; Sec. : #5,7; Sec. 8: #8; Sec. 0: # The due date for this assignment is //7. Sec. 6: #6. Use results in Sec. to verify that the function g z = ln r + iθ r >
More informationRezolvarea ecuaţiilor şi sistemelor de ecuaţii diferenţiale ordinare (II)
Rezolvarea ecuaţiilor şi sistemelor de ecuaţii diferenţiale ordinare (II) Metode multipas Prof.dr.ing. Universitatea "Politehnica" Bucureşti, Facultatea de Inginerie Electrică Suport didactic pentru disciplina
More informationChapter 3 Differentiation Rules (continued)
Chapter 3 Differentiation Rules (continued) Sec 3.5: Implicit Differentiation (continued) Implicit Differentiation What if you want to find the slope of the tangent line to a curve that is not the graph
More informationGradul de comutativitate al grupurilor finite 1
Gradul de comutativitate al grupurilor finite Marius TĂRNĂUCEANU Abstract The commutativity degree of a group is one of the most important probabilistic aspects of finite group theory In this survey we
More informationPROPRIETĂŢI GEOMETRICE ŞI ANALITICE ALE UNOR CLASE DE FUNCŢII UNIVALENTE
UNIVERSITATEA BABEŞ-BOLYAI CLUJ-NAPOCA FACULTATEA DE MATEMATICĂ ŞI INFORMATICĂ GABRIELA ROXANA ŞENDRUŢIU PROPRIETĂŢI GEOMETRICE ŞI ANALITICE ALE UNOR CLASE DE FUNCŢII UNIVALENTE Rezumatul tezei de doctorat
More informationPart D. Complex Analysis
Part D. Comple Analsis Chapter 3. Comple Numbers and Functions. Man engineering problems ma be treated and solved b using comple numbers and comple functions. First, comple numbers and the comple plane
More information18.04 Practice problems exam 1, Spring 2018 Solutions
8.4 Practice problems exam, Spring 8 Solutions Problem. omplex arithmetic (a) Find the real and imaginary part of z + z. (b) Solve z 4 i =. (c) Find all possible values of i. (d) Express cos(4x) in terms
More informationNorth MaharashtraUniversity ; Jalgaon.
North MaharashtraUniversity ; Jalgaon. Question Bank S.Y.B.Sc. Mathematics (Sem II) MTH. Functions of a omplex Variable. Authors ; Prof. M.D.Suryawanshi (o-ordinator) Head, Department of Mathematics, S.S.V.P.S.
More informationMA 242 Review Exponential and Log Functions Notes for today s class can be found at
MA 242 Review Exponential and Log Functions Notes for today s class can be found at www.xecu.net/jacobs/index242.htm Example: If y = x n If y = x 2 then then dy dx = nxn 1 dy dx = 2x1 = 2x Power Function
More informationSisteme cu logica fuzzy
Sisteme cu logica fuzzy 1/15 Sisteme cu logica fuzzy Mamdani Fie un sistem cu logică fuzzy Mamdani două intrări x şi y ieşire z x y SLF Structura z 2/15 Sisteme cu logica fuzzy Mamdani Baza de reguli R
More informationMath 185 Homework Exercises II
Math 185 Homework Exercises II Instructor: Andrés E. Caicedo Due: July 10, 2002 1. Verify that if f H(Ω) C 2 (Ω) is never zero, then ln f is harmonic in Ω. 2. Let f = u+iv H(Ω) C 2 (Ω). Let p 2 be an integer.
