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The integral transform (Kf)(x)=int_0^inftysqrt(xt)K_nu(xt)f(t)dt, where K_nu(x) is a modified Bessel function of the second kind. Note the lower limit of 0, not -infty as ...

The integral transform defined by (Kphi)(x)=int_0^inftyG_(pq)^(mn)(xt|(a_p); (b_q))phi(t)dt, where G_(pq)^(mn) is a Meijer G-function. Note the lower limit of 0, not -infty ...

The theory and applications of Laplace transforms and other integral transforms.

The integral transform (Kf)(x)=Gamma(p)int_0^infty(x+t)^(-p)f(t)dt. Note the lower limit of 0, not -infty as implied in Samko et al. (1993, p. 23, eqn. 1.101).

The sum rule for differentiation states d/(dx)[f(x)+g(x)]=f^'(x)+g^'(x), (1) where d/dx denotes a derivative and f^'(x) and g^'(x) are the derivatives of f(x) and g(x), ...

The W-transform of a function f(x) is defined by the integral where Gamma[(beta_m)+s, 1-(alpha_n)-s; (alpha_p^(n+1))+s, 1-(beta_q^(m+1))-s] =Gamma[beta_1+s, ..., beta_m+s, ...

The integral transform defined by (Kphi)(x) =int_(-infty)^inftyG_(p+2,q)^(m,n+2)(t|1-nu+ix,1-nu-ix,(a_p); (b_p))phi(t)dt, where G_(c,d)^(a,b) is the Meijer G-function.

The application of an apodization function.

The operator B^~ defined by B^~f(z)=int_D((1-|z|^2)^2)/(|1-zw^_|^4)f(w)dA(w) for z in D, where D is the unit open disk and w^_ is the complex conjugate (Hedenmalm et al. ...

A two-sided (doubly infinite) Laplace transform, L_t[f(t)](s)=int_(-infty)^inftyf(t)e^(-st)dt. While some authors use this as the primary definition of "the" Laplace ...