# Series

## Series Topics

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### Critical index

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).

### Nyquist frequency

In order to recover all Fourier components of a periodic waveform, it is necessary to use a sampling rate at least twice the highest waveform frequency. The Nyquist frequency, also called the Nyquist limit, is the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal, i.e.,

### Harmonic expansion

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, ...]

### Laurent series

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..

### Power series

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..

### Lagrange remainder

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).

### Picone's theorem

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

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 .

### Taylor series

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..

### Formal power series

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

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.

### False logarithmic series

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.

### Mercator series

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

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).

### Maclaurin series

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..

### Snake oil method

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).

### Logarithmic series

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.

### Logarithmic number

A coefficient of the Maclaurinseries of(OEIS A002206 and A002207), the multiplicative inverse of the Mercator series function .

### Bernstein expansion

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 .

### Schl&ouml;milch's series

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)

### Linear approximation

A linear approximation to a function at a point can be computed by taking the first term in the Taylor series

### Robertson conjecture

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.

### Leibniz series

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).

### Backhouse's constant

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.

### Cookson hills series

The Cookson Hills series is the series similar to the Flint Hills series, but with numerator instead of :(Pickover 2002, p. 268). It is not known if this series converges since can have sporadic large values. The plots above show its behavior up to . The positive integer values of giving incrementally largest values of are given by 1, 2, 5, 8, 11, 344, 699, 1054, 1409, 1764, 2119, ... (OEIS A004112), corresponding to the values 1.85082, 2.403, 3.52532, 6.87285, 225.953, 227.503, ....

### Tauberian theorem

A Tauberian theorem is a theorem that deduces the convergence of an series on the basis of the properties of the function it defines and any kind of auxiliary hypothesis which prevents the general term of the series from converging to zero too slowly. Hardy (1999, p. 46) states that "a 'Tauberian' theorem may be defined as a corrected form of the false converse of an 'Abelian theorem.' "Wiener's Tauberian theorem states that if , then the translates of span a dense subspace iff the Fourier transform is nonzero everywhere. This theorem is analogous with the theorem that if (for a Banach algebra with a unit), then spans the whole space if and only if the Gelfand transform is nonzero everywhere.

### Harmonic series of primes

Like the entire harmonic series, the harmonicseries(1)taken over all primes also diverges, as first shown by Euler in 1737 (Nagell 1951, p. 59; Hardy and Wright 1979, pp. 17 and 22; Wells 1986, p. 41; Havil 2003, pp. 28-31), although it does so very slowly. The sum exceeds 1, 2, 3, ... after 3, 59, 361139, ... (OEIS A046024) primes.Its asymptotic behavior is given by(2)where(3)(OEIS A077761) is the Mertens constant (Hardy and Wright 1979, p. 351; Hardy 1999, p. 50; Havil 2003, p. 64).

### Cauchy remainder

The Cauchy remainder is a different form of the remainder term than the Lagrange remainder. The Cauchy remainder after terms of the Taylor series for a function expanded about a point is given bywhere (Hamilton 1952).Note that the Cauchy 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).

### Series multisection

If(1)then(2)is given by(3)where .When applied to the generating function(4)it gives the identity(5)with integers (and where the sum can be taken only up to ).Other multisection examples are given by Somos (2006).

### Harmonic series

The series(1)is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function . The divergence, however, is very slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme (ca. 1323-1382), but was mislaid for several centuries (Havil 2003, p. 23; Derbyshire 2004, pp. 9-10). The result was proved again by Pietro Mengoli in 1647, by Johann Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter (Derbyshire 2004, pp. 9-10).Progressions of the form(2)are also sometimes called harmonic series (Beyer 1987).Oresme's proof groups the harmonic terms by taking 2, 4, 8, 16, ... terms (after the first two) and noting that each such block has a sum larger than 1/2,(3)(4)and since an infinite sum of 1/2's diverges, so does the harmonic series.The generalization of the harmonic series(5)is known as the Riemann zeta function.The sum of the first few terms of..

