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  2. Euler's identity - Wikipedia

    en.wikipedia.org/wiki/Euler's_identity

    It can be seen that as N gets larger (1 + ⁠ iπ / N ⁠) N approaches a limit of −1. Euler's identity asserts that e i π {\displaystyle e^{i\pi }} is equal to −1. The expression e i π {\displaystyle e^{i\pi }} is a special case of the expression e z {\displaystyle e^{z}} , where z is any complex number .

  3. atan2 - Wikipedia

    en.wikipedia.org/wiki/Atan2

    atan2(y, x) returns the angle θ between the positive x-axis and the ray from the origin to the point (x, y), confined to (−π, π].Graph of ⁡ (,) over /. In computing and mathematics, the function atan2 is the 2-argument arctangent.

  4. 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]

  5. Euler's formula - Wikipedia

    en.wikipedia.org/wiki/Euler's_formula

    r = | z | = √ x 2 + y 2 is the magnitude of z and; φ = arg z = atan2(y, x). φ is the argument of z, i.e., the angle between the x axis and the vector z measured counterclockwise in radians, which is defined up to addition of 2π. Many texts write φ = tan −1 ⁠ y / x ⁠ instead of φ = atan2(y, x), but the first equation needs ...

  6. Gaussian integral - Wikipedia

    en.wikipedia.org/wiki/Gaussian_integral

    A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.

  7. Wallis product - Wikipedia

    en.wikipedia.org/wiki/Wallis_product

    Comparison of the convergence of the Wallis product (purple asterisks) and several historical infinite series for π. S n is the approximation after taking n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times.

  8. Rectangular function - Wikipedia

    en.wikipedia.org/wiki/Rectangular_function

    Plot of normalized ⁡ function (i.e. ⁡ ()) with its spectral frequency components.. The unitary Fourier transforms of the rectangular function are [2] ⁡ = ⁡ = ⁡ (), using ordinary frequency f, where is the normalized form [10] of the sinc function and ⁡ = ⁡ (/) / = ⁡ (/), using angular frequency , where is the unnormalized form of the sinc function.

  9. Transcendental number - Wikipedia

    en.wikipedia.org/wiki/Transcendental_number

    For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x 22 = 0. The golden ratio (denoted or ) is another irrational number that is not transcendental, as it is a root of the polynomial equation x 2x1 = 0.