For instance, Mellin transform of function $f(x)=x$

$$\int_0^\infty f(x) x^{s-1} dx$$

returns the result $\delta(s)$ which is completely strange to me. Why only at $s=0$ the result is infinite? Why for instance for $s=2$ the result is $0$?

Moreover, for $f(x)=a^x$ the result is $\Gamma (s) (-\log (a))^{-s}$ which is again meaningless with $a\ge1$.

Why they just cannot use Delta Function or other infinite distributions?

What method do they use to compose these tables? Can I utilize the same method to obtain more sensible results?

Or maybe there is somewhere a more complete table?

`GenerateConditions -> True`

to find the range of validity of the Mellin transform. For example,`MellinTransform[a^x, x, s, GenerateConditions -> True]`

returns`ConditionalExpression[Gamma[s] (-Log[a])^-s, Re[Log[a]] < 0 && Re[s] > 0]`

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