### Series bias

The bias of a series is defined asA series is geometric iff . A series is artistic iff the bias is constant.

### Fej&eacute;r's integral

The integralwhich gives the th Cesàro mean of the Fourier series of .

### Series

A series is an infinite ordered set of terms combined together by the addition operator. The term "infinite series" is sometimes used to emphasize the fact that series contain an infinite number of terms. The order of the terms in a series can matter, since the Riemann series theorem states that, by a suitable rearrangement of terms, a so-called conditionally convergent series may be made to converge to any desired value, or to diverge.Conditions for convergence of a series can be determined in the Wolfram Language using SumConvergence[a, n].If the difference between successive terms of a series is a constant, then the series is said to be an arithmetic series. A series for which the ratio of each two consecutive terms is a constant function of the summation index is called a geometric series. The more general case of the ratio a rational function of produces a series called a hypergeometric series.A series may converge to a definite value,..

### Schl&ouml;milch remainder

A Taylor series remainder formula that gives after terms of the seriesfor and any (Blumenthal 1926, Beesack 1966), which Blumenthal (1926) ascribes to Roche (1858). The choices and give the Lagrange and Cauchy remainders, respectively (Beesack 1966).

### Glasser's master theorem

The identity(1)holds for any integrable function and of the form(2)with , , and arbitrary constants (Glasser 1983). Here, denotes a Cauchy principal value. This generalized the result known to Cauchy that(3)where .

### Binomial theorem

There are several closely related results that are variously known as the binomial theorem depending on the source. Even more confusingly a number of these (and other) related results are variously known as the binomial formula, binomial expansion, and binomial identity, and the identity itself is sometimes simply called the "binomial series" rather than "binomial theorem."The most general case of the binomial theorem is the binomialseries identity(1)where is a binomial coefficient and is a real number. This series converges for an integer, or . This general form is what Graham et al. (1994, p. 162). Arfken (1985, p. 307) calls the special case of this formula with the binomial theorem.When is a positive integer , the series terminates at and can be written in the form(2)This form of the identity is called the binomial theorem by Abramowitz and Stegun (1972, p. 10).The differing terminologies are..

### Binomial series

There are several related series that are known as the binomial series.The most general is(1)where is a binomial coefficient and is a real number. This series converges for an integer, or (Graham et al. 1994, p. 162). When is a positive integer , the series terminates at and can be written in the form(2)The theorem that any one of these (or several other related forms) holds is knownas the binomial theorem.Special cases give the Taylor series(3)(4)where is a Pochhammer symbol and . Similarly,(5)(6)which is the so-called negative binomial series.In particular, the case gives(7)(8)(9)(OEIS A001790 and A046161), where is a double factorial and is a binomial coefficient.The binomial series has the continued fractionrepresentation(10)(Wall 1948, p. 343).

### Geometric series

A geometric series is a series for which the ratio of each two consecutive terms is a constant function of the summation index . The more general case of the ratio a rational function of the summation index produces a series called a hypergeometric series.For the simplest case of the ratio equal to a constant , the terms are of the form . Letting , the geometric sequence with constant is given by(1)is given by(2)Multiplying both sides by gives(3)and subtracting (3) from (2) then gives(4)(5)so(6)For , the sum converges as ,in which case(7)Similarly, if the sums are taken starting at instead of ,(8)(9)the latter of which is valid for .

### Negative binomial series

The series which arises in the binomial theorem for negative integer ,(1)(2)for .For , the negative binomial series simplifies to(3)

### Flint hills series

The Flint Hills series is the series(Pickover 2002, p. 59). It is not known if this series converges, since can have sporadic large values. The plots above show its behavior up to . The positive integer values of giving incrementally largest values of are given by 1, 3, 22, 333, 355, 103993, ... (OEIS A046947), which are precisely the numerators of the convergents of , corresponding to the values 1.1884, 7.08617, 112.978, 113.364, 33173.7, ....Alekseyev (2011) has shown that the question of the convergence of the Flint Hill series is related to the irrationality measure of , and in particular, convergence would imply , which is much stronger than the best currently known upper bound.

