Fibonacci. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
|
|
- Κύρα Φωτινή Δοξαράς
- 6 χρόνια πριν
- Προβολές:
Transcript
1 Fiboacci Notatios Traditioal ame Fiboacci umber Traditioal otatio F Ν Mathematica StadardForm otatio FiboacciΝ Primary defiitio F Ν ΦΝ cosν Π Φ Ν Specific values Specialized values F ; F ; F Φ Φ ; F Φ Φ ; F F ;
2 F Φ Φ ; Φ Φ Values at fied poits F 0 0 F F F 3 F 4 3 F F F F F F 0 Values at ifiities F Geeral characteristics Domai ad aalyticity F Ν is a etire aalytical fuctio of Ν which is defied over the whole comple Ν-plae ΝF Ν Symmetries ad periodicities Parity
3 F F ; Mirror symmetry F Ν F Ν Periodicity No periodicity Poles ad essetial sigularities The fuctio F Ν has oly oe sigular poit at Ν. It is a essetial sigular poit ig Ν F Ν, Brach poits The fuctio F Ν does ot have brach poits Ν F Ν Brach cuts The fuctio F Ν does ot have brach cuts Ν F Ν Series represetatios Geeralized power series Epasios at geeric poit Ν Ν 0 For the fuctio itself F Ν F Ν0 Ν 0 csch Ν 0 4 Ν 0 csch cosπ Ν 0 Π siπ Ν 0 Ν Ν 0 0 Π Π cosπ Ν 0 csch siπ Ν 0 Ν 0 csch F Ν0 Ν Ν 0 ; Ν Ν 0
4 F Ν F Ν0 Ν 0 csch Ν 0 4 Ν 0 csch cosπ Ν 0 Π siπ Ν 0 Ν Ν 0 0 Π Π cosπ Ν 0 csch siπ Ν 0 Ν 0 csch F Ν0 Ν Ν 0 OΝ Ν F Ν F Ν 0 csch Ν 0 Ν 0 Π Ν 0 csch Π csch Π Ν 0 csch Π csch Ν Ν F Ν F Ν0 OΝ Ν 0 Epasios at Ν F Ν logφ Ν Π Ν log 3 Φ 3 Π logφ Ν3 ; Ν F Ν logφ Ν Π Ν log 3 Φ 3 Π logφ Ν3 OΝ F Ν csch Π csch Π csch Ν F Ν logφ Ν OΝ ; Ν 0 Asymptotic series epasios F Ν ΦΝ cosν Π Φ Ν ; Ν Φ Ν ImΝ 0 ReΝ Π ImΝ 0 F Ν Ν ΠΝ csch csch Ν Π Ν ImΝ 0 Π ImΝ ReΝ 0 ImΝ 0 ReΝ Π ImΝ 0 ; Ν Φ Ν cosν Π Φ Ν True
5 F Ν ΦΝ ; Ν Other series represetatios F F F ; ; ; F ; F ; F Π 4 ; F ep ta ; F F ; ; F ; Itegral represetatios
6 6 O the real ais Of the direct fuctio F Π 3 cost sitt ; Limit represetatios F lim ma loglogd Μ m loglogμ Μ,,m ; d m d Σ 0 m d 0 m Σ 0 m Geeratig fuctios t F t ; t t Differetial equatios Ordiary liear differetial equatios ad wrosias w 3 Ν logφ w Ν Π log Φ w Ν logφ log Φ Π wν 0 ; wν c F Ν c L Ν c 3 Φ Ν siπ Ν Trasformatios Additio formulas F m F F m F m F ; m F m F L m F m L ; m F m F m ; m F m F m F F F m ; m F m F m L F L m ; m
7 F m F m ; m F m F m ; m F m3 F m ; m F Ν F Ν L Ν F F F ; F F F ; 0 Cassii's formula Multiple argumets F Ν L Ν F Ν si Π Ν Φ Ν F Ν F Ν F Ν F Ν F Ν Φ Ν si Π Ν F Ν F Ν F Ν F Ν F Ν Φ Ν si Π Ν F F p F p F p F p ; p F L F ; F F ; F Ν 3 F Ν F Ν F m Ν L m F m Ν m F m Ν ; m
