DOCSLIB.ORG

1 Notes for Expansions/Series and Differential Equations in the Last

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 Notes for Expansions/Series and Differential Equations in the Last Notes for Expansions/Series and Differential Equations In the last discussion, we considered perturbation methods for constructing solutions/roots of algebraic equations. Three types of problems were illustrated starting from the simplest: regular (straightforward) expansions, non-uniform expansions requiring modification to the process through inner and outer expansions, and singular perturbations. Before proceeding further, we first more clearly define the various types of expansions of functions of variables. 1. Convergent and Divergent Expansions/Series Consider a series, which is the sum of the terms of a sequence of numbers. Given a sequence {a1,a2,a3,a4,a5,....,an..} , the nth partial sum Sn is the sum of the first n terms of the sequence, that is, n Sn = ak . (1) k =1 A series is convergent if the sequence of its partial sums {S1,S2,S3,....,Sn..} converges. In a more formal language, a series converges if there exists a limit such that for any arbitrarily small positive number ε>0, there is a large integer N such that for all n ≥ N , ^ Sn − ≤ ε . (2) A sequence that is not convergent is said to be divergent. Examples of convergent and divergent series: • The reciprocals of powers of 2 produce a convergent series: 1 1 1 1 1 1 + + + + + + ......... = 2 . (3) 1 2 4 8 16 32 • The reciprocals of positive integers produce a divergent series: 1 1 1 1 1 1 + + + + + + ......... (4) 1 2 3 4 5 6 • Alternating the signs of the reciprocals of positive integers produces a convergent series: 1 1 1 1 1 1 − + − + − + ......... = ln 2 . (5) 1 2 3 4 5 6 • The reciprocals of prime numbers produce a divergent series: 1 1 1 1 1 1 + + + + + ......... (6) 2 3 5 7 11 13 1 Convergence tests: There are a number of methods for determining whether a series is convergent or divergent. Comparison test: The terms of the sequence {a1,a2,a3,a4,a5,....,an..} are compared to those of another sequence {b1,b2,b3,b4,b5,....,bn..} . If, for all n, 0 ≤ an ≤ bn , and ∞ ∞ bn converges, then so does an . However, if, for all n, 0 ≤ bn ≤ an , and n=1 n=1 ∞ ∞ bn diverges, then so does an . n=1 n=1 Ratio test: Assume that for all n, an > 0. Suppose that there exists an r > 0 such that a lim n+1 = r . (7) n→∞ an If r <1, the series converges. If r >1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge. Root test or nth root test. Suppose that the terms of the sequence under consideration are non-negative, and that there exists r > 0 such that n lim an = r . (8) n→∞ If r <1, the series converges. If r >1, the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge. Root test is equivalent to ratio test. Integral test: The series can be compared to an integral to establish convergence or divergence. Let f(n) = an be a positive and monotone decreasing function. If ∞ t —f (x)dx =lim—f (x)dx < ∞ , (9) 1 t→∞1 then the series converges. If however, the integral diverges, the series does so as well. 2 an Limit comparison test: If {an}, {bn} >0, and the limit lim exists and is not zero, n→∞ bn ∞ ∞ then an converges if and only if bn converges. n=1 n=1 Alternating series test: Also known as the Leibniz criterion, the alternating series test ∞ n states that for an alternating series of the form an (−1) , if {an} is monotone n=1 decreasing, and has a limit of 0, then the series converges. Cauchy condensation test: If {an} is a monotone decreasing sequence, then ∞ ∞ a converges if and only if 2k a converges. n 2k n=1 k =1 Other tests for convergence include Dirichlet's test, Abel's test and Raabe's test. Conditional and absolute convergence: Note that for any sequence {a1,a2,a3,a4,a5,....,an..} , an ≤ an for all n. Therefore, ∞ ∞ ∞ ∞ an ≤ an . This means that if an converges, then an also converges (but n=1 n=1 n=1 n=1 not vice-versa). ∞ ∞ If the series an converges, then the series an is absolutely convergent. An n=1 n=1 absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long. The power series of the exponential function is absolutely convergent everywhere. ∞ ∞ ∞ If the series an converges but the series an diverges, then the series an is n=1 n=1 n=1 conditionally convergent. The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges. 3 Uniform convergence: ∞ Let { f1, f2, f3, f4,...., fn..