### Bessel function neumann series

A series of the form(1)where is a real and is a Bessel function of the first kind. Special cases are(2)where is the gamma function, and(3)where(4)and is the floor function.

### Exponential sum function

The exponential sum function , sometimes also denoted , is defined by(1)(2)where is the upper incomplete gamma function and is the (complete) gamma function.

### Multinomial series

A multinomial series is generalization of the binomial series discovered by Johann Bernoulli and Leibniz. The multinomial series arises in a generalization of the binomial distribution called the multinomial distribution.It is given bywhere .For example,

### Arithmetic series

An arithmetic series is the sum of a sequence , , 2, ..., in which each term is computed from the previous one by adding (or subtracting) a constant . Therefore, for ,(1)The sum of the sequence of the first terms is then given by(2)(3)(4)(5)(6)Using the sum identity(7)then gives(8)Note, however, that(9)so(10)or times the arithmetic mean of the first and last terms! This is the trick Gauss used as a schoolboy to solve the problem of summing the integers from 1 to 100 given as busy-work by his teacher. While his classmates toiled away doing the addition longhand, Gauss wrote a single number, the correct answer(11)on his slate (Burton 1989, pp. 80-81; Hoffman 1998, p. 207). When the answers were examined, Gauss's proved to be the only correct one.

### Lambert series

A Lambert series is a series of the form(1)for . Then(2)(3)where(4)The particular case is sometimes denoted(5)(6)(7)for (Borwein and Borwein 1987, pp. 91 and 95), where is a q-polygamma function. Special cases and related sums include(8)(9)(10)(11)(12)(Borwein and Borwein 1997, pp. 91-92), which arise in the reciprocalFibonacci and reciprocal Lucas constants.Some beautiful series of this type include(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)where is the Möbius function, is the totient function, is the number of divisors of , is the q-polygamma function, is the divisor function, is the number of representations of in the form where and are rational integers (Hardy and Wright 1979), is a Jacobi elliptic function (Bailey et al. 2006), is the Liouville function, and is the least significant bit of ...

### Arithmetic progression

An arithmetic progression, also known as an arithmetic sequence, is a sequence of numbers such that the differences between successive terms is a constant .An arithmetic progression can be generated in the Wolfram Language using the command Range[, , d].

### Kapteyn series

A Kapteyn series is a series of the form(1)where is a Bessel function of the first kind. Examples include Kapteyn's original series(2)and(3)

### Alternating series

A series of the form(1)or(2)where .A series with positive terms can be converted to an alternating series using(3)where(4)Explicit values for alternating series include(5)(6)(7)(8)where is Apéry's constant, and sums of the form (6) through (8) are special cases of the Dirichlet eta function.The following alternating series converges, but a closed form is apparently not known,(9)(10)(11)(OEIS A114884).

### Parseval's theorem

If a function has a Fourier series given by(1)then Bessel's inequality becomes an equalityknown as Parseval's theorem. From (1),(2)Integrating(3)so(4)For a generalized Fourier series of a complete orthogonal system , an analogous relationship holds.For a complex Fourierseries,(5)

### Fourier cosine series

If is an even function, then and the Fourier series collapses to(1)where(2)(3)(4)(5)where the last equality is true because(6)Letting the range go to ,(7)(8)

### Generalized fourier series

A generalized Fourier series is a series expansion of a function based on the special properties of a complete orthogonal system of functions. The prototypical example of such a series is the Fourier series, which is based of the biorthogonality of the functions and (which form a complete biorthogonal system under integration over the range . Another common example is the Laplace series, which is a double series expansion based on the orthogonality of the spherical harmonics over and .Given a complete orthogonal system of univariate functions over the interval , the functions satisfy an orthogonality relationship of the form(1)over a range , where is a weighting function, are given constants and is the Kronecker delta. Now consider an arbitrary function . Write it as a series(2)and plug this into the orthogonality relationships to obtain(3)Note that the order of integration and summation has been reversed in deriving the above equations...