More informationMath Spring 2014 Solutions to Assignment # 8 Completion Date: Friday May 30, 2014
Math 3 - Spring 4 Solutions to Assignment # 8 ompletion Date: Friday May 3, 4 Question. [p 49, #] By finding an antiderivative, evaluate each of these integrals, where the path is any contour between the
More informationPLANIFICAREA TEMELOR LA GRUPELE DE EXCELENȚĂ DISCIPLINA MATEMATICĂ AN ȘCOLAR
PLANIFICAREA TEMELOR LA GRUPELE DE EXCELENȚĂ DISCIPLINA MATEMATICĂ AN ȘCOLAR 0-0 Grupa V. Matematică Profesor coordonator: Aldescu Alina.0.0 Operatii in N-Teorema impartirii cu rest 0..0 Patrate perfecte,cuburi
More informationMTH 3102 Complex Variables Solutions: Practice Exam 2 Mar. 26, 2017
Name Last name, First name): MTH 31 omplex Variables Solutions: Practice Exam Mar. 6, 17 Exam Instructions: You have 1 hour & 1 minutes to complete the exam. There are a total of 7 problems. You must show
More informationLecture 4. Properties of Logarithmic Function (Contd ) y Log z tan constant x. It follows that
Lecture 4 Properties of Logarithmic Function (Contd ) Since, Logln iarg u Re Log ln( ) v Im Log tan constant It follows that u v, u v This shows that Re Logand Im Log are (i) continuous in C { :Re 0,Im
More informationMATH 417 Homework 4 Instructor: D. Cabrera Due July 7. z c = e c log z (1 i) i = e i log(1 i) i log(1 i) = 4 + 2kπ + i ln ) cosz = eiz + e iz
MATH 47 Homework 4 Instructor: D. abrera Due July 7. Find all values of each expression below. a) i) i b) cos i) c) sin ) Solution: a) Here we use the formula z c = e c log z i) i = e i log i) The modulus
More informationMTH 3102 Complex Variables Final Exam May 1, :30pm-5:30pm, Skurla Hall, Room 106
Name (Last name, First name): MTH 02 omplex Variables Final Exam May, 207 :0pm-5:0pm, Skurla Hall, Room 06 Exam Instructions: You have hour & 50 minutes to complete the exam. There are a total of problems.
More informationChapter 9: Complex Numbers
Chapter 9: Comple Numbers 9.1 Imaginary Number 9. Comple Number - definition - argand diagram - equality of comple number 9.3 Algebraic operations on comple number - addition and subtraction - multiplication
More information1.3. OPERAŢII CU NUMERE NEZECIMALE
1.3. OPERAŢII CU NUMERE NEZECIMALE 1.3.1 OPERAŢII CU NUMERE BINARE A. ADUNAREA NUMERELOR BINARE Reguli de bază: 0 + 0 = 0 transport 0 0 + 1 = 1 transport 0 1 + 0 = 1 transport 0 1 + 1 = 0 transport 1 Pentru
More informationMath Spring 2014 Solutions to Assignment # 6 Completion Date: Friday May 23, 2014
Math 11 - Spring 014 Solutions to Assignment # 6 Completion Date: Friday May, 014 Question 1. [p 109, #9] With the aid of expressions 15) 16) in Sec. 4 for sin z cos z, namely, sin z = sin x + sinh y cos
More informationMA 201 Complex Analysis Lecture 6: Elementary functions
MA 201 Complex Analysis : The Exponential Function Recall: Euler s Formula: For y R, e iy = cos y + i sin y and for any x, y R, e x+y = e x e y. Definition: If z = x + iy, then e z or exp(z) is defined
More informationSyllabus: for Complex variables
EE-2020, Spring 2009 p. 1/42 Syllabus: for omplex variables 1. Midterm, (4/27). 2. Introduction to Numerical PDE (4/30): [Ref.num]. 3. omplex variables: [Textbook]h.13-h.18. omplex numbers and functions,
More information1 Discussion on multi-valued functions
Week 3 notes, Math 7651 1 Discussion on multi-valued functions Log function : Note that if z is written in its polar representation: z = r e iθ, where r = z and θ = arg z, then log z log r + i θ + 2inπ
More informationFINAL - PART 1 MATH 150 SPRING 2017 KUNIYUKI PART 1: 135 POINTS, PART 2: 115 POINTS, TOTAL: 250 POINTS No notes, books, or calculators allowed.