8 m m F m F F m F ; m m F m m m m F L m ; m m m F m F F m L m m L m ; m F m m ; m m m F m m m F m m ; m Products, sums, ad powers of the direct fuctio Products of the direct fuctio F Ν F Ν F Ν cosν Π F F m L m L m ; m Powers of the direct fuctio F Ν F Ν F Ν cosν Π F L ; F 3 3 F F 3 ; F 4 4 F F 4 8 F F 4 6 ; F 4 F F F F ;
9 F m m m m m F m m F m ; m Related trasformatios F Ν L Ν L Ν Idetities Recurrece idetities Cosecutive eighbors F Ν F Ν F Ν F Ν F Ν F Ν F Ν F Ν Φ Ν Φ F Ν Φ F Ν Φ Ν Distat eighbors F Ν m U m 3 F mν m U m 3 F mν ; m F Ν m U m 3 F Νm m U m 3 F Νm ; m Fuctioal idetities Fuctioal equatios wz wz wz ; wz c F z c L z Relatios of special id F Ν F Ν F Ν cosν Π F F l F l F F F l ; l F F F F F ;
10 F F m F m m F m ; m F F F 3 F 4 F F 3 0 ; F a F b F c b c a ; F cb F ab F b F ac F bc F c F ba F cb F a a b c a b a c b c F Φ F ; F gcdm, gcdf m, F ; m F m m F F m F F F ; m ta F ta F ta F ; Comple characteristics Real part ReF y Φ Φ cosy logφ cosπ coshπ y cosy logφ siπ siy logφ sihπ y ReF y siπ siy csch sihπ y cosπ cosy csch coshπ y cosy csch Imagiary part ImF y Φ Φ siy logφ cosπ coshπ y siy logφ cosy logφ siπ sihπ y ImF y cosπ coshπ y siy csch cosy csch siπ sihπ y siy csch
11 Absolute value F y 0 Φ cosh Π y 4 Φ cosπ cos y logφ coshπ y Φ 4 sih Π y cos Π 4 Φ siπ si y logφ sihπ y F y cos Π cosh Π y 4 cosπ cos y csch coshπ y 4 siπ si y csch sihπ y Argumet argf y ta Φ Φ cosy logφ cosπ coshπ y cosy logφ siπ siy logφ sihπ y, Φ Φ siy logφ cosπ coshπ y siy logφ cosy logφ siπ sihπ y argf y ta siπ siy csch sihπ y cosπ cosy csch coshπ y cosy csch, cosπ coshπ y siy csch cosy csch siπ sihπ y siy csch Cojugate value F y Φ Φ cosy logφ siy logφ siπ sihπ ysiy logφ cosy logφ cosπ coshπ y cosy logφ siy logφ F y cosπ coshπ y cosy csch siy csch siπ siy csch cosy csch sihπ y cosy csch siy csch Sigum value
12 sgf y Φ cosπ coshπ y siy logφ cosy logφ Φ cosy logφ siy logφ siπ cosy logφ siy logφ sihπ y Φ cosh Π y 4 Φ cosπ cos y logφ coshπ y Φ 4 sih Π y cos Π 4 Φ siπ si y logφ sihπ y sgf y siπ cosy csch siy csch sihπ y cosy csch siy csch cosπ coshπ y cosy csch siy csch 3 4 cos Π cosh Π y 4 cosπ cos y csch coshπ y 4 siπ si y csch sihπ y Differetiatio Low-order differetiatio F Ν Ν ΦΝ Φ Ν logφ cosπ Ν logφ Π siπ Ν F Ν Ν Φ Ν cosπ Ν Π log Φ logφ Φ Ν logφ Π siπ Ν Symbolic differetiatio F Ν Ν Φ Ν log Φ Φ Ν Π Ν logφ Π Π Ν logφ Π ; F Ν Ν F Ν log Φ ΦΝ cosπ Ν log Φ Π cos Π Ν log Φ ; Fractioal itegro-differetiatio Α F Ν ΝΑ Ν Π csch Α ep Π csch ΝQΑ, Π csch Ν ep Π csch Ν Α Ν Ν Π csch Α QΑ, Π csch Ν Ν Α csch Α epν csch QΑ, Ν csch