### Fourier sine series

If is an odd function, then and the Fourier series collapses to(1)where(2)(3)for , 2, 3, .... The last equality is true because(4)(5)Letting the range go to ,(6)

### Fourier series--triangle wave

Consider a symmetric triangle wave of period . Since the function is odd,(1)(2)and(3)(4)(5)(6)The Fourier series for the triangle wave is therefore(7)Now consider the asymmetric triangle wave pinned an -distance which is ()th of the distance . The displacement as a function of is then(8)The coefficients are therefore(9)(10)(11)Taking gives the same Fourier series as before.

### Laplace series

The spherical harmonics form a complete orthogonal system, so an arbitrary real function can be expanded in terms of complex spherical harmonics by(1)or in terms of real spherical harmonics by(2)The representation of a function as such a double series is a generalized Fourier series known as a Laplace series.The process of determining the coefficients in (1) is analogous to that of determining the coefficients in a Fourier series, i.e., multiply both sides of (1) by , integrate, and use the orthogonality relationship (◇) to obtain(3)The following sequence of plots shows successive approximations to the function , which is illustrated in the final plot. Laplace series can also be written in terms real spherical harmonic as(4)Proceed as before, using the orthogonality relationships(5)So and are given by(6)(7)..

### Fourier series--square wave

Consider a square wave of length . Over the range , this can be written as(1)where is the Heaviside step function. Since , the function is odd, so , and(2)reduces to(3)(4)(5)(6)The Fourier series is therefore(7)

### Lanczos sigma factor

Writing a Fourier series aswhere is the last term, reduces the Gibbs phenomenon. The terms are the known as the Lanczos factors. Note that (Acton 1990, p. 228) incorrectly lists the upper index of the sum as , while Hamming (1986, p. 535) gives the correct form reproduced above.

### Fourier series--semicircle

Given a semicircular hump(1)(2)the Fourier coefficients are(3)(4)(5)where is a Bessel function of the first kind, so the Fourier series is therefore(6)

### Dini's test

A test for the convergence of Fourier series. Letthen ifis finite, the Fourier series converges to at .

### Fourier series--sawtooth wave

Consider a string of length plucked at the right end and fixed at the left. The functional form of this configuration is(1)The components of the Fourier series are thereforegiven by(2)(3)(4)(5)(6)(7)(8)(9)The Fourier series is therefore given by(10)(11)(12)

### Isospectral manifolds

Roughly speaking, isospectral manifolds are drums that sound the same, i.e., have the same eigenfrequency spectrum. Two drums with differing area, perimeter, or genus can always be distinguished. However, Kac (1966) asked if it was possible to construct differently shaped drums which have the same eigenfrequency spectrum. This question was answered in the affirmative by Gordon et al. (1992). Two such isospectral manifolds (which are 7-polyaboloes) are shown in the left figure above (Cipra 1992). The right figure above shows another pair obtained from the original ones by making a simple geometric substitution.Another example of isospectral manifolds is the pair of polyabolo configurations known as bilby (left figure) and hawk (right figure). The figures above show scaled displacements for a number of eigenmodes of these manifolds (M. Trott, pers.comm., Oct. 8, 2003).Furthermore, pairs of separate drums (having the..

### Fourier series--power

For a power function with on the interval and periodic with period , the coefficients of the Fourier series are given by(1)(2)(3)where is a generalized hypergeometric function.

### Fourier series

A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. Examples of successive approximations to common functions using Fourier series are illustrated above.In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function..