} be a sequence of functions. The series fn is said to n=1 converge uniformly to f if the sequence {Sn} of partial sums defined by ∞ Sn (x) = fn (x) (10) n=1 converges uniformly to f. Cauchy convergence criterion: The Cauchy convergence criterion states that a series ∞ an n=1 converges if and only if the sequence of partial sums is a Cauchy sequence. This means that for every ε > 0, there is a positive integer N such that for all n ≥ m ≥ N we have n ak < ε , (11) k =m which is equivalent to n+m lim an = 0. (12) n→∞ k =n m→∞ Radius of convergence: The radius of convergence of a power series is a non-negative quantity, either a real number or that represents a range (within the radius) in which the function will converge. For a complex power series f defined as: ∞ n f (z) = cn (z − a) (13) n=0 th where a is a constant, the center of the disk of convergence, cn is the n complex coefficient, and z is a complex variable. The radius of convergence r is a nonnegative real number or , such that the series converges if z − a < r , and diverges if z − a > r . In other words, the series converges if z is close enough to the center and diverges if it is too far away. The radius of convergence is infinite if the series converges for all complex numbers z. 4 The radius of convergence can be found by applying the root test to the terms of the series. The root test uses the number n C = limsup fn (14) n→∞ n where ƒn is the nth term cn(z − a) ("lim sup" denotes the limit superior). The root test states that the series converges if |C| < 1 and diverges if |C| > 1. It follows that the power series converges if the distance from z to the center a is less than n r =1/ limsup cn , (15) n→∞ and diverges if the distance exceeds that number. Note that r = 1/0 is interpreted as an infinite radius, meaning that ƒ is an entire function. The limit involved in the ratio test is usually easier to compute, but the limit may fail to exist, in which case the root test should be used. The ratio test uses the limit fn+1 L = lim . (16) n→∞ fn In the case of a power series, this can be used to find that cn r = lim . (17) n→∞ cn+1 2. Asymptotic Expansions An asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Let n be a sequence of continuous functions on some domain, and let L be a (possibly infinite) limit point of the domain. Then the sequence constitutes an asymptotic scale or guage functions if for every n, ϕn+1(x) = o(ϕn (x)) as x → L . If f is a continuous function on the domain of the guage functions, an asymptotic expansion of f ∞ with respect to the scale is a formal series anϕn (x) such that, for any fixed N, n=0 N f (x) = anϕn (x) + O(ϕn+1(x)) as x → L . (18) n=0 In this case, we write ∞ f (x) ~ anϕn (x) as x → L . (19) n=0 5 The most common type of an asymptotic expansion is a power series in either positive or negative terms. While a convergent Taylor series fits the definition as given, a non-convergent series is what is usually intended by the phrase. Methods of generating such expansions include the Euler-Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. Examples of asymptotic expansions: • Gamma function (20) • Exponential integral (21) • Riemann zeta function (22) where B2m are Bernoulli numbers and is a rising factorial. This expansion is valid for all complex s and is often used to compute the zeta function by using a large enough value of N, for instance N > | s |. • Error function (23) We now move on to differential equations, and proceed in the same manner, staring from the simplest to more complex. 6 1. Introduction to Perturbations Techniques for Differential Equations (http://www.sm.luth.se/~johanb/applmath/chap2en) Consider the example of the systems shown in the picture above. Let the mass of the earth be m.The motion of the earth around the sun is an ideal case, where we have no influences from other celestial bodies.
