Let be the Maclaurin series of a meromorphic function with a finite or infinite number of poles at points , indexed so thatthen a pole will occur as many times in the sequence as indicated by its order. Any index such thatholds is then called a critical index of (Henrici 1988, pp. 641-642).
A harmonic series is a continued fraction-like series defined by(Havil 2003, p. 99).Examples are given in the following table.OEISharmonic expansionA054977[2, 1, 1, 1, 1, 1, 1, ...]A096622[0, 1, 0, 1, 4, 1, 4, ...]A075874[3, 0, 0, 3, 1, 5, 6, 5, ...]
If is analytic throughout the annular region between and on the concentric circles and centered at and of radii and respectively, then there exists a unique series expansion in terms of positive and negative powers of ,(1)where(2)(3)(Korn and Korn 1968, pp. 197-198).Let there be two circular contours and , with the radius of larger than that of . Let be at the center of and , and be between and . Now create a cut line between and , and integrate around the path , so that the plus and minus contributions of cancel one another, as illustrated above. From the Cauchy integral formula,(4)(5)(6)Now, since contributions from the cut line in opposite directions cancel out,(7)(8)(9)For the first integral, . For the second, . Now use the Taylor series (valid for )(10)to obtain(11)(12)(13)where the second term has been re-indexed. Re-indexing again,(14)Since the integrands, including the function , are analytic in the annular region defined by and..
A power series in a variable is an infinite sum of the formwhere are integers, real numbers, complex numbers, or any other quantities of a given type.Pólya conjectured that if a function has a power series with integer coefficients and radius of convergence 1, then either the function is rational or the unit circle is a natural boundary (Pólya 1990, pp. 43 and 46). This conjecture was stated by G. Polya in 1916 and proved to be correct by Carlson (1921) in a result that is now regarded as a classic of early 20th century complex analysis.For any power series, one of the following is true: 1. The series converges only for . 2. The series converges absolutely for all . 3. The series converges absolutely for all in some finite open interval and diverges if or . At the points and , the series may converge absolutely, converge conditionally, or diverge. To determine the interval of convergence, apply the ratio test for absolute convergence..
Given a Taylor series(1)the error after terms is given by(2)Using the mean-value theorem, this can be rewrittenas(3)for some (Abramowitz and Stegun 1972, p. 880).Note that the Lagrange remainder is also sometimes taken to refer to the remainder when terms up to the st power are taken in the Taylor series, and that a notation in which , , and is sometimes used (Blumenthal 1926; Whittaker and Watson 1990, pp. 95-96).
Let be integrable in , let be of bounded variation in , let denote the least upper bound of in , and let denote the total variation of in . Given the function(1)then the terms of its Fourier-Legendre series(2)(3)where is a Legendre polynomial, satisfy the inequalities(4)for (Sansone 1991).
Taylor's theorem states that any function satisfying certain conditions may be representedby a Taylor series,Taylor's theorem (without the remainder term) was devised by Taylor in 1712 and published in 1715, although Gregory had actually obtained this result nearly 40 years earlier. In fact, Gregory wrote to John Collins, secretary of the Royal Society, on February 15, 1671, to tell him of the result. The actual notes in which Gregory seems to have discovered the theorem exist on the back of a letter Gregory had received on 30 January, 1671, from an Edinburgh bookseller, which is preserved in the library of the University of St. Andrews (P. Clive, pers. comm., Sep. 8, 2005).However, it was not until almost a century after Taylor's publication that Lagrange and Cauchy derived approximations of the remainder term after a finite number of terms (Moritz 1937). These forms are now called the Lagrange remainder and Cauchy remainder.Most..
If is a power series which is regular for except for poles within this circle and except for , at which points the function is assumed continuous when only points are considered, then at least a subsequence of the Padé approximants are uniformly bounded in the domain formed by removing the interiors of small circles with centers at these poles and uniformly continuous at for .
A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function about a point is given by(1)If , the expansion is known as a Maclaurin series.Taylor's theorem (actually discovered first by Gregory) states that any function satisfying certain conditions can be expressed as a Taylor series.The Taylor (or more general) series of a function about a point up to order may be found using Series[f, x, a, n]. The th term of a Taylor series of a function can be computed in the Wolfram Language using SeriesCoefficient[f, x, a, n] and is given by the inverse Z-transform(2)Taylor series of some common functions include(3)(4)(5)(6)(7)(8)To derive the Taylor series of a function , note that the integral of the st derivative of from the point to an arbitrary point is given by(9)where is the th derivative of evaluated at , and is therefore simply a constant. Now integrate a second time to obtain(10)where..