### Foxtrot series

The "Foxtrot series" is a mathematical sum that appeared in the June 2, 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007). It arose from a convergence testing problem in a calculus book by Anton, but was inadvertently converted into a summation problem on an alleged final exam by the strip's author:(1)The sum can be done using partial fraction decomposition to obtain(2)(3)(4)(5)(OEIS A127198), where and the last sums have been done in terms of the digamma function and symbolically simplified.

### Euler transform

There are (at least) three types of Euler transforms (or transformations). The first is a set of transformations of hypergeometric functions, called Euler's hypergeometric transformations.The second type of Euler transform is a technique for series convergence improvement which takes a convergent alternating series(1)into a series with more rapid convergence to the same value to(2)where the forward difference is defined by(3)(Abramowitz and Stegun 1972; Beeler et al. 1972). Euler's hypergeometric and convergence improvement transformations are related by the fact that when is taken in the second of Euler's hypergeometric transformations(4)where is a hypergeometric function, it gives Euler's convergence improvement transformation of the series (Abramowitz and Stegun 1972, p. 555).The third type of Euler transform is a relationship between certain types of integer sequences (Sloane and Plouffe 1995, pp. 20-21)...

### Series reversion

Series reversion is the computation of the coefficients of the inverse function given those of the forward function. For a function expressed in a series with no constant term (i.e., ) as(1)the series expansion of the inverse series is given by(2)By plugging (2) into (1), the following equationis obtained(3)Equating coefficients then gives(4)(5)(6)(7)(8)(9)(10)(Dwight 1961, Abramowitz and Stegun 1972, p. 16).Series reversion is implemented in the Wolfram Language as InverseSeries[s, x], where is given as a SeriesData object. For example, to obtain the terms shown above, With[{n = 7}, CoefficientList[ InverseSeries[SeriesData[x, 0, Array[a, n], 1, n + 1, 1]], x] ]A derivation of the explicit formula for the th term is given by Morse and Feshbach (1953),(11)where(12)

### Euler sum

In response to a letter from Goldbach, Euler considered sums ofthe form(1)(2)with and and where is the Euler-Mascheroni constant and is the digamma function. Euler found explicit formulas in terms of the Riemann zeta function for with , and E. Au-Yeung numerically discovered(3)where is the Riemann zeta function, which was subsequently rigorously proven true (Borwein and Borwein 1995). Sums involving can be re-expressed in terms of sums the form via(4)(5)(6)and(7)where is defined below.Bailey et al. (1994) subsequently considered sums ofthe forms(8)(9)(10)(11)(12)(13)(14)(15)where and have the special forms(16)(17)(18)where is a generalized harmonic number.A number of these sums can be expressed in terms of the multivariatezeta function, e.g.,(19)(Bailey et al. 2006a, p. 39, sign corrected; Bailey et al. 2006b).Special cases include(20)(P. Simone, pers. comm., Aug. 30, 2004).Analytic single..

### Teixeira's theorem

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).

### Lagrange inversion theorem

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).

### Weierstrass's double series theorem

Let all of the functions(1)with , 1, 2, ..., be regular at least for , and let(2)(3)be uniformly convergent for for every . Then the coefficients in any column form a convergent series. Furthermore, setting(4)for , 1, 2, ..., it then follows that(5)is the power series for , which converges at least for .

### Faltung

The term faltung is variously used to mean convolutionand a function of bilinear forms.Let and be bilinear forms(1)(2)and suppose that and are bounded in with bounds and . Then(3)where the series(4)is absolutely convergent, is called the faltung of and . is bounded in , and its bound does not exceed .

