Annotations to Abramowitz & Stegun
|
|
- Blake Henderson
- 5 years ago
- Views:
Transcription
1 Aotatios to Abraowitz & Stegu By Lias Vepstas 4 Jauary 4 corrected Dec 4, Dec The followig is a copediu of additios ad argi otes to the Hadbook of Matheatical Fuctios by Abraowitz & Stegu Dover 97 editio, culled fro persoal aotatios I have ade to that referece over the years. I have foud these forulas useful ad hady to have aroud. May are trivial restateets of what ca already be foud i the book, ad a few are deeper, o-trivial relatioships. Most of these are ot atheatically sigificat, but are useful if oe is just searchig for a itegral or soe such: ideed, this is what it eas to be a referece. They are put dow here to be of soe utility to the Iteret couity. It would be ice if future editios/revisios of the A&S referece were possible, ad were to iclude such updates. Sources & attributio: I derived all of these. I did ot copy ay of these fro soe other book/referece, except as oted. I ve tripped over these while solvig a large variety of other copletely urelated, but quite iterestig probles. These additioal forulas are ordered accordig to the relevat chapter/paragraph of that book. Parethetical coets justify the eed for the iclusio of the forula, but are ot eat to be added to the referece. Without further ado: 3. Eleetary Aalytical Methods a x α = α + k k x k hady restateet of i a o-obvious for 3.7.-a argx + iy = arctax/y
2 Just because arcta coes i a later chapter is o excuse to oit this very useful forula 4. Eleetary Trascedetal Fuctios 4..5-a Discotiuity across the Brach Cut l x + iε l x iε = πi + Oε for real x >, ad sall, real ε. This follows obviously fro 4..5 but is hady esp. for ovice b liε l iε = πi + Oε for sall, real ε. No-ituitive stateet about the liit o the iagiary axis. 4.7 Nuerical Methods A sequece of sies ad cosies ca be coputed very rapidly two ultiplicatios, oe additio each ad accurately with the followig recursio relatios: Let s = si ad c = cos. Defie s = siθ ad c = cosθ, the s = siθ + ca be coputed quickly, alog with c = cosθ +, by usig s = cs + sc ad c = cc ss. This ethod looses less tha 3 decials of floatig poit precisio after thousad iteratios. 5. Expoetial Itegral 5..5-a E x = x! [ = x E x e ]! x + This is related to 5.. ad but is easier to work with tha either; ad is uerically ore stable a Special Values E = A hady-dady value to have aroud
3 5..5-a Add Note: See also a Asyptotic Expasio The hypercoverget ca be obtaied fro the foral Euler Su =!w + = e x/w x dx 5..5-b For values of egative, see Gaa Fuctio 6..-a The followig itegrals look siilar but are i fact very differet: t z e t dt See Riea Zeta, sectio 3. + t z e t dt See Debye Fuctio, sectio 7. For iteger z, see b Γ + ε = e x x ε dx = + ε e x lxdx = εγ For sall, real ε >. Matheatically trivial, but hady if you just wated to look up this itegral. 6..-a k k = { f or x < where Θx = f or x = k + k + Θ k } is the Heaviside step fuctio. 3
