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For 0 < q < 1, the series converges to a function F(x) on an interval (0,A] if |f(x)x α | is bounded on the interval (0, A] for some 0 ≤ α < 1. The q-integral is a Riemann–Stieltjes integral with respect to a step function having infinitely many points of increase at the points q j..The jump at the point q j is q j.
Fatou's lemma: If {f k} k ∈ N is a sequence of non-negative measurable functions, then . Again, the value of any of the integrals may be infinite. Dominated convergence theorem : Suppose { f k } k ∈ N is a sequence of complex measurable functions with pointwise limit f , and there is a Lebesgue integrable function g (i.e., g belongs to the ...
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
A fundamental domain of the ring of integers of the field K obtained from Q by adjoining a root of x 3 − x 2 − 2x + 1. This fundamental domain sits inside K ⊗ Q R. The discriminant of K is 49 = 7 2. Accordingly, the volume of the fundamental domain is 7 and K is only ramified at 7.
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus.
where −a/d is not a natural number and k is a natural number. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. It is not possible for a harmonic progression (other than the trivial case where a = 1 and k = 0) to sum to an integer.
In number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as (,) = (= ())where V is a non-singular n-dimensional projective algebraic variety over the field F q with q elements and N k is the number of points of V defined over the finite field extension F q k of F q.
Depending on the columns removed, the answer will differ by multiplication by , where the power of is not necessarily the number of crossings in the knot. To resolve this ambiguity, divide out the largest possible power of t {\displaystyle t} and multiply by − 1 {\displaystyle -1} if necessary, so that the constant term is positive.