# A question on the operator e^D

• mhill
D and replacing a real number with it is valid in certain cases, as long as the functions it acts on are differentiable an infinite number of times. However, this may not always be properly justified. It is possible to replace real numbers with operators, but it is important to note that this may change the validity of certain statements, such as commutativity.

#### 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 ??

mhill said:
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