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  2. Jordan's lemma - Wikipedia

    en.wikipedia.org/wiki/Jordan's_lemma

    The path C is the concatenation of the paths C 1 and C 2.. Jordan's lemma yields a simple way to calculate the integral along the real axis of functions f(z) = e i a z g(z) holomorphic on the upper half-plane and continuous on the closed upper half-plane, except possibly at a finite number of non-real points z 1, z 2, …, z n.

  3. Summation by parts - Wikipedia

    en.wikipedia.org/wiki/Summation_by_parts

    The formula for an integration by parts is () ′ = [() ()] ′ (). Beside the boundary conditions , we notice that the first integral contains two multiplied functions, one which is integrated in the final integral ( g ′ {\displaystyle g'} becomes g {\displaystyle g} ) and one which is differentiated ( f {\displaystyle f} becomes f ...

  4. Lifting-the-exponent lemma - Wikipedia

    en.wikipedia.org/wiki/Lifting-the-exponent_lemma

    The exact origins of the LTE lemma are unclear; the result, with its present name and form, has only come into focus within the last 10 to 20 years. [1] However, several key ideas used in its proof were known to Gauss and referenced in his Disquisitiones Arithmeticae. [2]

  5. Farkas' lemma - Wikipedia

    en.wikipedia.org/wiki/Farkas'_lemma

    The lemma says that exactly one of the following two statements must be true (depending on b 1 and b 2): There exist x 1 ≥ 0, x 2 ≥ 0 such that 6x 1 + 4x 2 = b 1 and 3x 1 = b 2, or; There exist y 1, y 2 such that 6y 1 + 3y 2 ≥ 0, 4y 1 ≥ 0, and b 1 y 1 + b 2 y 2 < 0. Here is a proof of the lemma in this special case:

  6. Lemma (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Lemma_(mathematics)

    In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem " or an "auxiliary theorem".

  7. Burnside's lemma - Wikipedia

    en.wikipedia.org/wiki/Burnside's_lemma

    Burnside's lemma can compute the number of rotationally distinct colourings of the faces of a cube using three colours.. Let X be the set of 3 6 possible face color combinations that can be applied to a fixed cube, and let the rotation group G of the cube act on X by moving the colored faces: two colorings in X belong to the same orbit precisely when one is a rotation of the other.

  8. Hensel's lemma - Wikipedia

    en.wikipedia.org/wiki/Hensel's_lemma

    Hensel's original lemma concerns the relation between polynomial factorization over the integers and over the integers modulo a prime number p and its powers. It can be straightforwardly extended to the case where the integers are replaced by any commutative ring, and p is replaced by any maximal ideal (indeed, the maximal ideals of have the form , where p is a prime number).

  9. Layer cake representation - Wikipedia

    en.wikipedia.org/wiki/Layer_cake_representation

    Layer cake representation. In mathematics, the layer cake representation of a non-negative, real-valued measurable function defined on a measure space (,,) is the formula = (,) (),