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In such cases, the improper Riemann integral allows one to calculate the Lebesgue integral of the function. Specifically, the following theorem holds ( Apostol 1974 , Theorem 10.33): If a function f is Riemann integrable on [ a , b ] for every b ≥ a , and the partial integrals
To compute the integral, we set n to its value and use the reduction formula to express it in terms of the (n – 1) or (n – 2) integral. The lower index integral can be used to calculate the higher index ones; the process is continued repeatedly until we reach a point where the function to be integrated can be computed, usually when its index is 0 or 1.
An inexact differential is a differential for which the integral over some two paths with the same end points is different. Specifically, there exist integrable paths ,: [,] such that () = (), () = and In this case, we denote the integrals as | and | respectively to make explicit the path dependence of the change of the quantity we are considering as .
The path C is the concatenation of the paths C 1 and C 2.. Jordan's lemma yields a simple way to calculate the integral along the real axis of functions f(z) = e i a z g(z) holomorphic on the upper half-plane and continuous on the closed upper half-plane, except possibly at a finite number of non-real points z 1, z 2, …, z n.
Limits of integration can also be defined for improper integrals, with the limits of integration of both + and again being a and b. For an improper integral ∫ a ∞ f ( x ) d x {\displaystyle \int _{a}^{\infty }f(x)\,dx} or ∫ − ∞ b f ( x ) d x {\displaystyle \int _{-\infty }^{b}f(x)\,dx} the limits of integration are a and ∞, or − ...
Singular or weakly singular: An integral equation is called singular or weakly singular if the integral is an improper integral. [7] This could be either because at least one of the limits of integration is infinite or the kernel becomes unbounded, meaning infinite, on at least one point in the interval or domain over which is being integrated. [1]
This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g is here for insuring the continuity of the integral. This gives the following formulas (where a ≠ 0), which are valid over any interval where f is continuous (over larger intervals, the constant C must be replaced ...
The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, it is technically not a "principal value". The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function f ( z ) : z = x + i y , {\displaystyle f(z):z=x+i\,y\;,} with x , y ...