4 6.3.-a x lxdx = / + for Aother hady itegral deservig etio 6.3.-b t z lt dt = ψ ψ z + z + ad for iteger z = we have [ t lt dt = ] + + Aother hady dady itegral to have aroud a x Γ,x = Γe = x for iteger. See also 5..8! This is a special case that should be etioed explicitly a S u e x + xu u k! dx = k! = u e/u Γ +, u Occurs i certai types of stochastic equatios; uerically upleasat to evaluate See also Suatio of Ratioal Series Sectio 6.8 should really be broke out ito its ow, ad fortified with various utilitaria sus, e.g. the below. Sus occur i ay probles, ad should get a hady referece chapter, aalogous to chapter 3, o their ow.. 4
5 6.8. = + z = π si πz See = + z =! ψ z See = + z + z + = + z 6.9 Foral Sus, Spectral Asyetries Soe forally diverget sus ca be give eaigful values through regularizatio. For exaple, li t k k + e tk = 4 ad thus we write, forally, k k + = 4 with the uderstadig that regulatio took place. This is because other regulators, besides e tk ca be used: for exaple, e t k provides excellet uerical stability, while i the liit s is better suited to aalytical treatets. Geeral theories of series ks acceleratio ca be applied o forally diverget sus to get eaigful results s = = 6.9. = + s = s = + + s = s + 5
6 6.9.4 = s = s = p + s = p!s + p s... p = = = + s Follows fro above, & etc. = ss + 6. Fiite Sus I copied these sus fro soe other book; they belog here. 6.. k= k 4 = [ ]/3 6.. k= k 5 = [ + + ]/ 6..3 k= k 6 = [ ]/4 6
7 6..4 k= k 7 = [ ]/ k= k = 6..6 k= k = 4 / k= k 3 = 6..8 k= kk + = [ ]/ 6. Diverget Sus Forally diverget sus that ca be writte as liits of coverget sus. 6.. li t k e tk = 6.. li t k k + e tk = 3 4 7
8 6..3 li t k k + k + 3e tk = li t k k + k + 3k + 4e tk = li t k k + k + 3k + 4k + 5e tk = These are readily obtaied[] by cosiderig the bioial geeratig fuctio. That is, defie A x = Γk + + x k Γk + Γ + = x Γ + = x k= k x k + x + ad so the above sus are give by + A li A x = Γ + x + 7. Error Fuctio 7..4-a Itegral Represetatios er f z = z e t π e z t dt See also z 8
9 7. Repeated Itegrals i, rather, i stads for itegral. Usig i to stad for itegral was a poor choice of otatio for this etire sectio Repeated Itegrals, Recurrece Relatios Let I z = I tdt be the idefiite itegral of erf, that is, I z = e t dt π the I z = z I z + I z z! This looks like 7..5 but is the erf=-erfc versio of that relatio. The etire sectio 7. should be redoe with erf ad erfc versios of the repeated itegral. 7.4-a Defiite ad Idefiite Itegrals z [er f ct] e t dt = π [er f cz]+ + Just aother hady itegral 7.4-b a e a z π er f cbzdz = x er f caxer f cbx b er f caze b z dz x Sadly, there s o closed for for this beastie. 7.4-c z a t e a t er f cbtdt = b π a + b e a +b z ze a z er f cbz 7.4-d May of the itegrals i sectio 7.4 ca be obtaied by writig ad the doig the x itegral first. f xer f zxdx = dx f x x dze z x 9
10 . Bessel Fuctios of Fractioal Order..4-a Asyptotic Expasios For x real, x, j x = x six π/ + O x y x = cosx π/ + O x x..4-b For fixed, real x ad ex j x + By use of Sterlig s forula...-a Asyptotic Expasios f z = / z f z = + O z 3 for eve, positive or egative, ad + / z + O z 4 for odd, positive or egative...-b Thus, for k eve, k we have j k z = k/ z siz+ k/ ad, for k odd, k we have j k z = k+/ z k k + z cosz- k+/ k k + z cosz + O z 3 siz + O z 3 Although, see..4-a above for the correct treatet of the asyptotic phase agle. The phase agle is eeded for quatu scatterig probles.
