to take the Mellin transform. For example, if

then

Thus has a simple pole at and is thus defined for . Similarly, if

then

Thus has a simple pole at and is thus defined for .

Exponential functions

For , let . Then

Zeta function

It is possible to use the Mellin transform to produce one of the fundamental formulas for the Riemann zeta function, . Let . Then

Thus,

Generalized Gaussian

For , let