Load more

Recommended publications
  • Abel's Identity, 131 Abel's Test, 131–132 Abel's Theorem, 463–464 Absolute Convergence, 113–114 Implication of Condi
    INDEX Abel’s identity, 131 Bolzano-Weierstrass theorem, 66–68 Abel’s test, 131–132 for sequences, 99–100 Abel’s theorem, 463–464 boundary points, 50–52 absolute convergence, 113–114 bounded functions, 142 implication of conditional bounded sets, 42–43 convergence, 114 absolute value, 7 Cantor function, 229–230 reverse triangle inequality, 9 Cantor set, 80–81, 133–134, 383 triangle inequality, 9 Casorati-Weierstrass theorem, 498–499 algebraic properties of Rk, 11–13 Cauchy completeness, 106–107 algebraic properties of continuity, 184 Cauchy condensation test, 117–119 algebraic properties of limits, 91–92 Cauchy integral theorem algebraic properties of series, 110–111 triangle lemma, see triangle lemma algebraic properties of the derivative, Cauchy principal value, 365–366 244 Cauchy sequences, 104–105 alternating series, 115 convergence of, 105–106 alternating series test, 115–116 Cauchy’s inequalities, 437 analytic functions, 481–482 Cauchy’s integral formula, 428–433 complex analytic functions, 483 converse of, 436–437 counterexamples, 482–483 extension to higher derivatives, 437 identity principle, 486 for simple closed contours, 432–433 zeros of, see zeros of complex analytic on circles, 429–430 functions on open connected sets, 430–432 antiderivatives, 361 on star-shaped sets, 429 for f :[a, b] Rp, 370 → Cauchy’s integral theorem, 420–422 of complex functions, 408 consequence, see deformation of on star-shaped sets, 411–413 contours path-independence, 408–409, 415 Cauchy-Riemann equations Archimedean property, 5–6 motivation, 297–300
    [Show full text]
  • Theorems Wednesday, May 9, 2018 12:53 AM
    Theorems Wednesday, May 9, 2018 12:53 AM Theorem 1.11: Greatest-Lower-Bound Property • Suppose is an ordered set with the least-upper-bound property • Suppose , and is bounded below • Let be the set of lower bounds of • Then exists in and Theorem 1.20: The Archimedean property of • Given , and • There is a positive integer such that Theorem 1.20: is dense in • If , and , then there exists a s.t. • We can always find a rational number between two real numbers Theorem 1.21: -th Root of Real Numbers • For every real , and positive integer • There is one and only one positive real number s.t. • Theorem 1.31: Properties of Complex Numbers • If and are complex numbers, then • • • • is real and positive (except when ) Theorem 1.33: Properties of Complex Numbers • If and are complex numbers, then • unless in which case • • • • (Triangle Inequality) Theorem 1.37: Properties of Euclidean Spaces • Suppose , then • Page 1 • if and only if • • (Schwarz's Inequality) • (Triangle Inequality) • (Triangle Inequality) Theorem 2.8: Infinite Subset of Countable Set • Every infinite subset of a countable set is countable Theorem 2.12: Union of Countable Sets • Let be a sequence of countable sets, then • Theorem 2.13: Cartesian Product of Countable Sets • Let be a countable set • Let be the set of all -tuples where ○ for ○ may not be distinct • Then is countable Theorem 2.14: Cantor's Diagonalization Argument • Let be the set of all sequqnecse whose digits are 0 and 1 • Then is uncountable Theorem 2.19: Every Neighborhood is an Open Set • Every neighborhood
    [Show full text]
  • Cauchy's Cours D'analyse
    SOURCES AND STUDIES IN THE HISTORY OF MATHEMATICS AND PHYSICAL SCIENCES Robert E. Bradley • C. Edward Sandifer Cauchy’s Cours d’analyse An Annotated Translation Cauchy’s Cours d’analyse An Annotated Translation For other titles published in this series, go to http://www.springer.com/series/4142 Sources and Studies in the History of Mathematics and Physical Sciences Editorial Board L. Berggren J.Z. Buchwald J. Lutzen¨ Robert E. Bradley, C. Edward Sandifer Cauchy’s Cours d’analyse An Annotated Translation 123 Robert E. Bradley C. Edward Sandifer Department of Mathematics and Department of Mathematics Computer Science Western Connecticut State University Adelphi University Danbury, CT 06810 Garden City USA NY 11530 [email protected] USA [email protected] Series Editor: J.Z. Buchwald Division of the Humanities and Social Sciences California Institute of Technology Pasadena, CA 91125 USA [email protected] ISBN 978-1-4419-0548-2 e-ISBN 978-1-4419-0549-9 DOI 10.1007/978-1-4419-0549-9 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009932254 Mathematics Subject Classification (2000): 01A55, 01A75, 00B50, 26-03, 30-03 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