13 3 Itegratio Idefiite itegratio Ivolvig oly oe direct fuctio F a Ν Ν a Φ a Ν logφ cosπ a Ν Π siπ a Ν log Φ Π logφ F Ν Ν Φ Ν logφ cosπ Ν Π siπ Ν log Φ Π logφ Ivolvig oe direct fuctio ad elemetary fuctios Ivolvig power fuctio Φ Ν Φ a Ν Ν Α F a Ν Ν Ν Α a Ν Α Α, a Ν csch csch Α a Ν Π csch Α Α, a Ν Π csch a Ν Π csch Α Α, a Ν Π csch ΝΑ Ν Α F Ν Ν Ν Α Α, Ν csch csch Α Ν Π csch Α Α, Ν Π csch Ν Π csch Α Α, Ν Π csch Itegral trasforms Laplace trasforms t F t z z csch z csch z csch Π ; Rez logφ Summatio Fiite summatio F F
14 F F ; F F 3 ; F z z z z F F ; z z F q F pq z p F pq z p F qp z F pq z ; p q p z L p z F F L F ; F F F Ifiite summatio z F z z z F ϑ 0, F F si Π F cos Π F 0 ; F F F F z F F F z z z z 4 z as a formal power series
15 F Φ F Φ Multiple sums m 0 m 0 m 0,j m j j F m j j j 0 j j ; p p j p j j F j F,p ; p F,p p F,p p F,p F, F Operatios Limit operatio F Ν lim Ν L Ν F ΑΝ lim Φ Α Ν F Ν m F Ν lim Φ ; m Ν F mν F Ν Represetatios through more geeral fuctios Through hypergeometric fuctios Ivolvig F F Ν Ν Π Ν cos F Ν, Ν ; 3 ; 4 si Π Ν F Ν, Ν ; ; F Ν ΘΝ siν Π F Ν, Ν ; Ν; 4 ΘΝ siν Π cosν Π Ν F, Ν ; Ν ; Ν F Ν F, Ν ; Ν; 4 cosπ Ν Ν F, Ν ; Ν ; 4 ; Ν
16 6 F Ν Π Ν Ν siπ Ν cosπ Ν F Ν, Ν ; 3 ; 4 siπ Ν F Ν, Ν ; ; F Ν 0 Π Ν Ν cosπ Ν siπ Ν F Ν, Ν; 3 ; 4 siπ Ν F Ν, Ν; ; 4 F Ν Π Ν siπ Ν F Ν, Ν ; ; 4 Ν siπ Ν cosπ Ν F Ν, Ν ; 3 ; F F, ; 3 ; ; F F, ; ; 4 ; F F, ; 3 ; 4 ; F F, ; ; ; F 4 F, ; ; ; F F, ; ; 4 ; F F, ; ; 4 ; F F, ; ; 4 ; F 3 F F 3 8 F 7 8, 3 8, 3 8 ; ; 9 ; ; 9 ; ;
17 F F, ; 3 ; 4 ; Ivolvig p F q F 3F,, ; 3, ; 4 ; F 4 3F,, ; 3, ; 4 ; Through Meijer G Classical cases for the direct fuctio itself siπ Ν F Ν Π , G 3,3 4 Ν, Ν, Ν 0,, Ν ; Ν F Ν Ν Π G,, 4 Ν, Ν 0, Ν cosν Π Ν Π G,, 4 Ν, Ν 0, Ν ; Ν Geeralized cases for the direct fuctio itself siπ Ν F Ν Π , G 3,3, Ν, Ν, Ν 0,, Ν ; Ν Through other fuctios Ivolvig some hypergeometric-type fuctios F Ν F Ν F U ; Represetatios through equivalet fuctios With elemetary fuctios F Ν Ν logw Π logw Ν Π logw Ν ; w
18 F Ν ΦΝ Ν ΠlogΦ Ν ΠlogΦ F Ν ΦΝ Ν ΠlogΦ Ν ΠlogΦ F Ν epν csch ep Ν csch cosπ Ν F Ν Π Ν siπ Νcos Ν si cosπ Ν siπ Ν si Ν si F Ν cosπ Ν coshν logφ cosπ Ν sihν logφ F Ν si Π Ν si Ν csc Π Ν sihν csch F Ν Π Ν cosπ Ν siπ Ν si Ν si siπ Νcos Ν si F Ν Π Ν siπ Νcos Ν si siπ Ν cosπ Ν si Ν si F si z siz ; z log Π With Lucas umbers F Ν L Ν L Ν F m L m L m L m L m L m ; m F Ν L Ν L Ν Φ Ν si Π Ν L Ν L Ν
