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alpha(x) = 1/(sqrt(2pi))int_(-x)^xe^(-t^2/2)dt (1) = sqrt(2/pi)int_0^xe^(-t^2/2)dt (2) = 2Phi(x) (3) = erf(x/(sqrt(2))), (4) where Phi(x) is the normal distribution function ...

F(x) = Li_2(1-x) (1) = int_(1-x)^0(ln(1-t))/tdt, (2) where Li_2(x) is the dilogarithm.

Legendre and Whittaker and Watson's (1990) term for the beta integral int_0^1x^p(1-x)^qdx, whose solution is the beta function B(p+1,q+1).

The inverse of the Laplace transform F(t) = L^(-1)[f(s)] (1) = 1/(2pii)int_(gamma-iinfty)^(gamma+iinfty)e^(st)f(s)ds (2) f(s) = L[F(t)] (3) = int_0^inftyF(t)e^(-st)dt. (4)

Cauchy's integral formula states that f(z_0)=1/(2pii)∮_gamma(f(z)dz)/(z-z_0), (1) where the integral is a contour integral along the contour gamma enclosing the point z_0. It ...

The so-called generalized Fourier integral is a pair of integrals--a "lower Fourier integral" and an "upper Fourier integral"--which allow certain complex-valued functions f ...

Let alpha(x) be a monotone increasing function and define an interval I=(x_1,x_2). Then define the nonnegative function U(I)=alpha(x_2)-alpha(x_1). The Lebesgue integral with ...

If Omega subset= C is a domain and phi:Omega->C is a one-to-one analytic function, then phi(Omega) is a domain, and area(phi(Omega))=int_Omega|phi^'(z)|^2dxdy (Krantz 1999, ...

Let the elliptic modulus k satisfy 0<k^2<1, and the Jacobi amplitude be given by phi=amu with -pi/2<phi<pi/2. The incomplete elliptic integral of the first kind is then ...

If f(z) is analytic in some simply connected region R, then ∮_gammaf(z)dz=0 (1) for any closed contour gamma completely contained in R. Writing z as z=x+iy (2) and f(z) as ...