Approximants derived by expanding a function as a ratio of two power series and determining both the numerator and denominator coefficients. Padé approximations are usually superior to Taylor series when functions contain poles, because the use of rational functions allows them to be well-represented.The Padé approximant corresponds to the Maclaurin series. When it exists, the Padé approximant to any power series(1)is unique. If is a transcendental function, then the terms are given by the Taylor series about (2)The coefficients are found by setting(3)and equating coefficients. can be multiplied by an arbitrary constant which will rescale the other coefficients, so an additional constraint can be applied. The conventional normalization is(4)Expanding (3) gives(5)(6)These give the set of equations (7)(8)(9)(10)(11)(12)(13)(14)where for and for . Solving these directly gives(15)where sums are replaced..
A formal power series, sometimes simply called a "formal series" (Wilf 1994), of a field is an infinite sequence over . Equivalently, it is a function from the set of nonnegative integers to , . A formal power series is often writtenbut with the understanding that no value is assigned to the symbol .
Taylor's inequality is an estimate result for the value of the remainder term in any -term finite Taylor series approximation.Indeed, if is any function which satisfies the hypotheses of Taylor's theorem and for which there exists a real number satisfying on some interval , the remainder satisfieson the same interval .This result is an immediate consequence of the Lagrange remainder of and can also be deduced from the Cauchy remainder as well.
Euler (1738, 1753) considered the seriesHe showed that just like , for nonnegative integers , though is a different function from . (red) and (blue) for , showing their coincidence at positive integers.A closed form is given bywhere is the q-polygamma function.
The Mercator series, also called the Newton-Mercator series (Havil 2003, p. 33), is the Taylor series for the natural logarithm(1)(2)for , which was found by Newton, but independently discovered and first published by Mercator in 1668.Plugging in gives a beautiful series for the natural logarithm of 2,(3)also known as the alternating harmonic series and equal to , where is the Dirichlet eta function.
Darboux's formula is a theorem on the expansion of functions in infinite series and essentially consists of integration by parts on a specific integrand product of functions. Taylor series may be obtained as a special case of the formula, which may be stated as follows.Let be analytic at all points of the line joining to , and let be any polynomial of degree in . Then if , differentiation givesBut , so integrating over the interval 0 to 1 givesThe Taylor series follows by letting and letting (Whittaker and Watson 1990, p. 125).
A Maclaurin series is a Taylor series expansionof a function about 0,(1)Maclaurin series are named after the Scottish mathematician Colin Maclaurin.The Maclaurin series of a function up to order may be found using Series[f, x, 0, n]. The th term of a Maclaurin series of a function can be computed in the Wolfram Language using SeriesCoefficient[f, x, 0, n] and is given by the inverse Z-transform(2)Maclaurin series are a type of series expansion in which all terms are nonnegative integer powers of the variable. Other more general types of series include the Laurent series and the Puiseux series.Maclaurin series for common functions include (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)The explicit forms for some of these are (34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48)(49)(50)(51)(52)(53)where is a gamma function, is a Bernoulli..
Expanding the Riemann zeta function about gives(1)(Havil 2003, p. 118), where the constants(2)are known as Stieltjes constants.Another sum that can be used to define the constants is(3)These constants are returned by the WolframLanguage function StieltjesGamma[n].A generalization takes as the coefficient of is the Laurent series of the Hurwitz zeta function about . These generalized Stieltjes constants are implemented in the Wolfram Language as StieltjesGamma[n, a].The case gives the usual Euler-Mascheroni constant(4)A limit formula for is given by(5)where is the imaginary part and is the Riemann zeta function.An alternative definition is given by absorbing the coefficient of into the constant,(6)(e.g., Hardy 1912, Kluyver 1927).The Stieltjes constants are also given by(7)Plots of the values of the Stieltjes constants as a function of are illustrated above (Kreminski). The first few numerical values are given in the..
The expansion of the two sides of a sum equality in terms of polynomials in and , followed by closed form summation in terms of and . For an example of the technique, see Bloom (1995).
Infinite series of various simple functions of the logarithm include(1)(2)(3)(4)where is the Euler-Mascheroni constant and is the Riemann zeta function. Note that the first two of these are divergent in the classical sense, but converge when interpreted as zeta-regularized sums.
A coefficient of the Maclaurinseries of(OEIS A002206 and A002207), the multiplicative inverse of the Mercator series function .