19 9 Other idetities Idetities ivolvig determiats F if l if l 0 else l Theorems Zecedorf theorem Every positive iteger ca be decomposed i a uique way as a sum of Fiboacci umbers, such that o two of these umbers are cosecutive i the Fiboacci sequece. Fiboacci substitutio After actig o A times with the Fiboacci substitutio A A B, B A the resultig sequece cotais F As ad F Bs. A trascedetal umber F is a trascedetal umber. The umbers of primary ad secodary spirals i the positios of leaves The umbers of primary ad secodary spirals i the positios of leaves or scales alog a plat stem are early always two cosecutive Fiboacci umbers. Hirmer's cojecture The umber of the largest set of oitersectig circles arraged alog the circumferece of a give circle ad agle Π GoldeRatio betwee cosecutive midpoits is give by the Fiboacci umbers F. History J. Kepler (608) A. Girard (634); R. Simpso (73) É. Léger (837) É. Lucas (870, ) G.H. Hardy ad E.M. Wright (938)
20 0 Copyright This documet was dowloaded from fuctios.wolfram.com, a comprehesive olie compedium of formulas ivolvig the special fuctios of mathematics. For a ey to the otatios used here, see Please cite this documet by referrig to the fuctios.wolfram.com page from which it was dowloaded, for eample: To refer to a particular formula, cite fuctios.wolfram.com followed by the citatio umber. e.g.: This documet is curretly i a prelimiary form. If you have commets or suggestios, please commets@fuctios.wolfram.com , Wolfram Research, Ic.
Factorial. Notations. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values
Factorial Notatios Traditioal ame Factorial Traditioal otatio Mathematica StadardForm otatio Factorial Specific values Specialized values 06.0.0.000.0 k ; k 06.0.0.000.0 ; 06.0.0.000.0 p q q p q p k q
Διαβάστε περισσότεραHermiteHGeneral. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
HermiteHGeeral Notatios Traditioal ame Hermite fuctio Traditioal otatio H Mathematica StadardForm otatio HermiteH, Primary defiitio 07.0.0.000.0 H F ; ; F ; 3 ; Specific values Specialied values For fixed
Διαβάστε περισσότεραNotations. Primary definition. Traditional name. Traditional notation. Mathematica StandardForm notation. Generalized hypergeometric function
HyergeometricPFQ Notatios Traditioal ame Geeralied hyergeometric fuctio Traditioal otatio F a 1,, a ; b 1,, b ; Mathematica StadardForm otatio HyergeometricPFQa 1,, a, b 1,, b, Primary defiitio 07.31.0.0001.01
Διαβάστε περισσότεραZeta. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values
Zeta Notatios Traditioal ame Riema zeta fuctio Traditioal otatio Ζs Mathematica StadardForm otatio Zetas Primary defiitio... Ζs ; Res s k k Specific values Specialized values..3.. Ζ B ;..3.. Ζ ;..3.3.