11 . Itegrals of Bessel Fuctios. Siple Itegrals of Bessel Fuctios The z liit of these itegrals is o-trivial. See.4.6, a tjν tdt = z [ J ν z J ν zj ν z ] for Rν > This closed for is easier to work with tha the ifiite su give, ad also reduces the order o the RHS..3.3-b ν Jν tj ν t dt = z [ Jν z J ν+ zj ν z ] + J ν zj ν z t.3.34-a Special case of a See b J ν tj ν+ tdt = = J ν++ z Ulike.3.35, ν eed ot be iteger i this forula.3.36-a Cojecture: Itegrals of the type t J ξ tj ξ + tdt are solvable i closed for oly for + odd. Disproof of this cojecture would brig a iportat additio to this subsectio. Itegrals of the above for ca be attacked usig the recursio relatios J ν z = ν z J ν z + J ν z ad J ν+ z = ν z J ν z J ν z. A useful set of itegral recursio relatios, suitable for ueric evaluatio, are preseted below.
12 .3.36-b tjν tj ν+ tdt = ν ν zj ν z + ν + tjν tj ν tdt ν.3.36-c tjν tj ν+ tdt = z J ν z + ν + J ν tdt.3.36-d tjν tj ν+ tdt = zjν z + J ν tdt + tjν tj ν tdt.3.36-e J ν tdt = J ν zj ν z + J ν tdt Jν tj ν t dt t.3.36-f tj ν tj ν+ tdt = z J ν zj ν+ z + t Jν+ tdt J ν tdt.3.4-a Itegrals of Spherical Bessel Fuctios These occur i calculatios of wave fuctios ad are useful eough to deserve their ow sectio. The j zare spherical Bessel fuctios, of chapter..3.4-b µ + ν + t µ j ν tdt = Γ π µ for Rµ + ν > ad Rµ < ν µ + Γ.3.4-c t j t j t j + tdt = z j z follows fro.3.33
13 .3.4-d t [ j t j +t ] dt = z j z j + z.3.4-e t 3 j t j tdt = z3 4 [ + j z j + z j z ] 3. Beroulli ad Euler Polyoials - Riea Zeta Fuctio 3..3-a B = + + Σ k B k Hady for geeratig large B uerically a γ is Euler s costat, see γ i are called the Stieltjes costats. The first few are γ = ad γ =.9693 ad γ 3 =.5383 ad γ 4 = Sus of Riea Zeta Fuctios This is a ew sectio, ot i the curret A&S. Turs out these are a special case of I the below, ν ca be ay coplex value, ot ecessarily iteger ζ ν + = k + ν + k [ζ k + ν + ] 3.3. k + ν + k + [ζ k + ν + ] = See also for iteger ν. 3
14 k k + ν + k + k k + ν + k + k k + ν + k [ζ k + ν + ] = ν+ [ζ k + ν + ] = ν[ζ ν + ] ν [ζ k + ν + ] = ζ ν + ν S. p + p p= [ζ p + + ] = [ + k= k ζ k + ] For iteger Note that S = ad S = ζ ad S = ζ +ζ 3 ad i geeral S +S + = ζ +, which is to be used i Note li S = which is uerically satisfied for > T p + p p= [ζ p + + ] = + [ + ζ + k= k kζ k + For iteger. This follows fro the observatio T + T + = S whe used i ] The above trick ca be repeated to express fiite su, for ay iteger k. p= p + k p [ζ p + + ] as a For iteger >, k k + ν + k ζ k + ν + = j= j j ζ ν + j 4
15 3.3. k k + ζ k + = = k= k + = = ψ k + ζ k + = = k= k = = ψ Note this is a foral diverget su that ca be ade eaigful through regularizatio. XXX Need to do this. Refereces [] Stephe Crowley. Notes about the gkw operator. persoal couicatio,. 5
x !1! + 1!2!