    [Show full text]
  • The Infinity Theorem Is Presented Stating That There Is at Least One Multivalued Series That Diverge to Infinity and Converge to Infinite Finite Values
    Open Journal of Mathematics and Physics | Volume 2, Article 75, 2020 | ISSN: 2674-5747 https://doi.org/10.31219/osf.io/9zm6b | published: 4 Feb 2020 | https://ojmp.wordpress.com CX [microresearch] Diamond Open Access The infinity theorem Open Mathematics Collaboration∗† March 19, 2020 Abstract The infinity theorem is presented stating that there is at least one multivalued series that diverge to infinity and converge to infinite finite values. keywords: multivalued series, infinity theorem, infinite Introduction 1. 1, 2, 3, ..., ∞ 2. N =x{ N 1, 2,∞3,}... ∞ > ∈ = { } The infinity theorem 3. Theorem: There exists at least one divergent series that diverge to infinity and converge to infinite finite values. ∗All authors with their affiliations appear at the end of this paper. †Corresponding author: [email protected] | Join the Open Mathematics Collaboration 1 Proof 1 4. S 1 1 1 1 1 1 ... 2 [1] = − + − + − + = (a) 5. Sn 1 1 1 ... 1 has n terms. 6. S = lim+n + S+n + 7. A+ft=er app→ly∞ing the limit in (6), we have S 1 1 1 ... + 8. From (4) and (7), S S 2 = 0 + 2 + 0 + 2 ... 1 + 9. S 2 2 1 1 1 .+.. = + + + + + + 1 10. Fro+m (=7) (and+ (9+), S+ )2 2S . + +1 11. Using (6) in (10), limn+ =Sn 2 2 limn Sn. 1 →∞ →∞ 12. limn Sn 2 + = →∞ 1 13. From (6) a=nd (12), S 2. + 1 14. From (7) and (13), S = 1 1 1 ... 2. + = + + + = (b) 15. S 1 1 1 1 1 ... + 1 16. S =0 +1 +1 +1 +1 +1 1 ..
    [Show full text]
  • Mathematics Course: Analysis I
    Analysis I Carl Turner November 21, 2010 Abstract These are notes for an undergraduate course on analysis; please send corrections, suggestions and notes to [email protected] The author's homepage for all courses may be found on his website at SuchIdeas.com, which is where updated and corrected versions of these notes can also be found. The course materials are licensed under a permissive Creative Commons license: Attribution- NonCommercial-ShareAlike 3.0 Unported (see the CC website for more details). Thanks go to Professor G. Paternain for allowing me to use his Analysis I course (Lent 2010) as the basis for these notes. Contents 1 Limits and Convergence3 1.1 Fundamental Properties of the Real Line...........................3 1.2 Cauchy Sequences.......................................5 1.3 Series..............................................6 1.4 Sequences of Non-Negative Terms...............................7 1.5 Alternating Series and Absolute Convergence........................ 10 2 Continuity 13 2.1 Denitions and Basic Properties............................... 13 2.2 Intermediate Values, Extreme Values and Inverses..................... 15 3 Dierentiability 18 3.1 Denitions and Basic Properties............................... 18 3.2 The Mean Value Theorem and Taylor's Theorem...................... 20 3.3 Taylor Series and Binomial Series............................... 25 3.4 * Comments on Complex Dierentiability.......................... 26 4 Power Series 28 4.1 Fundamental Properties.................................... 28 4.2 Standard Functions....................................... 32 4.2.1 Exponentials and logarithms............................. 32 1 4.2.2 Trigonometric functions................................ 35 4.2.3 Hyperbolic functions.................................. 37 5 Integration 38 5.1 Denitions and Key Properties................................ 38 5.2 Computation of Integrals................................... 44 5.3 Mean Values and Taylor's Theorem Again, Improper Integrals and the Integral Test.
