W-Transform
The W-transform of a function f(x) is defined by the integral
| (Wf)(x)=(W_(pq)^(mn)|nu,(alpha)_p; (beta_q)|f(t))(x) =1/(2pii)int_sigmaGamma(nu-ix-s,nu+ix-s)Gamma[(beta_m)+s, 1-(alpha_n)-s; (alpha_p^(n+1))+s, 1-(beta_q^(m+1))-s]f^*(1-s)ds, |
where
![画像: 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, 1-alpha_1-s, ..., 1-alpha_n-s; alpha_(n+1)+s, ..., alpha_p+s, 1-beta_(m+1)-s, ... 1-beta_q-s] =(product_(j=1)^(m)Gamma(beta_j+s)product_(j=1)^(n)Gamma(1-alpha_j-s))/(product_(j=n+1)^(p)Gamma(alpha_j+s)product_(j=m+1)^(q)Gamma(1-beta_j-s)),](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/W-Transform/NumberedEquation2.svg){kind=link}
R[nu]>1/2, nu and the components of the vectors (alpha_p) and (beta_q) are complex numbers satisfying the conditions R[a_p]!=1/2, 3/2, 5/2, ... and R[b_q]!=-1/2, -3/2, -5/2, ..., f^*(s) is the Mellin transform of a function f(x) and sigma is the contour sigma={1/2-iinfty,1/2+iinfty}.
See also
G-TransformExplore with Wolfram|Alpha
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References
Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. "The W-Transform and Its Inversion." §37.5 in Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, pp. 752-758, 1993.Referenced on Wolfram|Alpha
W-TransformCite this as:
Weisstein, Eric W. "W-Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/W-Transform.html