Roger CotesRoger Cotes (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the Principia, before publication. He also invented the quadrature formulas known as Newton–Cotes formulas, and made a geometric argument that can be interpreted as a logarithmic version of Euler's formula. He was the first Plumian Professor at Cambridge University from 1707 until his death. Cotes was born in Burbage, Leicestershire.
Cis (mathematics)is a mathematical notation defined by cis x = cos x + i sin x, where cos is the cosine function, i is the imaginary unit and sin is the sine function. The notation is less commonly used in mathematics than Euler's formula, eix, which offers an even shorter notation for cos x + i sin x, but cis(x) is widely used as a name for this function in software libraries. The cis notation is a shorthand for the combination of functions on the right-hand side of Euler's formula: where i2 = −1. So, i.e.
Atan2In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, is the angle measure (in radians, with ) between the positive -axis and the ray from the origin to the point in the Cartesian plane. Equivalently, is the argument (also called phase or angle) of the complex number The function first appeared in the programming language Fortran in 1961. It was originally intended to return a correct and unambiguous value for the angle θ in converting from Cartesian coordinates (x, y) to polar coordinates (r, θ).
Ratio testIn mathematics, the ratio test is a test (or "criterion") for the convergence of a series where each term is a real or complex number and an is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. The usual form of the test makes use of the limit The ratio test states that: if L < 1 then the series converges absolutely; if L > 1 then the series diverges; if L = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.
Euler's identityIn mathematics, Euler's identity (also known as Euler's equation) is the equality where e is Euler's number, the base of natural logarithms, i is the imaginary unit, which by definition satisfies i2 = −1, and π is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula when evaluated for x = π. Euler's identity is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics.
Positive real numbersIn mathematics, the set of positive real numbers, is the subset of those real numbers that are greater than zero. The non-negative real numbers, also include zero. Although the symbols and are ambiguously used for either of these, the notation or for and or for has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians. In a complex plane, is identified with the positive real axis, and is usually drawn as a horizontal ray.
Characterizations of the exponential functionIn mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent of and equivalent to each other. As a special case of these considerations, it will be demonstrated that the three most common definitions given for the mathematical constant e are equivalent to each other.
Proofs of trigonometric identitiesThere are several equivalent ways for defining trigonometric functions, and the proof of the trigonometric identities between them depend on the chosen definition. The oldest and somehow the most elementary definition is based on the geometry of right triangles. The proofs given in this article use this definition, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions.
Pythagorean trigonometric identityThe Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions. The identity is As usual, means . Any similar triangles have the property that if we select the same angle in all of them, the ratio of the two sides defining the angle is the same regardless of which similar triangle is selected, regardless of its actual size: the ratios depend upon the three angles, not the lengths of the sides.