4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio
More informationBinomial transform of products
Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {
More informationBernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes
Beroulli Polyoials Tals give at LSBU, October ad Noveber 5 Toy Forbes Beroulli Polyoials The Beroulli polyoials B (x) are defied by B (x), Thus B (x) B (x) ad B (x) x, B (x) x x + 6, B (x) dx,. () B 3
More informationBertrand s postulate Chapter 2
Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are
More informationSphericalHarmonicY. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
SphericalHaroicY Notatios Traditioal ae Spherical haroic Traditioal otatio Y, φ Matheatica StadardFor otatio SphericalHaroicY,,, φ Priary defiitio 05.0.0.000.0 Y, φ φ P cos ; 05.0.0.000.0 Y 0 k, φ k k
More informationAutomated Proofs for Some Stirling Number Identities
Autoated Proofs for Soe Stirlig Nuber Idetities Mauel Kauers ad Carste Scheider Research Istitute for Sybolic Coputatio Johaes Kepler Uiversity Altebergerstraße 69 A4040 Liz, Austria Subitted: Sep 1, 2007;
More informationA PROBABILITY PROBLEM
A PROBABILITY PROBLEM A big superarket chai has the followig policy: For every Euros you sped per buy, you ear oe poit (suppose, e.g., that = 3; i this case, if you sped 8.45 Euros, you get two poits,
More informationThe Gamma function. Marco Bonvini. October 9, dt e t t z 1. (1) Γ(z + 1) = z Γ(z) : (2) = e t t z. + z dt e t t z 1. = z Γ(z).
The Gamma fuctio Marco Bovii October 9, 2 Gamma fuctio The Euler Gamma fuctio is defied as Γ() It is easy to show that Γ() satisfy the recursio relatio ideed, itegratig by parts, dt e t t. () Γ( + ) Γ()
More informationBinomial Notations Traditional name Traditional notation Mathematica StandardForm notation Primary definition
Bioial Notatios Traditioal ae Bioial coefficiet Traditioal otatio Matheatica StadardFor otatio Bioial, Priary defiitio 06.03.0.0001.01 1 1 1 ; 0 For Ν, Κ egative itegers with, the bioial coefficiet Ν caot
More informationHarmonic Number Identities Via Euler s Transform
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810
More informationIntroductions to LucasL
Itroductios to LucasL Itroductio to the Fiboacci ad Lucas ubers The sequece ow ow as Fiboacci ubers (sequece 0,,,, 3,, 8, 3...) first appeared i the wor of a aciet Idia atheaticia, Pigala (40 or 00 BC).
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More informationUncertainty Principle of Mathematics
Septeber 27 Ucertaity Priciple of Matheatics Shachter Mourici Israel, Holo ourici@walla.co.il Preface This short paper prove that atheatically, Reality is ot real. This short paper is ot about Heiseberg's
More informationChapter 2. Asymptotic Notation
Asyptotic Notatio 3 Chapter Asyptotic Notatio Goal : To siplify the aalysis of ruig tie by gettig rid of details which ay be affected by specific ipleetatio ad hardware. [1] The Big Oh (O-Notatio) : It
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationGenerating Functions and Their Applications
Geeratig Fuctios ad Their Applicatios Agustius Peter Sahaggau MIT Matheatics Departet Class of 2007 18.104 Ter Paper Fall 2006 Abstract. Geeratig fuctios have useful applicatios i ay fields of study. I
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 20
ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we
More informationx+ 2 + c p () x c p () x is an arbitrary function. ( sin x ) dx p f() d d f() dx = x dx p cosx = cos x+ 2 d p () x + x-a r (1.
Super Derivative (No-iteger ties Derivative). Super Derivative ad Super Differetiatio Defitio.. p () obtaied by cotiuig aalytically the ide of the differetiatio operator of Higher Derivative of a fuctio
More informationA string of not-so-obvious statements about correlation in the data. (This refers to the mechanical calculation of correlation in the data.
