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  2. Exponentiation by squaring - Wikipedia

    en.wikipedia.org/wiki/Exponentiation_by_squaring

    The method is based on the observation that, for any integer >, one has: = {() /, /,. If the exponent n is zero then the answer is 1. If the exponent is negative then we can reuse the previous formula by rewriting the value using a positive exponent.

  3. Knuth's up-arrow notation - Wikipedia

    en.wikipedia.org/wiki/Knuth's_up-arrow_notation

    People have adopted the linear notation for such environments; the up-arrow suggests 'raising to the power of'. If the character set does not contain an up arrow, the caret (^) is used instead. The superscript notation a b {\displaystyle a^{b}} doesn't lend itself well to generalization, which explains why Knuth chose to work from the inline ...

  4. Exponentiation - Wikipedia

    en.wikipedia.org/wiki/Exponentiation

    Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (). [ 26 ] This definition of exponentiation with negative exponents is the only one that allows extending the identity b m + n = b m ⋅ b n {\displaystyle b^{m+n}=b^{m}\cdot b^{n}} to negative exponents (consider the case m = − n ...

  5. Modular exponentiation - Wikipedia

    en.wikipedia.org/wiki/Modular_exponentiation

    Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = b e mod m. From the definition of division, it follows that 0 ≤ c < m. For example, given b = 5, e = 3 and m = 13, dividing 5 3 = 125 by 13 leaves a remainder of c = 8.

  6. Python syntax and semantics - Wikipedia

    en.wikipedia.org/wiki/Python_syntax_and_semantics

    Python supports normal floating point numbers, which are created when a dot is used in a literal (e.g. 1.1), when an integer and a floating point number are used in an expression, or as a result of some mathematical operations ("true division" via the / operator, or exponentiation with a negative exponent). Python also supports complex numbers ...

  7. Matrix exponential - Wikipedia

    en.wikipedia.org/wiki/Matrix_exponential

    It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let X be an n×n real or complex matrix. The exponential of X, denoted by e X or exp(X), is the n×n matrix given by the power series = =!

  8. Generating function - Wikipedia

    en.wikipedia.org/wiki/Generating_function

    Similar asymptotic analysis is possible for exponential generating functions; with an exponential generating function, it is ⁠ a n / n! ⁠ that grows according to these asymptotic formulae. Generally, if the generating function of one sequence minus the generating function of a second sequence has a radius of convergence that is larger than ...

  9. SymPy - Wikipedia

    en.wikipedia.org/wiki/SymPy

    SymPy is an open-source Python library for symbolic computation. It provides computer algebra capabilities either as a standalone application, as a library to other applications, or live on the web as SymPy Live [2] or SymPy Gamma. [3] SymPy is simple to install and to inspect because it is written entirely in Python with few dependencies.