Διαβάστε περισσότεραNotations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation. Mathematica StandardForm notation
KelvinKei Notations Traditional name Kelvin function of the second kind Traditional notation kei Mathematica StandardForm notation KelvinKei Primary definition 03.5.0.000.0 kei kei 0 Specific values Values
Διαβάστε περισσότεραDiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
Διαβάστε περισσότεραExpIntegralE. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
ExpIntegralE Notations Traditional name Exponential integral E Traditional notation E Mathematica StandardForm notation ExpIntegralE, Primary definition 06.34.0.000.0 E t t t ; Re 0 Specific values Specialied
Διαβάστε περισσότεραn r f ( n-r ) () x g () r () x (1.1) = Σ g() x = Σ n f < -n+ r> g () r -n + r dx r dx n + ( -n,m) dx -n n+1 1 -n -1 + ( -n,n+1)
8 Higher Derivative of the Product of Two Fuctios 8. Leibiz Rule about the Higher Order Differetiatio Theorem 8.. (Leibiz) Whe fuctios f ad g f g are times differetiable, the followig epressio holds. r
Διαβάστε περισσότεραNotations. Primary definition. Specific values. General characteristics. Series representations. Traditional name. Traditional notation
Pi Notations Traditional name Π Traditional notation Π Mathematica StandardForm notation Pi Primary definition.3... Π Specific values.3.3.. Π 3.5965358979338663383795889769399375589795937866868998683853
Διαβάστε περισσότεραOn Generating Relations of Some Triple. Hypergeometric Functions
It. Joural of Math. Aalysis, Vol. 5,, o., 5 - O Geeratig Relatios of Some Triple Hypergeometric Fuctios Fadhle B. F. Mohse ad Gamal A. Qashash Departmet of Mathematics, Faculty of Educatio Zigibar Ade
Διαβάστε περισσότεραGegenbauerC3General. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
GegenbauerC3General Notations Traditional name Gegenbauer function Traditional notation C Ν Λ z Mathematica StandardForm notation GegenbauerCΝ, Λ, z Primary definition 07.4.0.000.0 C Λ Ν z Λ Π Ν Λ F Ν,
Διαβάστε περισσότεραL.K.Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 4677 + {JEE Mai 04} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks:
Διαβάστε περισσότεραPresentation of complex number in Cartesian and polar coordinate system
1 a + bi, aεr, bεr i = 1 z = a + bi a = Re(z), b = Im(z) give z = a + bi & w = c + di, a + bi = c + di a = c & b = d The complex cojugate of z = a + bi is z = a bi The sum of complex cojugates is real:
Διαβάστε περισσότεραSUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6
SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES Readig: QM course packet Ch 5 up to 5. 1 ϕ (x) = E = π m( a) =1,,3,4,5 for xa (x) = πx si L L * = πx L si L.5 ϕ' -.5 z 1 (x) = L si
Διαβάστε περισσότεραIntroduction of Numerical Analysis #03 TAGAMI, Daisuke (IMI, Kyushu University)
Itroductio of Numerical Aalysis #03 TAGAMI, Daisuke (IMI, Kyushu Uiversity) web page of the lecture: http://www2.imi.kyushu-u.ac.jp/~tagami/lec/ Strategy of Numerical Simulatios Pheomea Error modelize
Διαβάστε περισσότεραSolve the difference equation
Solve the differece equatio Solutio: y + 3 3y + + y 0 give tat y 0 4, y 0 ad y 8. Let Z{y()} F() Taig Z-trasform o both sides i (), we get y + 3 3y + + y 0 () Z y + 3 3y + + y Z 0 Z y + 3 3Z y + + Z y
Διαβάστε περισσότεραNotations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values
PolyGamma Notations Traditional name Digamma function Traditional notation Ψz Mathematica StandardForm notation PolyGammaz Primary definition 06.4.02.000.0 Ψz k k k z Specific values Specialized values
Διαβάστε περισσότεραIIT JEE (2013) (Trigonomtery 1) Solutions
L.K. Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 677 (+) PAPER B IIT JEE (0) (Trigoomtery ) Solutios TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE
Διαβάστε περισσότερα1. For each of the following power series, find the interval of convergence and the radius of convergence:
Math 6 Practice Problems Solutios Power Series ad Taylor Series 1. For each of the followig power series, fid the iterval of covergece ad the radius of covergece: (a ( 1 x Notice that = ( 1 +1 ( x +1.
Διαβάστε περισσότεραFREE VIBRATION OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM Revision B
FREE VIBRATION OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM Revisio B By Tom Irvie Email: tomirvie@aol.com February, 005 Derivatio of the Equatio of Motio Cosier a sigle-egree-of-freeom system. m x k c where m
Διαβάστε περισσότεραHomework for 1/27 Due 2/5
Name: ID: Homework for /7 Due /5. [ 8-3] I Example D of Sectio 8.4, the pdf of the populatio distributio is + αx x f(x α) =, α, otherwise ad the method of momets estimate was foud to be ˆα = 3X (where
Διαβάστε περισσότεραBetaRegularized. Notations. Primary definition. Traditional name. Traditional notation. Mathematica StandardForm notation.
