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  2. Risch algorithm - Wikipedia

    en.wikipedia.org/wiki/Risch_Algorithm

    It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch called it a decision procedure , because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that ...

  3. Rational function - Wikipedia

    en.wikipedia.org/wiki/Rational_function

    Every Laurent polynomial can be written as a rational function while the converse is not necessarily true, i.e., the ring of Laurent polynomials is a subring of the rational functions. The rational function f ( x ) = x x {\displaystyle f(x)={\tfrac {x}{x}}} is equal to 1 for all x except 0, where there is a removable singularity .

  4. Polynomial and rational function modeling - Wikipedia

    en.wikipedia.org/wiki/Polynomial_and_rational...

    A polynomial function is one that has the form = + + + + + where n is a non-negative integer that defines the degree of the polynomial. A polynomial with a degree of 0 is simply a constant function; with a degree of 1 is a line; with a degree of 2 is a quadratic; with a degree of 3 is a cubic, and so on.

  5. Function (mathematics) - Wikipedia

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

    Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function , whose graph is a hyperbola, and whose domain is the whole real line except for 0.

  6. Function of a real variable - Wikipedia

    en.wikipedia.org/wiki/Function_of_a_real_variable

    A real-valued function of a real variable is a function that takes as input a real number, commonly represented by the variable x, for producing another real number, the value of the function, commonly denoted f(x). For simplicity, in this article a real-valued function of a real variable will be simply called a function. To avoid any ambiguity ...

  7. Descartes' rule of signs - Wikipedia

    en.wikipedia.org/wiki/Descartes'_rule_of_signs

    The rule states that if the nonzero terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign changes between consecutive (nonzero) coefficients, or is less than it by an even number.

  8. Rank of an elliptic curve - Wikipedia

    en.wikipedia.org/wiki/Rank_of_an_elliptic_curve

    In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve defined over the field of rational numbers or more generally a number field K. Mordell's theorem (generalized to arbitrary number fields by André Weil) says the group of rational points on an elliptic curve has a finite basis. This means that ...

  9. Dirichlet function - Wikipedia

    en.wikipedia.org/wiki/Dirichlet_function

    The Dirichlet function provides a counterexample showing that the monotone convergence theorem is not true in the context of the Riemann integral. Proof Using an enumeration of the rational numbers between 0 and 1, we define the function f n (for all nonnegative integer n ) as the indicator function of the set of the first n terms of this ...