Math 150 Name: FINAL - PART 1 MATH 150 SPRING 2017 KUNIYUKI PART 1: 135 POINTS, PART 2: 115 POINTS, TOTAL: 250 POINTS No notes, books, or calculators allowed. 135 points: 45 problems, 3 pts. each. You
More informationSummary: Primer on Integral Calculus:
Physics 2460 Electricity and Magnetism I, Fall 2006, Primer on Integration: Part I 1 Summary: Primer on Integral Calculus: Part I 1. Integrating over a single variable: Area under a curve Properties of
More informationEE2007: Engineering Mathematics II Complex Analysis
EE2007: Engineering Mathematics II omplex Analysis Ling KV School of EEE, NTU ekvling@ntu.edu.sg V4.2: Ling KV, August 6, 2006 V4.1: Ling KV, Jul 2005 EE2007 V4.0: Ling KV, Jan 2005, EE2007 V3.1: Ling
More informationMth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.
For Test # study these problems, the examples in your notes, and the homework. Derivative Rules D [u n ] = nu n 1 du D [ln u] = du u D [log b u] = du u ln b D [e u ] = e u du D [a u ] = a u ln a du D [sin
More information90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.
90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y
More informationPhysics 307. Mathematical Physics. Luis Anchordoqui. Wednesday, August 31, 16
Physics 307 Mathematical Physics Luis Anchordoqui 1 Bibliography L. A. Anchordoqui and T. C. Paul, ``Mathematical Models of Physics Problems (Nova Publishers, 2013) G. F. D. Duff and D. Naylor, ``Differential
More informationTWO BOUNDARY ELEMENT APPROACHES FOR THE COMPRESSIBLE FLUID FLOW AROUND A NON-LIFTING BODY
U.P.B. Sci. Bull., Series A, Vol. 7, Iss., 9 ISSN 3-77 TWO BOUNDARY ELEMENT APPROACHES FOR THE COMPRESSIBLE FLUID FLOW AROUND A NON-LIFTING BODY Luminiţa GRECU, Gabriela DEMIAN, Mihai DEMIAN 3 În lucrare
More information2016 FAMAT Convention Mu Integration 1 = 80 0 = 80. dx 1 + x 2 = arctan x] k2
6 FAMAT Convention Mu Integration. A. 3 3 7 6 6 3 ] 3 6 6 3. B. For quadratic functions, Simpson s Rule is eact. Thus, 3. D.. B. lim 5 3 + ) 3 + ] 5 8 8 cot θ) dθ csc θ ) dθ cot θ θ + C n k n + k n lim
More informationDespre AGC cuasigrupuri V. Izbaș
Despre AGC cuasigrupuri V Izbaș 1 Introducere Se ştie că grupurile au apărut în matematică ca grupuri de automorfisme Rolul automorfismelor este remarcabil şi bine cunoscut La studierea diverselor structuri
More informationHyperbolics. Scott Morgan. Further Mathematics Support Programme - WJEC A-Level Further Mathematics 31st March scott3142.
Hyperbolics Scott Morgan Further Mathematics Support Programme - WJEC A-Level Further Mathematics 3st March 208 scott342.com @Scott342 Topics Hyperbolic Identities Calculus with Hyperbolics - Differentiation
More informationMTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.
MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:
More informationFINDING THE TRACES OF A GIVEN PLANE: ANALYTICALLY AND THROUGH GRAPHICAL CONSTRUCTIONS
BULETINUL INSTITUTULUI POLITEHNI DIN IŞI Publicat de Universitatea Tehnică Gheorghe sachi din Iaşi Tomul LVII (LXI), Fasc. 3, 20 Secţia ONSTRUŢII DE MŞINI FINDING THE TRES OF GIVEN PLNE: NLYTILLY ND THROUGH
More informationMugur Acu OPERATORUL INTEGRAL LIBERA-PASCU ŞI PROPRIETĂŢILE ACESTUIA CU PRIVIRE LA FUNCŢIILE UNIFORM STELATE, CONVEXE, APROAPE CONVEXE ŞI
Mugur Acu OPERATORUL INTEGRAL LIBERA-PASCU ŞI PROPRIETĂŢILE ACESTUIA CU PRIVIRE LA FUNCŢIILE UNIFORM STELATE, CONVEXE, APROAPE CONVEXE ŞI α-uniform CONVEXE Editura Universităţii Lucian Blaga din Sibiu
More informationCHAPTER 4. Elementary Functions. Dr. Pulak Sahoo
CHAPTER 4 Elementary Functions BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India E-mail : sahoopulak1@gmail.com 1 Module-4: Multivalued Functions-II
More informationATTENUATION OF THE ACOUSTIC SCREENS IN CLOSED SPACES
U.P.B. Sci. Bull., Series D, Vol. 69, No. 3, 007 ISSN 15-358 ATTENUATION OF THE ACOUSTIC SCREENS IN CLOSED SPACES Ioan MAGHEŢI 1, Mariana SAVU Lucrarea prezintă calculul atenuării acustice a unui ecran
More informationSolutions to Selected Exercises. Complex Analysis with Applications by N. Asmar and L. Grafakos
Solutions to Selected Exercises in Complex Analysis with Applications by N. Asmar and L. Grafakos Section. Complex Numbers Solutions to Exercises.. We have i + i. So a and b. 5. We have So a 3 and b 4.
More informationIntegration in the Complex Plane (Zill & Wright Chapter 18)
Integration in the omplex Plane Zill & Wright hapter 18) 116-4-: omplex Variables Fall 11 ontents 1 ontour Integrals 1.1 Definition and Properties............................. 1. Evaluation.....................................
More informationCompletion Date: Monday February 11, 2008
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #4 Completion Date: Monday February, 8 Department of Mathematical and Statistical Sciences University of Alberta Question. [Sec..9,
More informationDefinite integrals. We shall study line integrals of f (z). In order to do this we shall need some preliminary definitions.
5. OMPLEX INTEGRATION (A) Definite integrals Integrals are extremely important in the study of functions of a complex variable. The theory is elegant, and the proofs generally simple. The theory is put
More informationSubiecte geometrie licenta matematica-informatica 4 ani
Class: Date: Subiecte geometrie licenta matematica-informatica 4 ani Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. Complementara unui subspatiu
More informationPOLAR CHARACTERISTIC OF ENERGETIC INTENSITY EMITTED BY AN ANISOTROPIC THERMAL SOURCE IRREGULARLY SHAPED
Annals of the Academ of Romanian Scientists Series on Science and Technolog of nformation SSN 066-856 Volume 1, Number 1/008 43 POLAR CHARACTERSTC OF ENERGETC NTENSTY EMTTED BY AN ANSOTROPC THERMAL SOURCE
More informationLecture Notes Complex Analysis. Complex Variables and Applications 7th Edition Brown and Churchhill
Lecture Notes omplex Analysis based on omplex Variables and Applications 7th Edition Brown and hurchhill Yvette Fajardo-Lim, Ph.D. Department of Mathematics De La Salle University - Manila 2 ontents THE
More informationARTICOLE ŞI NOTE MATEMATICE
S. Rădulescu, M. Drăgan, I. V. Maftei, On W. J. Blundon s inequality 3 ARTICOLE ŞI NOTE MATEMATICE SOME CONSEQUENCES OF W.J.BLUNDON S INEQUALITY Sorin Rădulescu 1), Marius Drăgan 2), I.V.Maftei 3) Abstract.