BetaRegularized Notations Traditional name Regularized incomplete beta function Traditional notation I z a, b Mathematica StandardForm notation BetaRegularizedz, a, b Primary definition Basic definition
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραBinet Type Formula For The Sequence of Tetranacci Numbers by Alternate Methods
DOI: 545/mjis764 Biet Type Formula For The Sequece of Tetraacci Numbers by Alterate Methods GAUTAMS HATHIWALA AND DEVBHADRA V SHAH CK Pithawala College of Eigeerig & Techology, Surat Departmet of Mathematics,
Διαβάστε περισσότεραCHAPTER 103 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES
CHAPTER 3 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES EXERCISE 364 Page 76. Determie the Fourier series for the fuctio defied by: f(x), x, x, x which is periodic outside of this rage of period.
Διαβάστε περισσότεραα β
6. Eerg, Mometum coefficiets for differet velocit distributios Rehbock obtaied ) For Liear Velocit Distributio α + ε Vmax { } Vmax ε β +, i which ε v V o Give: α + ε > ε ( α ) Liear velocit distributio
Διαβάστε περισσότεραBessel function for complex variable
Besse fuctio for compex variabe Kauhito Miuyama May 4, 7 Besse fuctio The Besse fuctio Z ν () is the fuctio wich satisfies + ) ( + ν Z ν () =. () Three kids of the soutios of this equatio are give by {
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραB.A. (PROGRAMME) 1 YEAR
Graduate Course B.A. (PROGRAMME) YEAR ALGEBRA AND CALCULUS (PART-A : ALGEBRA) CONTENTS Lesso Lesso Lesso Lesso Lesso Lesso : Complex Numbers : De Moivre s Theorem : Applicatios of De Moivre s Theorem 4
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραMATHEMATICS. 1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81
1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81 We know that KA = A If A is n th Order 3AB =3 3 A. B = 27 1 3 = 81 3 2. If A= 2 1 0 0 2 1 then
Διαβάστε περισσότεραΣτα επόμενα θεωρούμε ότι όλα συμβαίνουν σε ένα χώρο πιθανότητας ( Ω,,P) Modes of convergence: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ.
Στα πόμνα θωρούμ ότι όλα συμβαίνουν σ ένα χώρο πιθανότητας ( Ω,,). Modes of covergece: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ. { } ίναι οι ξής: σ μια τ.μ.. Ισχυρή σύγκλιση strog covergece { } lim = =.
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραΨηφιακή Επεξεργασία Εικόνας
ΠΑΝΕΠΙΣΤΗΜΙΟ ΙΩΑΝΝΙΝΩΝ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΪΚΑ ΜΑΘΗΜΑΤΑ Ψηφιακή Επεξεργασία Εικόνας Φιλτράρισμα στο πεδίο των συχνοτήτων Διδάσκων : Αναπληρωτής Καθηγητής Νίκου Χριστόφορος Άδειες Χρήσης Το παρόν εκπαιδευτικό
Διαβάστε περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότεραOn Inclusion Relation of Absolute Summability
It. J. Cotemp. Math. Scieces, Vol. 5, 2010, o. 53, 2641-2646 O Iclusio Relatio of Absolute Summability Aradhaa Dutt Jauhari A/66 Suresh Sharma Nagar Bareilly UP) Idia-243006 aditya jauhari@rediffmail.com
Διαβάστε περισσότεραThe Heisenberg Uncertainty Principle
Chemistry 460 Sprig 015 Dr. Jea M. Stadard March, 015 The Heiseberg Ucertaity Priciple A policema pulls Werer Heiseberg over o the Autobah for speedig. Policema: Sir, do you kow how fast you were goig?