STAT-UB.003 NOTES for Wedesday 0.MAY.0 We will use the file JulieApartet.tw. We ll give the regressio of Price o SqFt, show residual versus fitted plot, save residuals ad fitted. Give plot of (Resid, Price,
More informationRiemann Hypothesis Proof
Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. Riea Hypothesis Proof H. Vic Dao vic0@cocast.et March, 009 Revised Deceber, 009 Abstract
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationOrthogonal Functions
Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationREGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS
REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x]
More informationThe natural exponential function
The atural expoetial fuctio Attila Máté Brookly College of the City Uiversity of New York December, 205 Cotets The atural expoetial fuctio for real x. Beroulli s iequality.....................................2
More informationMultinomial. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
Multioial Notatios Traditioal ae Multioial coefficiet Traditioal otatio 1 2 ; 1, 2,, Matheatica StadardFor otatio Multioial 1, 2,, Priary defiitio 06.04.02.0001.01 1 2 ; 1, 2,, 06.04.02.0002.01 1 k k 1
More informationA Pair of Operator Summation Formulas and Their Applications
A Pair of Operator Suatio Forulas ad Their Applicatios Tia-Xiao He 1, Leetsch C. Hsu, ad Dogsheg Yi 3 1 Departet of Matheatics ad Coputer Sciece Illiois Wesleya Uiversity Blooigto, IL 6170-900, USA Departet
More informationMath 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version]
Math 4707 Sprig 08 Darij Griberg: idter page Math 4707 Sprig 08 Darij Griberg: idter with solutios [preliiary versio] Cotets 0.. Coutig first-eve tuples......................... 3 0.. Coutig legal paths
More informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
More informationSequences, Series, and All That
Chapter Te Sequeces, Series, ad All That. Itroductio Suppose we wat to compute a approximatio of the umber e by usig the Taylor polyomial p for f ( x) = e x at a =. This polyomial is easily see to be 3
More informationX. Perturbation Theory
X. Perturbatio Theory I perturbatio theory, oe deals with a ailtoia that is coposed Ĥ that is typically exactly solvable of two pieces: a referece part ad a perturbatio ( Ĥ ) that is assued to be sall.
More informationREVIEW OF CALCULUS Herman J. Bierens Pennsylvania State University (January 28, 2004) x 2., or x 1. x j. ' ' n i'1 x i well.,y 2
REVIEW OF CALCULUS Hera J. Bieres Pesylvaia State Uiversity (Jauary 28, 2004) 1. Suatio Let x 1,x 2,...,x e a sequece of uers. The su of these uers is usually deoted y x 1 % x 2 %...% x ' j x j, or x 1
More information... and realizing that as n goes to infinity the two integrals should be equal. This yields the Wallis result-
INFINITE PRODUTS Oe defies a ifiite product as- F F F... F x [ F ] Takig the atural logarithm of each side oe has- l[ F x] l F l F l F l F... So that the iitial ifiite product will coverge oly if the sum
More informationThe r-generalized Fibonacci Numbers and Polynomial Coefficients
It. J. Cotemp. Math. Scieces, Vol. 3, 2008, o. 24, 1157-1163 The r-geeralized Fiboacci Numbers ad Polyomial Coefficiets Matthias Schork Camillo-Sitte-Weg 25 60488 Frakfurt, Germay mschork@member.ams.org,
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationEvaluation of Some Non-trivial Integrals from Finite Products and Sums
Turkish Joural of Aalysis umber Theory 6 Vol. o. 6 7-76 Available olie at http://pubs.sciepub.com/tjat//6/5 Sciece Educatio Publishig DOI:.69/tjat--6-5 Evaluatio of Some o-trivial Itegrals from Fiite Products
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationMAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
More informationThe minimum value and the L 1 norm of the Dirichlet kernel
The miimum value ad the L orm of the Dirichlet kerel For each positive iteger, defie the fuctio D (θ + ( cos θ + cos θ + + cos θ e iθ + + e iθ + e iθ + e + e iθ + e iθ + + e iθ which we call the (th Dirichlet
More informationTHE ZETA FUNCTION AND THE RIEMANN HYPOTHESIS. Contents 1. History 1
