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