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραp n r.01.05.10.15.20.25.30.35.40.45.50.55.60.65.70.75.80.85.90.95
r r Table 4 Biomial Probability Distributio C, r p q This table shows the probability of r successes i idepedet trials, each with probability of success p. p r.01.05.10.15.0.5.30.35.40.45.50.55.60.65.70.75.80.85.90.95
Διαβάστε περισσότεραFourier Series. constant. The ;east value of T>0 is called the period of f(x). f(x) is well defined and single valued periodic function
Fourier Series Periodic uctio A uctio is sid to hve period T i, T where T is ve costt. The ;est vlue o T> is clled the period o. Eg:- Cosider we kow tht, si si si si si... Etc > si hs the periods,,6,..
Διαβάστε περισσότεραphysicsandmathstutor.com
physicsadmathstuto.com physicsadmathstuto.com Jauay 009 blak 3. The ectagula hypebola, H, has paametic equatios x = 5t, y = 5 t, t 0. (a) Wite the catesia equatio of H i the fom xy = c. Poits A ad B o
Διαβάστε περισσότεραINTEGRATION OF THE NORMAL DISTRIBUTION CURVE
INTEGRATION OF THE NORMAL DISTRIBUTION CURVE By Tom Irvie Email: tomirvie@aol.com March 3, 999 Itroductio May processes have a ormal probability distributio. Broadbad radom vibratio is a example. The purpose
Διαβάστε περισσότεραEN40: Dynamics and Vibrations
EN40: Dyamics a Vibratios School of Egieerig Brow Uiversity Solutios to Differetial Equatios of Motio for Vibratig Systems Here, we summarize the solutios to the most importat ifferetial equatios of motio
Διαβάστε περισσότερα1. Matrix Algebra and Linear Economic Models
Matrix Algebra ad Liear Ecoomic Models Refereces Ch 3 (Turkigto); Ch 4 5 (Klei) [] Motivatio Oe market equilibrium Model Assume perfectly competitive market: Both buyers ad sellers are price-takers Demad:
Διαβάστε περισσότεραB.A. (PROGRAMME) 1 YEAR
Graduate Course B.A. (PROGRAMME) YEAR ALGEBRA AND CALCULUS (PART-A : ALGEBRA) CONTENTS Lesso Lesso Lesso Lesso Lesso Lesso : Complex Numbers : De Moivre s Theorem : Applicatios of De Moivre s Theorem 4
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραPotential Dividers. 46 minutes. 46 marks. Page 1 of 11
Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π 2, π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραLAD Estimation for Time Series Models With Finite and Infinite Variance
LAD Estimatio for Time Series Moels With Fiite a Ifiite Variace Richar A. Davis Colorao State Uiversity William Dusmuir Uiversity of New South Wales 1 LAD Estimatio for ARMA Moels fiite variace ifiite
Διαβάστε περισσότεραOutline. Detection Theory. Background. Background (Cont.)
Outlie etectio heory Chapter7. etermiistic Sigals with Ukow Parameters afiseh S. Mazloum ov. 3th Backgroud Importace of sigal iformatio Ukow amplitude Ukow arrival time Siusoidal detectio Classical liear
Διαβάστε περισσότεραCommutative Monoids in Intuitionistic Fuzzy Sets
Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,
Διαβάστε περισσότεραECE 308 SIGNALS AND SYSTEMS FALL 2017 Answers to selected problems on prior years examinations
ECE 308 SIGNALS AND SYSTEMS FALL 07 Answers to selected problems on prior years examinations Answers to problems on Midterm Examination #, Spring 009. x(t) = r(t + ) r(t ) u(t ) r(t ) + r(t 3) + u(t +
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραTrigonometric Formula Sheet
Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραPg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is
Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραA study on generalized absolute summability factors for a triangular matrix
Proceedigs of the Estoia Acadey of Scieces, 20, 60, 2, 5 20 doi: 0.376/proc.20.2.06 Available olie at www.eap.ee/proceedigs A study o geeralized absolute suability factors for a triagular atrix Ere Savaş
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραΕγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα
[ 1 ] Πανεπιστήµιο Κύπρου Εγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα Νίκος Στυλιανόπουλος, Πανεπιστήµιο Κύπρου Λευκωσία, εκέµβριος 2009 [ 2 ] Πανεπιστήµιο Κύπρου Πόσο σηµαντική είναι η απόδειξη
Διαβάστε περισσότεραTrigonometry Functions (5B) Young Won Lim 7/24/14
Trigonometry Functions (5B 7/4/14 Copyright (c 011-014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραBiorthogonal Wavelets and Filter Banks via PFFS. Multiresolution Analysis (MRA) subspaces V j, and wavelet subspaces W j. f X n f, τ n φ τ n φ.