THE ZETA FUNCTION AND THE RIEMANN HYPOTHESIS VIKTOR MOROS Abstract. The zeta fuctio has bee studied for ceturies but mathematicias are still learig about it. I this paper, I will discuss some of the zeta
More informationSeries with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers
3 47 6 3 Joural of Iteger Sequeces, Vol. 5 (0), Article..7 Series with Cetral Biomial Coefficiets, Catala Numbers, ad Harmoic Numbers Khristo N. Boyadzhiev Departmet of Mathematics ad Statistics Ohio Norther
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More information(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1
ROOT LOCUS TECHNIQUE 93 should be desiged differetly to eet differet specificatios depedig o its area of applicatio. We have observed i Sectio 6.4 of Chapter 6, how the variatio of a sigle paraeter like
More informationENGI Series Page 5-01
ENGI 344 5 Series Page 5-01 5. Series Cotets: 5.01 Sequeces; geeral term, limits, covergece 5.0 Series; summatio otatio, covergece, divergece test 5.03 Series; telescopig series, geometric series, p-series
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationA PROOF OF THE THUE-SIEGEL THEOREM ABOUT THE APPROXIMATION OF ALGEBRAIC NUMBERS FOR BINOMIAL EQUATIONS
A PROO O THE THUE-SIEGEL THEOREM ABOUT THE APPROXIMATION O ALGEBRAIC NUMBERS OR BINOMIAL EQUATIONS KURT MAHLER, TRANSLATED BY KARL LEVY I 98 Thue () showed that algebraic ubers of the special for = p a
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s) that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationA New Type of q-szász-mirakjan Operators
Filoat 3:8 07, 567 568 https://doi.org/0.98/fil7867c Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat A New Type of -Szász-Miraka Operators
More informationInformal Notes: Zeno Contours, Parametric Forms, & Integrals. John Gill March August S for a convex set S in the complex plane.
Iformal Notes: Zeo Cotours Parametric Forms & Itegrals Joh Gill March August 3 Abstract: Elemetary classroom otes o Zeo cotours streamlies pathlies ad itegrals Defiitio: Zeo cotour[] Let gk ( z = z + ηk
More informationSome results on the Apostol-Bernoulli and Apostol-Euler polynomials
Soe results o the Apostol-Beroulli ad Apostol-Euler polyoials Weipig Wag a, Cagzhi Jia a Tiaig Wag a, b a Departet of Applied Matheatics, Dalia Uiversity of Techology Dalia 116024, P. R. Chia b Departet
More informationPart I: Covers Sequence through Series Comparison Tests
Part I: Covers Sequece through Series Compariso Tests. Give a example of each of the followig: (a) A geometric sequece: (b) A alteratig sequece: (c) A sequece that is bouded, but ot coverget: (d) A sequece
More information1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.
.3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(
More informationName Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions
Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve
More informationDiscrete Mathematics: Lectures 8 and 9 Principle of Inclusion and Exclusion Instructor: Arijit Bishnu Date: August 11 and 13, 2009
Discrete Matheatics: Lectures 8 ad 9 Priciple of Iclusio ad Exclusio Istructor: Arijit Bishu Date: August ad 3, 009 As you ca observe by ow, we ca cout i various ways. Oe such ethod is the age-old priciple
More informationC. Complex Numbers. x 6x + 2 = 0. This equation was known to have three real roots, given by simple combinations of the expressions
C. Complex Numbers. Complex arithmetic. Most people thik that complex umbers arose from attempts to solve quadratic equatios, but actually it was i coectio with cubic equatios they first appeared. Everyoe
More information6.4 Binomial Coefficients
64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteeth-cetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter
More informationAMS Mathematics Subject Classification : 40A05, 40A99, 42A10. Key words and phrases : Harmonic series, Fourier series. 1.
J. Appl. Math. & Computig Vol. x 00y), No. z, pp. A RECURSION FOR ALERNAING HARMONIC SERIES ÁRPÁD BÉNYI Abstract. We preset a coveiet recursive formula for the sums of alteratig harmoic series of odd order.