More informationO V E R V I E W. This study suggests grouping of numbers that do not divide the number
MSCN(2010) : 11A99 Author : Barar Stelian Liviu Adress : Israel e-mail : stelibarar@yahoo.com O V E R V I E W This study suggests grouping of numbers that do not divide the number 3 and/or 5 in eight collumns.
More informationProcedeu de demonstrare a unor inegalităţi bazat pe inegalitatea lui Schur
Procedeu de demonstrare a unor inegalităţi bazat pe inegalitatea lui Schur Andi Gabriel BROJBEANU Abstract. A method for establishing certain inequalities is proposed and applied. It is based upon inequalities
More informationUNIVERSITY OF SOUTHAMPTON
UNIVERSITY OF SOUTHAMPTON MATH03W SEMESTER EXAMINATION 0/ MATHEMATICS FOR ELECTRONIC & ELECTRICAL ENGINEERING Duration: 0 min This paper has two parts, part A and part B. Answer all questions from part
More informationComplex Variables & Integral Transforms
Complex Variables & Integral Transforms Notes taken by J.Pearson, from a S4 course at the U.Manchester. Lecture delivered by Dr.W.Parnell July 9, 007 Contents 1 Complex Variables 3 1.1 General Relations
More informationA GENERALIZATION OF A CLASSICAL MONTE CARLO ALGORITHM TO ESTIMATE π
U.P.B. Sci. Bull., Series A, Vol. 68, No., 6 A GENERALIZATION OF A CLASSICAL MONTE CARLO ALGORITHM TO ESTIMATE π S.C. ŞTEFĂNESCU Algoritmul Monte Carlo clasic A1 estimeazează valoarea numărului π bazându-se
More informationMath 122 Test 3. April 17, 2018
SI: Math Test 3 April 7, 08 EF: 3 4 5 6 7 8 9 0 Total Name Directions:. No books, notes or April showers. You may use a calculator to do routine arithmetic computations. You may not use your calculator
More informationComplex Derivative and Integral
hapter 19 omplex Derivative and Integral So far we have concerned ourselves with the algebra of the complex numbers. The subject of complex analysis is extremely rich and important. The scope and the level
More informationSOME INVARIANTS CONNECTED WITH EULER-LAGRANGE EQUATIONS
U.P.B. Sci. Bull., Series A, Vol. 71, Iss.2, 29 ISSN: 1223-727 SOME INVARIANTS CONNECTED WITH EULER-LAGRANGE EQUATIONS Irena ČOMIĆ Lucrarea descrie mai multe tipuri de omogenitate definite în spaţiul Osc
More informationInverse Trig Functions
6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)
More informationCOMPARATIVE DISCUSSION ABOUT THE DETERMINING METHODS OF THE STRESSES IN PLANE SLABS
74 COMPARATIVE DISCUSSION ABOUT THE DETERMINING METHODS OF THE STRESSES IN PLANE SLABS Codrin PRECUPANU 3, Dan PRECUPANU,, Ștefan OPREA Correspondent Member of Technical Sciences Academy Gh. Asachi Technical
More informationChapter 3 Elementary Functions
Chapter 3 Elementary Functions In this chapter, we will consier elementary functions of a complex variable. We will introuce complex exponential, trigonometric, hyperbolic, an logarithmic functions. 23.
More informationBeautiful Images from Some Simple Formulas
Department of Mathematics and Computer Science Mount St. Mary s University Ordinary plot of f(x, y) = x 2 + y 2 Contour plot of f(x, y) = x 2 + y 2 Looking down from above Colored contour map Temperatures
More informationTest one Review Cal 2
Name: Class: Date: ID: A Test one Review Cal 2 Short Answer. Write the following expression as a logarithm of a single quantity. lnx 2ln x 2 ˆ 6 2. Write the following expression as a logarithm of a single
More information1 z n = 1. 9.(Problem) Evaluate each of the following, that is, express each in standard Cartesian form x + iy. (2 i) 3. ( 1 + i. 2 i.
