Leibnitz's Theorem

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  Theoretical Introduction to Mathematical Theorems, A

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  The first part of the theorem, sometimes called the first fundamental theorem of calculus , shows that an indefinite integration can be reversed by a differentiation. The first part is also important because it guarantees the existence of antiderivatives for continuous functions. The second part, sometimes called the second fundamental theorem of calculus , allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals. The first published statement and proof of a restricted version of the fundamental theorem was by James Gregory (1638–1675). Isaac Barrow (1630–1677) proved the first completely general version of the theorem, while Barrow's student Isaac Newton (1643– 1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities. Physical intuition Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time (or over some other quantity) adds up to the net change in the quantity. In the case of a particle traveling in a straight line, its position, x , is given by x ( t ) where t is time and x ( t ) means that x is a function of t . The derivative of this function is equal to the infinitesimal change in quantity, d x , per infinitesimal change in time, d t (of course, the derivative itself is dependent on time). This change in displacement per change in time is the velocity v of the particle. In Leibniz's notation: Rearranging this equation, it follows that: By the logic above, a change in x (or Δ x ) is the sum of the infinitesimal changes d x . It is also equal to the sum of the infinitesimal products of the derivative and time.

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  Series and Products in the Development of Mathematics: Volume 1

  • Ranjan Roy(Author)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  . . . The resulting series is Taylor’s expansion. This proof is not rigorous, but the same argument was given by de Moivre in his letter to Bernoulli and then again by Taylor in his Methodus Incrementorum Directa et Inversa of 1715. 12 Leibniz too started with a formula involving finite differences to derive Bernoulli’s series. On the other hand, the unpublished argument of Newton, also independently found by Stirling and Maclaurin, assumed that the function had a series expansion and then, by repeated differentiation of the equation, showed that the coefficients of the series were the derivatives of the function computed at specific values. This is called the method of undetermined coefficients. We can see that there were three different methods by which the early researchers on the Taylor series discovered the expansion: (a) the method of taking the limit of an appropriate finite difference formula, by Gregory, Leibniz, de Moivre, and Taylor, (b) the method of undetermined coefficients, by Newton, Stirling, and Maclaurin, and (c) repeated integration by parts, or, equivalently, repeated use of the product rule, by Johann Bernoulli. Infinite series, including power series, were used extensively in the eighteenth century for numerical calculations. On the basis of considerable experience, mathe- maticians usually had a good idea of the accuracy of their results even though they did not perform error analyses. It was only in the second half of the eighteenth century that a few mathematicians started considering an explicit error term. In the specific case of a binomial series, Jean d’Alembert (1717–1783) obtained bounds for the remainder 10 Bernoulli and Leibniz (1745) vol. 1, pp. 13–16. 11 See Feigenbaum (1981) chapter 4. 12 Taylor (1715). 11.1 Preliminary Remarks 251 of the series after the first n terms.

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  Sources in the Development of Mathematics

  Series and Products from the Fifteenth to the Twenty-first Century

  • Ranjan Roy(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  Hilbert commented that the establishment of this last theorem created an indispensable tool for refined analytical and arithmetical investigations. In modern calculus books, the remainder term for the Taylor series of a function f (x) is used to determine the values of x for which the series represents the function. This approach is due to Cauchy; in his courses at the École Polytechnique in the 1820s, he used this method to find series for the elementary functions. Cauchy’s use of the remainder term for this purpose was consistent with his pursuit of rigor; we also note that in 1822 he discovered and published the fact that all the derivatives at zero of the function f (x) = e −1/x 2 when x  = 0 and f (0) = 0 were equal to zero. Thus, the Taylor series at x = 0 was identically zero; it therefore represented the function only at x = 0. This example would have come as a great surprise to Lagrange who believed that all functions could be represented as series and even attempted to prove it. He built the whole theory of calculus on this basis. He defined the derivative of f (x), for example, as the coefficient of h in the series expansion of f (x + h). He was thereby attempting to eliminate vague concepts such as fluxions, infinitesimals, and limits in order to reduce all computations to the algebraic analysis of finite quantities. Cauchy, by contrast, rejected Lagrange’s foundations for analysis but accepted with small changes some of Lagrange’s proofs. The proof of Taylor’s theorem based on Rolle’s theorem, now commonly given in textbooks, seems to have first been given in J. A. Serret’s 1868 text on calculus; he attributed the result to Pierre Ossian Bonnet (1819–1892). In fact, Rolle proved the theorem only for polynomials. Serret did not mention Rolle explicitly in the course of his proof, but did mention him in his algebra book. Michel Rolle (1652–1719) was a paid member of the Academy of Sciences of Paris.