More informationThe Gamma function Michael Taylor. Abstract. This material is excerpted from 18 and Appendix J of [T].
The Gamma fuctio Michael Taylor Abstract. This material is excerpted from 8 ad Appedix J of [T]. The Gamma fuctio has bee previewed i 5.7 5.8, arisig i the computatio of a atural Laplace trasform: 8. ft
More informationECE 901 Lecture 4: Estimation of Lipschitz smooth functions
ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationEE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course
Sigal-EE Postal Correspodece Course 1 SAMPLE STUDY MATERIAL Electrical Egieerig EE / EEE Postal Correspodece Course GATE, IES & PSUs Sigal System Sigal-EE Postal Correspodece Course CONTENTS 1. SIGNAL
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More information18.S34 (FALL, 2007) GREATEST INTEGER PROBLEMS. n + n + 1 = 4n + 2.
18.S34 (FALL, 007) GREATEST INTEGER PROBLEMS Note: We use the otatio x for the greatest iteger x, eve if the origial source used the older otatio [x]. 1. (48P) If is a positive iteger, prove that + + 1
More informationThe Higher Derivatives Of The Inverse Tangent Function Revisited
Alied Mathematics E-Notes, 0), 4 3 c ISSN 607-50 Available free at mirror sites of htt://www.math.thu.edu.tw/ame/ The Higher Derivatives Of The Iverse Taget Fuctio Revisited Vito Lamret y Received 0 October
More informationCOMPUTING THE EULER S CONSTANT: A HISTORICAL OVERVIEW OF ALGORITHMS AND RESULTS
COMPUTING THE EULER S CONSTANT: A HISTORICAL OVERVIEW OF ALGORITHMS AND RESULTS GONÇALO MORAIS Abstract. We preted to give a broad overview of the algorithms used to compute the Euler s costat. Four type
More informationAfter the completion of this section the student
Chapter II CALCULUS II. Liits ad Cotiuity 55 II. LIMITS AND CONTINUITY Objectives: After the copletio of this sectio the studet - should recall the defiitios of the it of fuctio; - should be able to apply
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationAcoustic Field inside a Rigid Cylinder with a Point Source
Acoustic Field iside a Rigid Cylider with a Poit Source 1 Itroductio The ai objectives of this Deo Model are to Deostrate the ability of Coustyx to odel a rigid cylider with a poit source usig Coustyx
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
More informationSome sufficient conditions of a given. series with rational terms converging to an irrational number or a transcdental number
Some sufficiet coditios of a give arxiv:0807.376v2 [math.nt] 8 Jul 2008 series with ratioal terms covergig to a irratioal umber or a trascdetal umber Yu Gao,Jiig Gao Shaghai Putuo college, Shaghai Jiaotog
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationFACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS. Dedicated to the memory of Paul Erdős. 1. Introduction. n k. f n,a =
FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS NEIL J. CALKIN Abstract. We prove divisibility properties for sus of powers of bioial coefficiets ad of -bioial coefficiets. Dedicated to the eory of
More informationWHAT ARE THE BERNOULLI NUMBERS? 1. Introduction
WHAT ARE THE BERNOULLI NUMBERS? C. D. BUENGER Abstract. For the "What is?" semiar today we will be ivestigatig the Beroulli umbers. This surprisig sequece of umbers has may applicatios icludig summig powers
More informationA talk given at Institut Camille Jordan, Université Claude Bernard Lyon-I. (Jan. 13, 2005), and University of Wisconsin at Madison (April 4, 2006).
