List of mathematical series

Last updated

This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums.

Contents

Sums of powers

See Faulhaber's formula.

The first few values are:

See zeta constants.

The first few values are:

Power series

Low-order polylogarithms

Finite sums:

Infinite sums, valid for (see polylogarithm):

The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form:

Exponential function

where is the Touchard polynomials.

Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship

Modified-factorial denominators

Binomial coefficients

Harmonic numbers

(See harmonic numbers, themselves defined , and generalized to the real numbers)

Binomial coefficients

Trigonometric functions

Sums of sines and cosines arise in Fourier series.

Rational functions

Exponential function

Numeric series

These numeric series can be found by plugging in numbers from the series listed above.

Alternating harmonic series

Sum of reciprocal of factorials

Trigonometry and π

Reciprocal of tetrahedral numbers

Where

Exponential and logarithms

See also

Notes

  1. Weisstein, Eric W. "Haversine". MathWorld . Wolfram Research, Inc. Archived from the original on 2005-03-10. Retrieved 2015-11-06.
  2. 1 2 3 4 Wilf, Herbert R. (1994). generatingfunctionology (PDF). Academic Press, Inc.
  3. 1 2 3 4 "Theoretical computer science cheat sheet" (PDF).
  4. Calculate the Fourier expansion of the function on the interval :
  5. "Bernoulli polynomials: Series representations (subsection 06/02)". Wolfram Research. Retrieved 2 June 2011.
  6. Hofbauer, Josef. "A simple proof of 1 + 1/22 + 1/32 + ··· = π2/6 and related identities" (PDF). Retrieved 2 June 2011.
  7. Sondow, Jonathan; Weisstein, Eric W. "Riemann Zeta Function (eq. 52)". MathWorld—A Wolfram Web Resource.
  8. Abramowitz, Milton; Stegun, Irene (1964). "6.4 Polygamma functions". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . p.  260. ISBN   0-486-61272-4.

Related Research Articles

<span class="mw-page-title-main">Bessel function</span> Families of solutions to related differential equations

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation

<span class="mw-page-title-main">Trigonometric functions</span> Functions of an angle

In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

<span class="mw-page-title-main">Chebyshev polynomials</span> Polynomial sequence

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions:

Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form

<span class="mw-page-title-main">Inverse trigonometric functions</span> Inverse functions of sin, cos, tan, etc.

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

<span class="mw-page-title-main">Clausen function</span> Transcendental single-variable function

In mathematics, the Clausen function, introduced by Thomas Clausen, is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function.

<span class="mw-page-title-main">Theta function</span> Special functions of several complex variables

In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory.

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi. Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later.

The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up more than a century later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.

<span class="mw-page-title-main">Sinc function</span> Special mathematical function defined as sin(x)/x

In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized.

In mathematics, the Legendre chi function is a special function whose Taylor series is also a Dirichlet series, given by

In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions. This function is of great importance in the description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents and the Weber modular functions, that are used for solving equations of higher degrees.

<span class="mw-page-title-main">Lemniscate elliptic functions</span> Mathematical functions

In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others.

<span class="mw-page-title-main">Sine and cosine</span> Fundamental trigonometric functions

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted as and .

<span class="mw-page-title-main">Wrapped Cauchy distribution</span>

In probability theory and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the unit circle. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution.

<span class="mw-page-title-main">Bickley–Naylor functions</span>

In physics, engineering, and applied mathematics, the Bickley–Naylor functions are a sequence of special functions arising in formulas for thermal radiation intensities in hot enclosures. The solutions are often quite complicated unless the problem is essentially one-dimensional. These functions have practical applications in several engineering problems related to transport of thermal or neutron, radiation in systems with special symmetries. W. G. Bickley was a British mathematician born in 1893.

References