. 5(b). (Problem) Show that z n = z n and z n = z n for n =,,... (b) Use polar form, i.e. let z = re iθ, then z n = r n = z n. Note e iθ = cos θ + i sin θ =. 9.(Problem) Evaluate each of the following,
More informationMath 122 Test 3. April 15, 2014
SI: Math 1 Test 3 April 15, 014 EF: 1 3 4 5 6 7 8 Total Name Directions: 1. No books, notes or 6 year olds with ear infections. You may use a calculator to do routine arithmetic computations. You may not
More informationLinear DifferentiaL Equation
Linear DifferentiaL Equation Massoud Malek The set F of all complex-valued functions is known to be a vector space of infinite dimension. Solutions to any linear differential equations, form a subspace
More informationChapter 9. Analytic Continuation. 9.1 Analytic Continuation. For every complex problem, there is a solution that is simple, neat, and wrong.
Chapter 9 Analytic Continuation For every complex problem, there is a solution that is simple, neat, and wrong. - H. L. Mencken 9.1 Analytic Continuation Suppose there is a function, f 1 (z) that is analytic
More informationNATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part I
NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Comple Analsis II Lecture Notes Part I Chapter 1 Preliminar results/review of Comple Analsis I These are more detailed notes for the results
More informationIn this note we will evaluate the limits of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0. f(x)
L Hôpital s Rule In this note we will evaluate the its of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0 f() Suppose a f() = 0 and a g() = 0. Then a g() the indeterminate
More information8.a: Integrating Factors in Differential Equations. y = 5y + t (2)
8.a: Integrating Factors in Differential Equations 0.0.1 Basics of Integrating Factors Until now we have dealt with separable differential equations. Net we will focus on a more specific type of differential
More informationComplex Integration Line Integral in the Complex Plane CHAPTER 14
HAPTER 14 omplex Integration hapter 13 laid the groundwork for the study of complex analysis, covered complex numbers in the complex plane, limits, and differentiation, and introduced the most important
More informationCristalul cu N atomi = un sistem de N oscilatori de amplitudini mici;
Curs 8 Caldura specifica a retelei Cristalul cu N atomi = un sistem de N oscilatori de amplitudini mici; pentru tratarea cuantica, se inlocuieste tratamentul clasic al oscilatorilor cuplati, cu cel cuantic
More informationLogică avansată pentru informatică Master Anul I, Semestrul I
Logică avansată pentru informatică Master Anul I, Semestrul I 2017-2018 Laurenţiu Leuştean Pagina web: http:unibuc.ro/~lleustean/ În prezentarea acestui curs sunt folosite parţial slideurile Ioanei Leuştean
More information= 2πi Res. z=0 z (1 z) z 5. z=0. = 2πi 4 5z
MTH30 Spring 07 HW Assignment 7: From [B4]: hap. 6: Sec. 77, #3, 7; Sec. 79, #, (a); Sec. 8, #, 3, 5, Sec. 83, #5,,. The due date for this assignment is 04/5/7. Sec. 77, #3. In the example in Sec. 76,
More informationIntroduction to Differential Equations
Chapter 1 Introduction to Differential Equations 1.1 Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
More information3 Elementary Functions
3 Elementary Functions 3.1 The Exponential Function For z = x + iy we have where Euler s formula gives The note: e z = e x e iy iy = cos y + i sin y When y = 0 we have e x the usual exponential. When z
More informationMath 132 Exam 3 Fall 2016
Math 3 Exam 3 Fall 06 multiple choice questions worth points each. hand graded questions worth and 3 points each. Exam covers sections.-.6: Sequences, Series, Integral, Comparison, Alternating, Absolute
More information6.6 Inverse Trigonometric Functions
6.6 6.6 Inverse Trigonometric Functions We recall the following definitions from trigonometry. If we restrict the sine function, say fx) sinx, π x π then we obtain a one-to-one function. π/, /) π/ π/ Since
More informationJuly 21 Math 2254 sec 001 Summer 2015
July 21 Math 2254 sec 001 Summer 2015 Section 8.8: Power Series Theorem: Let a n (x c) n have positive radius of convergence R, and let the function f be defined by this power series f (x) = a n (x c)
More informationPhysics 2400 Midterm I Sample March 2017
Physics 4 Midterm I Sample March 17 Question: 1 3 4 5 Total Points: 1 1 1 1 6 Gamma function. Leibniz s rule. 1. (1 points) Find positive x that minimizes the value of the following integral I(x) = x+1
More informationMATH115. Infinite Series. Paolo Lorenzo Bautista. July 17, De La Salle University. PLBautista (DLSU) MATH115 July 17, / 43
MATH115 Infinite Series Paolo Lorenzo Bautista De La Salle University July 17, 2014 PLBautista (DLSU) MATH115 July 17, 2014 1 / 43 Infinite Series Definition If {u n } is a sequence and s n = u 1 + u 2
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK. Summer Examination 2009.