The th order Bernstein expansion of a function in terms of a variable is given by(1)(Gzyl and Palacios 1997, Mathé 1999), where is a binomial coefficient and(2)is a Bernstein polynomial.Letting gives the identity(3)for and .
A Fourier series-like expansion of a twice continuouslydifferentiable function(1)for , where is a zeroth order Bessel function of the first kind. The coefficients are then given by(2)(3)(Gradshteyn and Ryzhik 2000, p. 926), where and care should be taken to avoid the two typos of Iyanaga and Kawada (1980) and Itô (1986).As an example, consider , which has and therefore(4)(5)(6)(7)(8)so(9)(Whittaker and Watson 1990, p. 378; Gradshteyn and Ryzhik 2000, p. 926). This is illustrated above with 1 (red), 2 (green), 3 (blue), and 4 terms (violet) included.Similarly, for ,(10)
A linear approximation to a function at a point can be computed by taking the first term in the Taylor series
A conjecture due to M. S. Robertson in 1936 which treats a univalent power series containing only odd powers within the unit disk. This conjecture implies the Bieberbach conjecture and follows in turn from the Milin conjecture. de Branges' proof of the Bieberbach conjecture proceeded by proving the Milin conjecture, thus establishing the Robertson conjecture and hence implying the truth of the Bieberbach conjecture.
The series for the inversetangent,Plugging in gives Gregory's formulaThis series is intimately connected with the number of representations of by squares , and also with Gauss's circle problem (Hilbert and Cohn-Vossen 1999, pp. 27-39).
Let be defined as the power series whose th term has a coefficient equal to the th prime ,(1)(2)The function has a zero at (OEIS A088751). Now let be defined by(3)(4)(5)(OEIS A030018).Then N. Backhouse conjectured that(6)(7)(OEIS A072508). This limit was subsequently shown to exist by P. Flajolet. Note that , which follows from the radius of convergence of the reciprocal power series.The continued fraction of Backhouse's constant is [1, 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, ...] (OEIS A074269), which is also the same as the continued fraction of except for a leading 0 in the latter.
Stirling's approximation gives an approximate value for the factorial function or the gamma function for . The approximation can most simply be derived for an integer by approximating the sum over the terms of the factorial with an integral, so that(1)(2)(3)(4)(5)(6)The equation can also be derived using the integral definition of the factorial,(7)Note that the derivative of the logarithm of the integrandcan be written(8)The integrand is sharply peaked with the contribution important only near . Therefore, let where , and write(9)(10)Now,(11)(12)(13)so(14)(15)(16)Taking the exponential of each side thengives(17)(18)Plugging into the integral expression for then gives(19)(20)Evaluating the integral gives(21)(22)(Wells 1986, p. 45). Taking the logarithm of bothsides then gives(23)(24)This is Stirling's series with only the first term retained and, for large , it reduces to Stirling's approximation(25)Taking successive..
An extended form of Bürmann's theorem. Let be a function of analytic in a ring-shaped region , bounded by another curve and an inner curve . Let be a function analytic on and inside having only one zero (which is simple) within the contour. Further let be a given point within . Finally, let(1)for all points of , and(2)for all points of . Then(3)where(4)(5)(Whittaker and Watson 1990, pp. 131-132).
Let be defined as a function of in terms of a parameter by(1)Then Lagrange's inversion theorem, also called a Lagrange expansion, states that any function of can be expressed as a power series in which converges for sufficiently small and has the form(2)The theorem can also be stated as follows. Let and where , then(3)(4)Expansions of this form were first considered by Lagrange (1770; 1868, pp. 680-693).
Given a series of the formthe notation is used to indicate the coefficient (Sedgewick and Flajolet 1996). This corresponds to the Wolfram Language functions Coefficient[A[z], z, k] and SeriesCoefficient[series, k].
Bürmann's theorem deals with the expansion of functions in powers of another function. Let be a function of which is analytic in a closed region , of which is an interior point, and let . Suppose also that . Then Taylor's theorem gives the expansion(1)and, if it is legitimate to revert this series, the expression(2)is obtained which expresses as an analytic function of the variable for sufficiently small values of . If is then analytic near , it follows that is an analytic function of when is sufficiently small, and so there will be an expansion in the form(3)(Whittaker and Watson 1990, p. 129).The actual coefficients in the expansion are given by the following theorem, generally known as Bürmann's theorem (Whittaker and Watson 1990, p. 129). Let be a function of defined by the equation(4)Then an analytic function can, in a certain domain of values of , be expanded in the form(5)where the remainder term is(6)and is a contour..