Search results
Results From The WOW.Com Content Network
The converse does not hold; not all Lebesgue-integrable functions are Riemann integrable. The Lebesgue–Vitali theorem does not imply that all type of discontinuities have the same weight on the obstruction that a real-valued bounded function be Riemann integrable on [a, b].
A version holds for Fourier series as well: if is an integrable function on a bounded interval, then the Fourier coefficients ^ of tend to 0 as . This follows by extending f {\displaystyle f} by zero outside the interval, and then applying the version of the Riemann–Lebesgue lemma on the entire real line.
The region that is bounded can be seen as the area inside and . For example, the function f ( x ) = x 3 {\displaystyle f(x)=x^{3}} is defined on the interval [ 2 , 4 ] {\displaystyle [2,4]} ∫ 2 4 x 3 d x {\displaystyle \int _{2}^{4}x^{3}\,dx} with the limits of integration being 2 {\displaystyle 2} and 4 {\displaystyle 4} .
A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval [a, b] and can be generalized to other notions of integral (Lebesgue and Daniell). Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval.
The Lebesgue criterion for integrability states that a bounded function is Riemann integrable if and only if the set of all discontinuities has measure zero. [5] Every countable subset of the real numbers - such as the rational numbers - has measure zero, so the above discussion shows that Thomae's function is Riemann integrable on any interval.
{(x, y) : P(x, y) = 0} ⊆ C 2 defines a Riemann surface provided there are no points on this locus with ∂P/∂x, ∂P/∂y = 0 (or we restrict to an open subset containing no such points). This is an example of an algebraic curve. Every elliptic curve is an algebraic curve, given by (the compactification of) the locus y 2 = x 3 + ax + b
The Riemann curvature tensor is given by [15] [16] (,,,) = ((,),). To check independence of K it suffices to note that it does not change under elementary transformations sending (X, Y) to (Y, X), (λX, Y) and (X + Y, Y). That in turn relies on the fact that the operator R(X,Y) is skew-adjoint. [17] Skew-adjointness entails that (R(X,Y)Z,Z) = 0 ...
The harmonic function U. If X is a Riemann surface and P is a point on X with local coordinate z, there is a unique real-valued harmonic function U on X \ {P} such that U(z) – Re z −1 is harmonic near z = 0 (the point P) and dU is square integrable on the complement of a neighbourhood of P.