Integral ($\int$)

An integral accumulates a function over a domain and may be defined as a limit of finite sums.

An integral representation expresses a function through an integral of specified data and an integral kernel. Green-function representations solve differential equations this way, while the Bromwich inversion formula reconstructs a function from its Laplace transform.

Lebesgue integration constructs an integral from the measure of level sets or from approximation by simple functions. It is linear on integrable functions and is stable under the principal convergence theorems of measure theory.

If $0<\theta <1$ and nonnegative measurable functions satisfy $h((1-\theta )x+\theta y)\ge f(x{)}^{1-\theta }g(y{)}^{\theta }$ for all $x,y\in {\mathbb{R}}^n$, then their Lebesgue integrals obeyThe quantile derivative identity and monotone substitution inequality prove the one-dimensional case; Tonelli theorem extends it by induction. It implies the Brunn–Minkowski inequality.

If $f\in L^1$, then for every $\epsilon >0$ there is $\delta >0$ such that ${\int }_E|f|<\epsilon$ whenever the measure of $E$ is below $\delta$. In particular, integrals over a sequence of shrinking measurable sets tend to zero.

A time integral accumulates an integrand over a time coordinate. In perturbation theory it commonly sums the effect of an interaction over all possible vertex times.

A double integral integrates a function over a two-dimensional region. A change of variables formula can replace the region and area element using a Jacobian determinant.

If measurable functions satisfy $f\le g$ almost everywhere, thenwhenever the two sides are defined.

An antiderivative of $g$ is a differentiable function $p$ satisfying $p^{\prime }=g$.

Any two antiderivatives of the same function on a connected interval differ by an additive constant.

The Gaussian integral isSquaring the integral and changing to polar coordinates proves the formula.

The Dawson function obeys $F_D^{\prime }+2z{F}_D=1$ by differentiation. Its large-real-argument asymptotic expansion begins $F_D(z)=1/(2z)+1/(4z^3)+O(z^{-5})$. This inverse-power behavior contrasts with the separate exponentially growing factors in its definition.

Define $E_0(z)={\int }_0^z{e}^{-t^2}{\int }_0^t{e}^{u^2}dudt$ and $E_1(z)={\int }_0^z{e}^{-t^2}{\int }_0^t{e}^{u^2}\mathrm{erf}ududt$. They satisfy $(D^2+2zD)E_0=1$ and $(D^2+2zD)E_1=\mathrm{erf}z$. The first is an even function, the second an odd function, and their real tails are respectively $\frac{1}{2}\log |z|+C_1+o(1)$ and $\mathrm{sgn}z[\frac{1}{2}\log |z|+C_2+o(1)]$. Their derivatives have inverse-linear tails, explaining the logarithmic matching constants in a singular perturbation.

The error function isIt is odd and tends to $\pm 1$ as $x\to \pm \infty$.

The complementary error function avoids subtracting a quantity close to one when describing a small Gaussian integral tail. For real $x>0$, $\mathrm{erfc}x=(2/\sqrt{\pi }){\int }_x^{\infty }{e}^{-t^2}dt$; analytic continuation defines it for complex arguments.

The Faddeeva function is an entire function built from the complementary error function. Its integral representation for $\mathrm{Im}z>0$ is $w(z)=(i\pi {)}^{-1}{\int }_{-\infty }^{\infty }{e}^{-t^2}/(t-z)dt$, as normalized in NIST's integral representation. Its reflection identity is $w(-z)=2e^{-z^2}-w(z)$, directly from the odd function property of the error function.

For a pole off the real axis, substitution $s=2t$ into the Faddeeva function integral gives $G(a)=i\pi w(a/2)$ when $\mathrm{Im}a>0$, and $G(a)=-i\pi w(-a/2)$ when $\mathrm{Im}a<0$. The difference of boundary values is the residue theorem jump $2\pi i{e}^{-a^2/4}$. In particular, for a lower-half-plane pole, $G(a)+2\pi i{e}^{-a^2/4}=i\pi w(a/2)$, which packages a crossed residue and a nearby saddle into one smooth expression.