A tal give at Istitut Caille Jorda, Uiversité Claude Berard Lyo-I (Ja. 13, 005, ad Uiversity of Wiscosi at Madiso (April 4, 006. SOME CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS Zhi-Wei Su Departet
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationProof of Fermat s Last Theorem by Algebra Identities and Linear Algebra
Proof of Fermat s Last Theorem by Algebra Idetities ad Liear Algebra Javad Babaee Ragai Youg Researchers ad Elite Club, Qaemshahr Brach, Islamic Azad Uiversity, Qaemshahr, Ira Departmet of Civil Egieerig,
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationLecture 4 Conformal Mapping and Green s Theorem. 1. Let s try to solve the following problem by separation of variables
Lecture 4 Coformal Mappig ad Gree s Theorem Today s topics. Solvig electrostatic problems cotiued. Why separatio of variables does t always work 3. Coformal mappig 4. Gree s theorem The failure of separatio
More informationA new sequence convergent to Euler Mascheroni constant
You ad Che Joural of Iequalities ad Applicatios 08) 08:7 https://doi.org/0.86/s3660-08-670-6 R E S E A R C H Ope Access A ew sequece coverget to Euler Mascheroi costat Xu You * ad Di-Rog Che * Correspodece:
More informationDirichlet s Theorem on Arithmetic Progressions
Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty
More informationTopic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist.
Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si
More informationNotes 12 Asymptotic Series
ECE 6382 Fall 207 David R. Jackso otes 2 Asymptotic Series Asymptotic Series A asymptotic series (as ) is of the form a ( ) f as = 0 or f a + a a + + ( ) 2 0 2 ote the asymptotically equal to sig. The
More informationInteresting Series Associated with Central Binomial Coefficients, Catalan Numbers and Harmonic Numbers
3 47 6 3 Joural of Iteger Sequeces Vol. 9 06 Article 6.. Iterestig Series Associated with Cetral Biomial Coefficiets Catala Numbers ad Harmoic Numbers Hogwei Che Departmet of Mathematics Christopher Newport
More informationSummer MA Lesson 13 Section 1.6, Section 1.7 (part 1)
Suer MA 1500 Lesso 1 Sectio 1.6, Sectio 1.7 (part 1) I Solvig Polyoial Equatios Liear equatio ad quadratic equatios of 1 variable are specific types of polyoial equatios. Soe polyoial equatios of a higher
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationarxiv: v1 [math.ca] 17 Dec 2017
Fiite ad ifiite product trasforatios Marti Nicholso Several ifiite products are studied alog with their fiite couterparts. For certai values of the paraeters these ifiite products reduce to odular fors.
More informationMath 113 Exam 4 Practice
Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationLecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More informationLebesgue Constant Minimizing Bivariate Barycentric Rational Interpolation
Appl. Math. If. Sci. 8, No. 1, 187-192 (2014) 187 Applied Matheatics & Iforatio Scieces A Iteratioal Joural http://dx.doi.org/10.12785/ais/080123 Lebesgue Costat Miiizig Bivariate Barycetric Ratioal Iterpolatio
More informationIntegrals of Functions of Several Variables
Itegrals of Fuctios of Several Variables We ofte resort to itegratios i order to deterie the exact value I of soe quatity which we are uable to evaluate by perforig a fiite uber of additio or ultiplicatio
More informationLecture 7: Fourier Series and Complex Power Series
Math 1d Istructor: Padraic Bartlett Lecture 7: Fourier Series ad Complex Power Series Week 7 Caltech 013 1 Fourier Series 1.1 Defiitios ad Motivatio Defiitio 1.1. A Fourier series is a series of fuctios
More informationAppendix: The Laplace Transform
Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio
More informationTHE CLOSED FORMS OF CONVERGENT INFINITE SERIES ESTIMATION OF THE SERIES SUM OF NON-CLOSED FORM ALTERNATING SERIES TO A HIGH DEGREE OF PRECISION.
THE CLSED FRMS F CNERGENT INFINITE SERIES ESTIMATIN F THE SERIES SUM F NN-CLSED FRM ALTERNATING SERIES T A HIGH DEGREE F PRECISIN. Peter G.Bass. PGBass M er..0.0. www.relativityoais.co May 0 Abstract This
More information