### Double series

A double sum is a series having terms depending on twoindices,(1)A finite double series can be written as a product of series(2)(3)(4)(5)An infinite double series can be written in terms of a single series(6)by reordering as follows,(7)(8)(9)(10)Many examples exists of simple double series that cannot be computed analytically,such as the Erdős-Borwein constant(11)(12)(13)(OEIS A065442), where is a q-polygamma function.Another series is(14)(15)(OEIS A091349), where is a harmonic number and is a cube root of unity.A double series that can be done analytically is given by(16)where is the Riemann zeta function zeta(2) (B. Cloitre, pers. comm., Dec. 9, 2004).The double series(17)can be evaluated by interchanging and and averaging,(18)(19)(20)(21)(Borwein et al. 2004, p. 54).Identities involving double sums include the following:(22)where(23)is the floor function, and(24)Consider the series(25)over..

### Leibniz criterion

Also known as the alternating series test.Given a serieswith , if is monotonic decreasing as andthen the series converges.

### Kummer's test

Given a series of positive terms and a sequence of finite positive constants , let1. If , the series converges. 2. If and the series diverges, the series diverges. 3. If , the series may converge or diverge.The test is a general case of Bertrand's test, the root test, Gauss's test, and Raabe's test. With and , the test becomes Raabe's test.

### Conditional convergence

A series is said to be conditionally convergent iff it is convergent, the series of its positive terms diverges to positive infinity, and the series of its negative terms diverges to negative infinity.Examples of conditionally convergent series include the alternating harmonic seriesand the logarithmic serieswhere is the Euler-Mascheroni constant.The Riemann series theorem states that, by a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge. The Riemann series theorem can be proved by first taking just enough positive terms to exceed the desired limit, then taking just enough negative terms to go below the desired limit, and iterating this procedure. Since the terms of the original series tend to zero, the rearranged series converges to the desired limit. A slight variation works to make the new series diverge to positive infinity or to negative infinity...

### Weak convergence

Weak convergence is usually either denoted or . A sequence of vectors in an inner product space is called weakly convergent to a vector in ifEvery strongly convergent sequence is also weakly convergent (but the opposite does not usually hold). This can be seen as follows. Consider the sequence that converges strongly to , i.e., as . Schwarz's inequality now givesThe definition of weak convergence is therefore satisfied.

### Comparison test

Let and be a series with positive terms and suppose , , .... 1. If the bigger series converges, then thesmaller series also converges. 2. If the smaller series diverges, then the biggerseries also diverges.

### Vitali's convergence theorem

Let be a sequence of functions, each regular in a region , let for every and in , and let tend to a limit as at a set of points having a limit point inside . Then tends uniformly to a limit in any region bounded by a contour interior to , the limit therefore being an analytic function of .

### Integral test

Let be a series with positive terms and let be the function that results when is replaced by in the formula for . If is decreasing and continuous for and(1)then(2)and(3)both converge or diverge, where . The test is also called the Cauchy integral test or Maclaurin integral test.

### Uniform convergence

A sequence of functions , , 2, 3, ... is said to be uniformly convergent to for a set of values of if, for each , an integer can be found such that(1)for and all .A series converges uniformly on if the sequence of partial sums defined by(2)converges uniformly on .To test for uniform convergence, use Abel's uniform convergence test or the Weierstrass M-test. If individual terms of a uniformly converging series are continuous, then the following conditions are satisfied. 1. The series sum(3)is continuous. 2. The series may be integrated term by term(4)For example, a power series is uniformly convergent on any closed and bounded subset inside its circle of convergence. 3. The situation is more complicated for differentiation since uniform convergence of does not tell anything about convergence of . Suppose that converges for some , that each is differentiable on , and that converges uniformly on . Then converges uniformly on to a function , and for each..

### Herschfeld's convergence theorem

For real, nonnegative terms and real with , the expressionconverges iff is bounded.

### Cauchy criterion

A necessary and sufficient condition for a sequence to converge. The Cauchy criterion is satisfied when, for all , there is a fixed number such that for all .

