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# Leibniz formula for π

In mathematics, the Leibniz formula for pi, named after Gottfried Leibniz, states that It is also called Madhava–Leibniz series as it is a special case of a more general series expansion for the inverse tangent function, first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century, the specific case first published by Leibniz around 1676. [1]

## Abel–Plana formula

In mathematics, the Abel–Plana formula is a summation formula discovered independently by and.

## Asymptotic expansion

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) 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.

## Binomial transform

In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences.

## Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

## Decimal representation

A decimal representation of a non-negative real number r is an expression in the form of a series, traditionally written as a sum where a0 is a nonnegative integer, and a1, a2,...

## Dirichlet beta function

In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function.

## Dirichlet character

In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z. Dirichlet characters are used to define Dirichlet ''L''-functions, which are meromorphic functions with a variety of interesting analytic properties.

## Dirichlet series

In mathematics, a Dirichlet series is any series of the form where s is complex, and a_n is a complex sequence.

## Euler number

In mathematics, the Euler numbers are a sequence En of integers defined by the Taylor series expansion where is the hyperbolic cosine.

## Euler product

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.

## Euler–Maclaurin formula

In mathematics, the Euler–Maclaurin formula provides a powerful connection between integrals (see calculus) and sums.

## Gottfried Wilhelm Leibniz

Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.

## Gregory's series

Gregory's series, also known as the Madhava–Gregory series or Leibniz's series, is an infinite Taylor series expansion of the inverse tangent function.

## Infinite product

In mathematics, for a sequence of complex numbers a1, a2, a3,...

## Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains).

## Jonathan Borwein

Jonathan Michael Borwein (20 May 1951 &ndash; 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia.

## List of formulae involving π

The following is a list of significant formulae involving the mathematical constant pi.

## List of things named after Gottfried Leibniz

Gottfried Wilhelm Leibniz (1646–1716) was a German philosopher and mathematician.

Mādhava of Sangamagrāma, was a mathematician and astronomer from the town of Sangamagrama (believed to be present-day Aloor, Irinjalakuda in Thrissur District), Kerala, India.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Numerical integration

In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations.

## Pi

The number is a mathematical constant.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Rate of convergence

In numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence.

## Richardson extrapolation

In numerical analysis, Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence.

## Series acceleration

In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series.

## Shanks transformation

In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence.

## Squeeze theorem

In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, and sometimes the squeeze lemma, is a theorem regarding the limit of a function.

## Superparticular ratio

In mathematics, a superparticular ratio, also called a superparticular number or epimoric ratio, is the ratio of two consecutive integer numbers.

## Van Wijngaarden transformation

In mathematics and numerical analysis, in order to accelerate convergence of an alternating series, Euler's transform can be computed as follows.

## References

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