    [Show full text]
  • SFS / M.Sc. (Mathematics with Computer Science)
    Mathematics Syllabi for Entrance Examination M.Sc. (Mathematics)/ M.Sc. (Mathematics) SFS / M.Sc. (Mathematics with Computer Science) Algebra.(5 Marks) Symmetric, Skew symmetric, Hermitian and skew Hermitian matrices. Elementary Operations on matrices. Rank of a matrices. Inverse of a matrix. Linear dependence and independence of rows and columns of matrices. Row rank and column rank of a matrix. Eigenvalues, eigenvectors and the characteristic equation of a matrix. Minimal polynomial of a matrix. Cayley Hamilton theorem and its use in finding the inverse of a matrix. Applications of matrices to a system of linear (both homogeneous and non–homogeneous) equations. Theoremson consistency of a system of linear equations. Unitary and Orthogonal Matrices, Bilinear and Quadratic forms. Relations between the roots and coefficients of general polynomial equation in one variable. Solutions of polynomial equations having conditions on roots. Common roots and multiple roots. Transformation of equations. Nature of the roots of an equation Descarte’s rule of signs. Solutions of cubic equations (Cardon’s method). Biquadratic equations and theirsolutions. Calculus.(5 Marks) Definition of the limit of a function. Basic properties of limits, Continuous functions and classification of discontinuities. Differentiability. Successive differentiation. Leibnitz theorem. Maclaurin and Taylor series expansions. Asymptotes in Cartesian coordinates, intersection of curve and its asymptotes, asymptotes in polar coordinates. Curvature, radius of curvature for Cartesian curves, parametric curves, polar curves. Newton’s method. Radius of curvature for pedal curves. Tangential polar equations. Centre of curvature. Circle of curvature. Chord of curvature, evolutes. Tests for concavity and convexity. Points of inflexion. Multiplepoints. Cusps, nodes & conjugate points. Type of cusps.
    [Show full text]
  • 1 X = 1 Ln 2 Ln 2
    Let’s consider that power series for ln x about x 1 x12 x 1 3 x 1 4 x 1 5 x 1 6 x 1 7 x 1 8 lnxx 1 ... 2 3 4 5 6 7 8 So if this is true, then what is ln2 ? 212 21 3 21 4 21 5 21 6 21 7 21 8 ln2 2 1 ... 2 3 4 5 6 7 8 12 1 3 1 4 1 5 1 6 1 7 1 8 ln2 1 ... 2 3 4 5 6 7 8 n1 1 1 1 1 1 1 1 1 ln2 1 ... ... = the alternating harmonic series! 2 3 4 5 6 7 8 n This series is Conditionally Convergent since the non-alternating harmonic series diverges (by integral test), but the alternating series converges (by alternating series test). Now we know that the alternating harmonic series converges to . We can also show that this series converges when 02x . 111111111111111 So ln2 1 ... 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Let’s rearrange these terms: 1111111111111111 ln2 1 ... 2 4 3 6 8 5 10 12 7 14 16 9 18 20 11 22 Do you agree that this series will include all of the terms of the power series for ? Let’s group certain terms together: 1111111111 111 1 11 ln2 1 ... 2 4 3 6 8 5 10 12 7 14 16 9 18 20 11 22 11111111111 = ... 2 4 6 8 10 12 14 16 18 20 22 Look at each term in this series, compared with each term in our original series for ln2.