### Strong convergence

Strong convergence is the type of convergence usually associated with convergence of a sequence. More formally, a sequence of vectors in a normed space (and, in particular, in an inner product space )is called convergent to a vector in if

### Gauss's test

If and given a bounded function of as , express the ratio of successive terms asfor . The series converges for and diverges for (Arfken 1985, p. 287; Courant and John 1999, p. 567).Equivalently, with the same conditions as above, giventhe series converges absolutely iff (Zwillinger 1996, p. 32).

### Cauchy condensation test

Let be a series of positive terms with . Then converges iffconverges.

### Root test

Let be a series with positive terms, and let1. If , the series converges. 2. If or , the series diverges. 3. If , the series may converge or diverge. This test is also called the Cauchy root test (Zwillinger 1996, p. 32).

### Bertrand's test

A convergence test also called "de Morgan's and Bertrand's test." If the ratio of terms of a series can be written in the formthen the series converges if and diverges if , where is the lower limit and is the upper limit.

### Riemann series theorem

By a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge.For example,(1)(2)(3)converges to , but the same series can be rearranged to(4)(5)(6)(7)so the series now converges to half of itself.

### Euler's series transformation

Euler's series transformation is a transformation that sometimes accelerates the rate of convergence for an alternating series. Given a convergent alternating series with sum(1)Abramowitz and Stegun (1972, p. 16) define Euler's transformation as(2)where is the forward difference operator(3)and is a binomial coefficient.An alternate formulation due to Knopp (1990, p. 244) instead defines the transformation as(4)where is the backward difference operator(5)Knopp (1990, p. 263) gives examples of different types of convergence behavior upon application of the transformation:(6)gives faster convergence,(7)gives same rate of convergence, and(8)gives slower convergence.To see why the Euler transformation works, consider Knopp's convention for difference operator and write(9)(10)Now repeat the process on the series in brackets to obtain(11)and continue to infinity. This proves each finite step in..

Suppose the harmonic series converges to :Then rearranging the terms in the sum giveswhich is a contradiction.

### Ratio test

Let be a series with positive terms and supposeThen 1. If , the series converges. 2. If or , the series diverges. 3. If , the series may converge or diverge. The test is also called the Cauchy ratio test or d'Alembert ratio test.

### Divergent series

A series which is not convergent. Series may diverge by marching off to infinity or by oscillating. Divergent series have some curious properties. For example, rearranging the terms of gives both and .The Riemann series theorem states that, by a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge.No less an authority than N. H. Abel wrote "The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever" (Gardner 1984, p. 171; Hoffman 1998, p. 218). However, divergent series can actually be "summed" rigorously by using extensions to the usual summation rules (e.g., so-called Abel and Cesàro sums). For example, the divergent series has both Abel and Cesàro sums of 1/2...

A power series will converge only for certain values of . For instance, converges for . In general, there is always an interval in which a power series converges, and the number is called the radius of convergence (while the interval itself is called the interval of convergence). The quantity is called the radius of convergence because, in the case of a power series with complex coefficients, the values of with form an open disk with radius .A power series always converges absolutely within its radius of convergence. This can be seen by fixing and supposing that there exists a subsequence such that is unbounded. Then the power series does not converge (in fact, the terms are unbounded) because it fails the limit test. Therefore, for with , the power series does not converge, where(1)(2)and denotes the supremum limit.Conversely, suppose that . Then for any radius with , the terms satisfy(3)for large enough (depending on ). It is sufficient to fix a value..

### Raabe's test

Given a series of positive terms and a sequence of positive constants , use Kummer's test(1)with , giving(2)(3)Defining(4)then gives Raabe's test: 1. If , the series converges. 2. If , the series diverges. 3. If , the series may converge or diverge.

### Absolute convergence

A series is said to converge absolutely if the series converges, where denotes the absolute value. If a series is absolutely convergent, then the sum is independent of the order in which terms are summed. Furthermore, if the series is multiplied by another absolutely convergent series, the product series will also converge absolutely.

