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  2. Euler's identity - Wikipedia

    en.wikipedia.org/wiki/Euler's_identity

    Euler's identity is also a special case of the more general identity that the n th roots of unity, for n > 1, add up to 0: = = Euler's identity is the case where n = 2. A similar identity also applies to quaternion exponential: let {i, j, k} be the basis quaternions; then,

  3. Euler's formula - Wikipedia

    en.wikipedia.org/wiki/Euler's_formula

    Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.

  4. Euler's totient function - Wikipedia

    en.wikipedia.org/wiki/Euler's_totient_function

    Thus, it is often called Euler's phi function or simply the phi function. In 1879, J. J. Sylvester coined the term totient for this function, [14] [15] so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient. Jordan's totient is a generalization of Euler's. The cototient of n is defined as n − φ(n).

  5. Contributions of Leonhard Euler to mathematics - Wikipedia

    en.wikipedia.org/wiki/Contributions_of_Leonhard...

    Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and made distinct contributions to the Lagrange's four-square theorem. He also invented the totient function φ(n) which assigns to a positive integer n the number of positive integers less than n and coprime to n.

  6. List of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/List_of_trigonometric...

    In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles .

  7. Rotation matrix - Wikipedia

    en.wikipedia.org/wiki/Rotation_matrix

    Euler angles can also be used, though not with each angle uniformly distributed (Murnaghan 1962; Miles 1965). For the axis–angle form, the axis is uniformly distributed over the unit sphere of directions, S 2, while the angle has the nonuniform distribution over [0, π] noted previously .

  8. Basel problem - Wikipedia

    en.wikipedia.org/wiki/Basel_problem

    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, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2]

  9. Euler function - Wikipedia

    en.wikipedia.org/wiki/Euler_function

    The coefficient in the formal power series expansion for / gives the number of partitions of k.That is, = = ()where is the partition function.. The Euler identity, also known as the Pentagonal number theorem, is