    [Show full text]
  • If Riemann's Zeta Function Is True, It Contradicts Zeta's Dirichlet Series, Causing
    If Riemann’s Zeta Function is True, it Contradicts Zeta’s Dirichlet Series, Causing “Explosion”. If it is False, it Causes Unsoundness. Ayal I. Sharon ∗y September 4, 2019 Abstract Riemann’s "analytic continuation" produces a second definition of the Zeta function, that Riemann claimed is convergent throughout half-plane s 2 C, Re(s) ≤ 1, (except at s = 1). This contradicts the original definition of the Zeta function (the Dirichlet series), which is proven divergent there. Moreover, a function can- not be both convergent and divergent at any domain value. In physics and in other mathematics conjectures and assumed-proven theorems, the Riemann Zeta func- tion (or the class of L-functions that generalizes it) is assumed to be true. Here the author shows that the two contradictory definitions of Zeta violate Aristotle’s Laws of Identity, Non-Contradiction, and Excluded Middle. Non-Contradiction is an axiom of classical and intuitionistic logics, and an inherent axiom of Zermelo- Fraenkel set theory (which was designed to avoid paradoxes). If Riemann’s defini- tion of Zeta is true, then the Zeta function is a contradiction that causes deductive "explosion", and the foundation logic of mathematics must be replaced with one that is paradox-tolerant. If Riemann’s Zeta is false, it renders unsound all theo- rems and conjectures that falsely assume that it is true. Riemann’s Zeta function appears to be false, because its derivation uses the Hankel contour, which violates the preconditions of Cauchy’s integral theorem. ∗Patent Examiner, U.S. Patent and Trademark Office (USPTO). This research received no funding, and was conducted in the author’s off-duty time.
    [Show full text]
  • Real Analysis Honors Exam 1 Do the Best That You Can to Respond to The
    Real Analysis Honors Exam 1 Do the best that you can to respond to the following problems. It is not required that you respond to every problem, and, in fact, it is much better to provide complete and clearly explained responses to a subset of the given problems than to provide hurried and incomplete responses to all of them. When a complete response is not possible you are encouraged to clearly explain the partial progress that you can achieve. x2 y2 z2 4 1. Prove that the ellipsoid a2 + b2 + c2 = 1 has volume 3 πabc. You may assume that this formula is correct for the spherical case where a = b = c. 2. Newton's Method: Suppose that f : R ! R is continuously differen- tiable. Newton's method is an iterative algorithm designed to approx- imate the zeros of f. Start with a first guess x0, then for each integer n > 0 find the equation of the tangent line to the graph of f at the point (xn−1; f(xn−1))), and solve for the x intercept of that line which is named xn. Show that the sequence (xn) is guaranteed to converge under the following conditions: (a) f has a zero in the interval (a; b). (b) f is twice continuous differentiable with f 0(x) > 0 and f 00(x) > 0 on (a; b). 2 3. A function f : R ! R is Gateux differentiable at x0 if the limit f(x + hv) − f(x ) lim 0 0 ; h!0 h exists for all v 2 R2, i.e.
    [Show full text]
  • Of Triangles, Gas, Price, and Men
    OF TRIANGLES, GAS, PRICE, AND MEN Cédric Villani Univ. de Lyon & Institut Henri Poincaré « Mathematics in a complex world » Milano, March 1, 2013 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemannian (= non-Euclidean) geometry At each location, the units of length and angles may change Shortest path (= geodesics) are curved!! Geodesics can tend to get closer (positive curvature, fat triangles) or to get further apart (negative curvature, skinny triangles) Hyperbolic surfaces Bernhard Riemann 1826-1866 List of topics named after Bernhard Riemann From Wikipedia, the free encyclopedia Riemann singularity theorem Cauchy–Riemann equations Riemann solver Compact Riemann surface Riemann sphere Free Riemann gas Riemann–Stieltjes integral Generalized