### Dirichlet's test

Let(1)where is independent of . Then if and(2)it follows that(3)converges.

### Abel's uniform convergence test

Let be a sequence of functions. If 1. can be written , 2. is convergent, 3. is a monotonic decreasing sequence (i.e., ) for all , and 4. is bounded in some region (i.e., for all ) then, for all , the series converges uniformly.

### Limit test

The limit test, also sometimes known as the th term test, says that if or this limit does not exist as tends to infinity, then the series does not converge. For example, does not converge by the limit test. The limit test is inconclusive when the limit is zero.

### Convergent series

A series is said to be convergent if it approaches some limit(D'Angelo and West 2000, p. 259).Formally, the infinite series is convergent if the sequence of partial sums(1)is convergent. Conversely, a series is divergent if the sequence of partial sums is divergent. If and are convergent series, then and are convergent. If , then and both converge or both diverge. Convergence and divergence are unaffected by deleting a finite number of terms from the beginning of a series. Constant terms in the denominator of a sequence can usually be deleted without affecting convergence. All but the highest power terms in polynomials can usually be deleted in both numerator and denominator of a series without affecting convergence.If the series formed by taking the absolute values of its terms converges (in which case it is said to be absolutely convergent), then the original series converges.Conditions for convergence of a series can be determined..

### Limit comparison test

Let and be two series with positive terms and supposeIf is finite and , then the two series both converge or diverge.

### Abel's convergence theorem

Given a Taylor series(1)where the complex number has been written in the polar form , examine the real and imaginary parts(2)(3)Abel's theorem states that, if and are convergent, then(4)Stated in words, Abel's theorem guarantees that, if a real power series converges for some positive value of the argument, the domain of uniform convergence extends at least up to and including this point. Furthermore, the continuity of the sum function extends at least up to and including this point.

### Borel's expansion

Let be any function for which the integralconverges. Then the expansionwhere is the gamma function, is usually an asymptotic series for .

### Stirling's series

The asymptotic series for the gammafunction is given by(1)(OEIS A001163 and A001164).The coefficient of can given explicitly by(2)where is the number of permutations of with permutation cycles all of which are (Comtet 1974, p. 267). Another formula for the s is given by the recurrence relation(3)with , then(4)where is the double factorial (Borwein and Corless 1999).The series for is obtained by adding an additional factor of ,(5)(6)The expansion of is what is usually called Stirling's series. It is given by the simple analytic expression(7)(8)(OEIS A046968 and A046969), where is a Bernoulli number. Interestingly, while the numerators in this expansion are the same as those of for the first several hundred terms, they differ at , 1185, 1240, 1269, 1376, 1906, 1910, ... (OEIS A090495), with the corresponding ratios being 37, 103, 37, 59, 131, 37, 67, ... (OEIS A090496)...

### Asymptotic series

An asymptotic series is a series expansion of a function in a variable which may converge or diverge (Erdélyi 1987, p. 1), but whose partial sums can be made an arbitrarily good approximation to a given function for large enough . To form an asymptotic series of(1)take(2)where(3)The asymptotic series is defined to have the properties(4)(5)Therefore,(6)in the limit . If a function has an asymptotic expansion, the expansion is unique. The symbol is also used to mean directly similar.Asymptotic series can be computed by doing the change of variable and doing a series expansion about zero. Many mathematical operations can be performed on asymptotic series. For example, asymptotic series can be added, subtracted, multiplied, divided (as long as the constant term of the divisor is nonzero), and exponentiated, and the results are also asymptotic series (Gradshteyn and Ryzhik 2000, p. 20)...

### Abelian theorem

A theorem which asserts that if a sequence or function behaves regularly, then some average of it behaves regularly. For example,impliesfor any . The converse is false, but can be made into a correct Tauberian theorem if is subjected to an appropriate additional condition (Hardy 1999, p. 46).

### Coefficient notation

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].

### B&uuml;rmann's theorem

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..