an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the...

several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of...

ordinary improper Riemann integral (f∗ is a strictly decreasing positive function, and therefore has a well-defined improper Riemann integral). For a suitable...

not have an improper Riemann integral, its Lebesgue integral is also undefined (it equals ∞ − ∞). Unfortunately, the improper Riemann integral is not powerful...

non-negative monotonically decreasing function, then the integral of fg is a convergent improper integral. Démonstration d’un théorème d’Abel. Journal de mathématiques...

divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given series or integral to one whose convergence properties...

positive real part, the gamma function is defined via a convergent improper integral: Γ ( z ) = ∫ 0 ∞ t z − 1 e − t  d t ,   ℜ ( z ) > 0 . {\displaystyle...

{\displaystyle \eta (2)={\frac {\pi ^{2}}{12}}} The improper integral of the Complete Elliptic Integral of first kind K takes the value of twice the Catalan...

and only if the improper integral ∫ N ∞ f ( x ) d x {\displaystyle \int _{N}^{\infty }f(x)\,dx} is finite. In particular, if the integral diverges, then...

monotonically non-increasing. The Lebesgue integral may then be defined as the improper Riemann integral of f∗: ∫ E f d μ   = def   ∫ 0 ∞ f ∗ ( t ) d...