Riemann hypothesis Riemann sum Generalized Riemann integral Riemann surface Grand Riemann hypothesis Riemann theta function Riemann bilinear relations Riemann–von Mangoldt formula Riemann–Cartan geometry Riemann Xi function Riemann conditions Riemann zeta function Riemann curvature tensor Zariski–Riemann space Riemann form Riemannian bundle metric Riemann function Riemannian circle Riemann–Hilbert correspondence Riemannian cobordism Riemann–Hilbert problem Riemannian connection Riemann–Hurwitz formula Riemannian cubic polynomials Riemann hypothesis Riemannian foliation Riemann hypothesis for finite fields Riemannian geometry Riemann integral Riemannian graph Bernhard
    [Show full text]
  • Bibliography
    Bibliography 1. E. Artin, The Gamma Function, Holt, Rinehart and Winston, New York, 1964 (translated by M. Butler from the German original, Einfubrung in Die Theorie Der Gamma Function, B. G. Teubner, 1931) 2. F. Bowman, Introduction to Elliptic Functions with Applications, Dover, New York, 1961 3. N. Bourbaki, Function D'une Variable Reel/e (Theorie Elementaire), Hermann, Editeur Des Sciences et Des Arts, Paris, 1949 4. T. J. I. Bromwich, Infinite Series (Second Edition Revised), Macmillan & Co. Limited, London, 1942 5. Burrill and Knudsen, Real Variables, Holt, Rinehart and Winston, New York, 1969 6. G. Chrystal, Textbook of Algebra, Vol. II, Dover Edition 7. Courant and John, Introduction to Calculus and Analysis, Interscience Divi­ sion, John Wiley and Sons, New York, 1965 8. Phillip Franklin, Treatise on Advanced Calculus, John Wiley and Sons, New York, 1940 9. Avmer Friedman, Advanced Calculus, Holt, Rinehart and Winston, New York, 1971 10. Watson Fulks, Advanced Calculus (Third Edition), John Wiley and Sons, New York, 1978 11. Eimar Hille, Analysis, Vol. I, Blaisdell Publishing Co., 1964 Ila. E. W. Hobson, The Theory of Functions of a Real Variable, Vol. II, Dover Edition, New York, 1957 12. Konrad Knopf, Theory and Application of Infinite Series, Blackie and Sons, London, 1951 13. Konrad Knopf, Infinite Sequences and Series, Dover Edition, 1956 14. D. Mitronovic, Analytic Inequalities, Springer-Verlag, New York 15. Poly and Azego, Aufgabe Der Analysis (2 vols.) Dover 16. John Olmsted, Advanced Calculus, Appleton, Century, Crofts, New York, 1956 16a. James Pierpont, Theory of Functions of Real Variables, Vol. II, Ginn and Company, Boston, New York, Chicago, London, 1912 17.
    [Show full text]
  • Analytic Continuation of Ζ(S) Violates the Law of Non-Contradiction (LNC)
    Analytic Continuation of ζ(s) Violates the Law of Non-Contradiction (LNC) Ayal Sharon ∗y July 30, 2019 Abstract The Dirichlet series of ζ(s) was long ago proven to be divergent throughout half-plane Re(s) ≤ 1. If also Riemann’s proposition is true, that there exists an "expression" of ζ(s) that is convergent at all s (except at s = 1), then ζ(s) is both divergent and convergent throughout half-plane Re(s) ≤ 1 (except at s = 1). This result violates all three of Aristotle’s "Laws of Thought": the Law of Identity (LOI), the Law of the Excluded Middle (LEM), and the Law of Non- Contradition (LNC). In classical and intuitionistic logics, the violation of LNC also triggers the "Principle of Explosion" / Ex Contradictione Quodlibet (ECQ). In addition, the Hankel contour used in Riemann’s analytic continuation of ζ(s) violates Cauchy’s integral theorem, providing another proof of the invalidity of Riemann’s ζ(s). Riemann’s ζ(s) is one of the L-functions, which are all in- valid due to analytic continuation. This result renders unsound all theorems (e.g. Modularity, Fermat’s last) and conjectures (e.g. BSD, Tate, Hodge, Yang-Mills) that assume that an L-function (e.g. Riemann’s ζ(s)) is valid. We also show that the Riemann Hypothesis (RH) is not "non-trivially true" in classical logic, intuitionistic logic, or three-valued logics (3VLs) that assign a third truth-value to paradoxes (Bochvar’s 3VL, Priest’s LP ). ∗Patent Examiner, U.S. Patent and Trademark Office (USPTO).
    [Show full text]