A question on the operator e^D

[mhill]

if D means the derivative respect to x , is always licit to use this operator ??

for example Ramanujan obtained his Master theorem from the integral equality

\int_{0}^{\infty} dx e^{-ax} x^{m-1} = \Gamma (m) a^{-m]

by just replacing the 'a' number by the operator exp(D) and expanding exp(-exp(x) but is this valid ?? , can we always replace real numbers by operators ??

 

[Pere Callahan]

if D means the derivative respect to x , is always licit to use this operator ??

for example Ramanujan obtained his Master theorem from the integral equality

\int_{0}^{\infty} dx e^{-ax} x^{m-1} = \Gamma (m) a^{-m]

by just replacing the 'a' number by the operator exp(D) and expanding exp(-exp(x) but is this valid ?? , can we always replace real numbers by operators ??

For your first questions If you want to use the operator e^D the functions it acts on must certainly be C^\infty or in some other weak sense differentiable an infinite number of times.

To your second question: in this special case it seems to have been correct (maybe not properly justified, though).

Your last question: In principle: yes, but a true statement may become false if you replace numbers by operators. Consider ab=ba which is true for real numbers a und b but not true for non-commuting operators.

Pere