The 
-transform of a function 
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,](/images/equations/W-Transform/NumberedEquation1.svg) |
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, 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)),](/images/equations/W-Transform/NumberedEquation2.svg) |
![R[nu]>1/2](/images/equations/W-Transform/Inline3.svg)
, 
and the components of the vectors 
and 
are complex numbers satisfying the conditions ![R[a_p])!=1/2](/images/equations/W-Transform/Inline7.svg)
, 3/2, 5/2, ... and ![R[b_q]!=-1/2](/images/equations/W-Transform/Inline8.svg)
, 
, 
, ..., 
is the Mellin transform
of a function 
and 
is the contour 
.