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  Tangled Origins Of The Leibnizian Calculus, The: A Case Study Of A Mathematical Revolution

  A Case Study of a Mathematical Revolution

  • Richard C Brown(Author)
  • 2012(Publication Date)
  • World Scientific

    (Publisher)

  If we are to believe his Historia et Origo Leibniz used these finite sum and difference ideas to derive a great many formulas concerning series and differences involving their terms. Suppose, for instance, we have a series with terms a k that are constantly deceasing toward a limiting value ω . If the successive differences written in modern notation are Δ a j = a j − a j +1 ..., Δ k a j = Δ(Δ k − 1 a j ) , 40 As we have already noted, this was written to answer the charges that Leibniz had plagiarized the calculus from Newton published by the Royal Society in its 1712 report Commercium Epistolicum D. Johannis Collins et aliorum . See Chapter 10 for a brief discussion of this famous controversy. First Steps in Mathematics 103 Leibniz gives tables stating that a 1 − ω = Δ a 1 + (Δ a 1 − Δ 2 a 1 ) + · · · + k summationdisplay j =1 ( − 1) j − 1 parenleftbigg k − 1 j − 1 parenrightbigg Δ j a 1 + ... and that a 1 − ω = 1 · Δ 2 a 1 + 2 · Δ 2 a 2 + ...j · Δ 2 a j + ... = 1 · Δ 3 a 1 + (1 + 2) · Δ 3 a 2 + · · · + (1 + 2 + · · · + j ) · Δ 3 a j + ... = ∞ summationdisplay j =1 β jk · Δ k a j where β jk = ∑ j i =1 β i ( k − 1) . In terms of the “ d ” and “ integraltext ” notation with y standing for a term of the series, x the natural numbers, and integraltext k repeated summation for k ≥ 3 we have also the formulas y − ω = d ( xy ) − dd ( y integraldisplay x ) + d 3 ( y integraldisplay integraldisplay x ) − d 4 ( y integraldisplay 3 x ) + ... integraldisplay y = yx − dy · integraldisplay x + ddy · integraldisplay integraldisplay x − d 3 · integraldisplay 3 x + ..., the latter formula holding when ω = 0.

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  Real and Complex Analysis

  • Christopher Apelian, Steve Surace(Authors)
  • 2009(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  But we also have 1 - 0 1 1 + x 2 dx = tan -1 x (1 - ) 0 = tan -1 (1 - ), and so overall we obtain tan -1 (1 - ) = ∞ j =0 ( -1) j (1 - ) 2 j +1 2 j + 1 . (9.10) Applying Abel’s theorem on page 463, we may take the limit as → 0 + in (9.10) . This yields tan -1 (1) = π 4 on the left-hand side and ∑ ∞ j =0 ( -1) j 2 j +1 on the right-hand side, obtaining what is known as the Leibniz series, π 4 = ∞ j =0 ( -1) j 2 j + 1 . Multiplying the above equality by 4 obtains what is referred to as the Leibniz formula for π , π = 4 ( 1 -1 3 + 1 5 -1 7 + · · · ) . Example 1.20 In this example we will find a power series centered at x 0 = 1 and convergent on N 1 (1) = (0, 2) for the real natural logarithm ln x . To do so, we will exploit term-by-term integration of power series. Note that since 1 x = 1 1 -(1 -x ) = ∞ j =0 (1 -x ) j 472 TAYLOR SERIES, LAURENT SERIES, AND THE RESIDUE CALCULUS converges on N 1 (1), it follows from Theorem 1.18 that for 0 < x < 2, ln x = x 1 1 y dy = x 1 ∞ j =0 (1 -y ) j dy = ∞ j =0 x 1 (1 -y ) j dy = ∞ j =0 ( -1) j j + 1 ( x -1) j +1 for 0 < x < 2. (9.11) 9.19 From the results of the previous example, justify letting x → 2 -to show that ln 2 = ∑ ∞ j =0 ( -1) j j +1 . We now develop a term-by-term integration theorem for complex integrals of complex power series. We will need the following result, analogous to Theorem 3.10 in Chapter 7. Theorem 1.21 For each n ∈ N , let f n : D → C be a continuous function on D and suppose there exists a function f : D → C such that lim n →∞ f n ( z ) = f ( z ) uniformly on D . Then for any contour C ⊂ D , lim n →∞ C f n ( z ) dz = C lim n →∞ f n ( z ) dz = C f ( z ) dz . P ROOF Since each f n for n ≥ 1 is continuous on D , it follows from Theo-rem 3.8 in Chapter 5 that f is continuous on D . Let L C be the length of the contour C , and let > 0. Then there exists N ∈ N such that for all z ∈ D , n > N ⇒ | f n ( z ) -f ( z ) | < 2 L C .

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