The definition of as the solution of the 'initial value problem'...

(1)

... is [in principle...] perfecly acceptable. The 'initial condition' guarantees that is analytic in , so that we can write...

(2)

From (1) we can easily derive that , so that is...

(3)

The 'more traditional' alternative is to define...

(4)

... and from (4) is possible to derive all the properties of , including the fundamental identity...

(5)

... so that defintions (3) and (4) are fully equivalent...

Kind regards