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK Summer Examination 2009 First Engineering MA008 Calculus and Linear Algebra
More informationU.P.B. Sci. Bull., Series A, Vol. 74, Iss. 3, 2012 ISSN SCALAR OPERATORS. Mariana ZAMFIR 1, Ioan BACALU 2
U.P.B. ci. Bull., eries A, Vol. 74, Iss. 3, 212 IN 1223-727 A CALAR OPERATOR Mariana ZAMFIR 1, Ioan BACALU 2 În această lucrare studiem o clasă nouă de operatori numiţi -scalari. Aceştia apar în mod natural,
More information7.1. Calculus of inverse functions. Text Section 7.1 Exercise:
Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential
More informationToday: 5.6 Hyperbolic functions
Toay: 5.6 Hyerbolic functions Warm u: Let f() = (e ) an g() = (e + ) Verify the following ientities: () f 0 () =g() () g 0 () =f() (3) f() is an o function (i.e. f(-) = -f()) (4) g() is an even function
More informationMath Test #3 Info and Review Exercises
Math 181 - Test #3 Info and Review Exercises Fall 2018, Prof. Beydler Test Info Date: Wednesday, November 28, 2018 Will cover sections 10.1-10.4, 11.1-11.7. You ll have the entire class to finish the test.
More informationPROBLEME DIVERSE lecţie susţinută la lotul de 13 de Andrei ECKSTEIN Bucureşti, 25 mai 2015
PROBLEME DIVERSE lecţie susţinută la lotul de 13 de Andrei ECKSTEIN Bucureşti, 5 mai 015 I. SUBSTITUŢIA TAIWANEZĂ 1. Fie a, b, c > 0 astfel încât a bc, b ca şi c ab. Determinaţi valoarea maximă a expresiei
More informationcos t 2 sin 2t (vi) y = cosh t sinh t (vii) y sin x 2 = x sin y 2 (viii) xy = cot(xy) (ix) 1 + x = sin(xy 2 ) (v) g(t) =
MATH1003 REVISION 1. Differentiate the following functions, simplifying your answers when appropriate: (i) f(x) = (x 3 2) tan x (ii) y = (3x 5 1) 6 (iii) y 2 = x 2 3 (iv) y = ln(ln(7 + x)) e 5x3 (v) g(t)
More informationArc Length and Surface Area in Parametric Equations
Arc Length and Surface Area in Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2011 Background We have developed definite integral formulas for arc length
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION 4 (Products and quotients) & (Logarithmic differentiation) A.J.Hobson
JUST THE MATHS UNIT NUMBER 104 DIFFERENTIATION 4 (Products and quotients) & (Logarithmic differentiation) by AJHobson 1041 Products 1042 Quotients 1043 Logarithmic differentiation 1044 Exercises 1045 Answers
More information