Chapter 3. Biorthogoal Wavelets ad Filter Baks via PFFS 3.0 PFFS applied to shift-ivariat subspaces Defiitio: X is a shift-ivariat subspace if h X h( ) τ h X. Ex: Multiresolutio Aalysis (MRA) subspaces
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραΗ ΨΥΧΙΑΤΡΙΚΗ - ΨΥΧΟΛΟΓΙΚΗ ΠΡΑΓΜΑΤΟΓΝΩΜΟΣΥΝΗ ΣΤΗΝ ΠΟΙΝΙΚΗ ΔΙΚΗ
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΝΟΜΙΚΗ ΣΧΟΛΗ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΤΟΜΕΑΣ ΙΣΤΟΡΙΑΣ ΦΙΛΟΣΟΦΙΑΣ ΚΑΙ ΚΟΙΝΩΝΙΟΛΟΓΙΑΣ ΤΟΥ ΔΙΚΑΙΟΥ Διπλωματική εργασία στο μάθημα «ΚΟΙΝΩΝΙΟΛΟΓΙΑ ΤΟΥ ΔΙΚΑΙΟΥ»
Διαβάστε περισσότεραThe Neutrix Product of the Distributions r. x λ
ULLETIN u. Maaysia Math. Soc. Secod Seies 22 999 - of the MALAYSIAN MATHEMATICAL SOCIETY The Neuti Poduct of the Distibutios ad RIAN FISHER AND 2 FATMA AL-SIREHY Depatet of Matheatics ad Copute Sciece
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότερα6.003: Signals and Systems. Modulation
6.003: Signals and Systems Modulation May 6, 200 Communications Systems Signals are not always well matched to the media through which we wish to transmit them. signal audio video internet applications
Διαβάστε περισσότεραω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
Διαβάστε περισσότεραΠαραμετρικές εξισώσεις καμπύλων. ΗΥ111 Απειροστικός Λογισμός ΙΙ
ΗΥ-111 Απειροστικός Λογισμός ΙΙ Παραμετρικές εξισώσεις καμπύλων Παραδείγματα ct (): U t ( x ( t), x ( t)) 1 ct (): U t ( x ( t), x ( t), x ( t)) 3 1 3 Θέσης χρόνου ταχύτητας χρόνου Χαρακτηριστικού-χρόνου
Διαβάστε περισσότεραHomework 4.1 Solutions Math 5110/6830
Homework 4. Solutios Math 5/683. a) For p + = αp γ α)p γ α)p + γ b) Let Equilibria poits satisfy: p = p = OR = γ α)p ) γ α)p + γ = α γ α)p ) γ α)p + γ α = p ) p + = p ) = The, we have equilibria poits
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραAn Inventory of Continuous Distributions
Appendi A An Inventory of Continuous Distributions A.1 Introduction The incomplete gamma function is given by Also, define Γ(α; ) = 1 with = G(α; ) = Z 0 Z 0 Z t α 1 e t dt, α > 0, >0 t α 1 e t dt, α >
Διαβάστε περισσότεραA Hierarchy of Theta Bodies for Polynomial Systems
A Hierarchy of Theta Bodies for Polynomial Systems Rekha Thomas, U Washington, Seattle Joint work with João Gouveia (U Washington) Monique Laurent (CWI) Pablo Parrilo (MIT) The Theta Body of a Graph G
Διαβάστε περισσότεραΑλγόριθμοι και πολυπλοκότητα NP-Completeness (2)
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Αλγόριθμοι και πολυπλοκότητα NP-Completeness (2) Ιωάννης Τόλλης Τμήμα Επιστήμης Υπολογιστών NP-Completeness (2) x 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 12 22 32 11 13 21
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότερα