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  2. Binomial theorem - Wikipedia

    en.wikipedia.org/wiki/Binomial_theorem

    In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power ⁠ (+) ⁠ expands into a polynomial with terms of the form ⁠ ⁠, where the exponents ⁠ ⁠ and ⁠ ⁠ are nonnegative integers satisfying ⁠ + = ⁠ and the coefficient ⁠ ⁠ of each term is a specific positive integer ...

  3. Binomial approximation - Wikipedia

    en.wikipedia.org/wiki/Binomial_approximation

    The binomial approximation for the square root, + + /, can be applied for the following expression, + where and are real but .. The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.

  4. Binomial series - Wikipedia

    en.wikipedia.org/wiki/Binomial_series

    where the power series on the right-hand side of is expressed in terms of the (generalized) binomial coefficients ():= () (+)!.Note that if α is a nonnegative integer n then the x n + 1 term and all later terms in the series are 0, since each contains a factor of (n − n).

  5. Binomial coefficient - Wikipedia

    en.wikipedia.org/wiki/Binomial_coefficient

    The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. Visualisation of binomial expansion up to the 4th power. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.

  6. Fractional calculus - Wikipedia

    en.wikipedia.org/wiki/Fractional_calculus

    Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator = (), and of the integration operator J {\displaystyle J} [ Note 1 ] J f ( x ) = ∫ 0 x f ( s ) d s , {\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,}

  7. Heaviside cover-up method - Wikipedia

    en.wikipedia.org/wiki/Heaviside_cover-up_method

    This separation can be accomplished by the Heaviside cover-up method, another method for determining the coefficients of a partial fraction. Case one has fractional expressions where factors in the denominator are unique. Case two has fractional expressions where some factors may repeat as powers of a binomial.

  8. Method of matched asymptotic expansions - Wikipedia

    en.wikipedia.org/wiki/Method_of_matched...

    The appropriate form of these expansions is not always clear: while a power-series expansion in may work, sometimes the appropriate form involves fractional powers of , functions such as ⁡, et cetera. As in the above example, we will obtain outer and inner expansions with some coefficients which must be determined by matching.

  9. General Leibniz rule - Wikipedia

    en.wikipedia.org/wiki/General_Leibniz_rule

    Relationship to the binomial theorem [ edit ] The Leibniz rule bears a strong resemblance to the binomial theorem , and in fact the binomial theorem can be proven directly from the Leibniz rule by taking f ( x ) = e a x {\displaystyle f(x)=e^{ax}} and g ( x ) = e b x , {\displaystyle g(x)=e^{bx},} which gives