It is well-known that, for any real "x", the following series is equal to "e":
(in other words, this is the Taylor series for the real exponential function, and it has an infinite radius of convergence). This invites the following "definition" of "e" for complex "z":
This can be proven to be well-defined; in particular, the series converges for any "z".

Using ordinary differential equations
टेलर श्रेणी
समिश्र चरघातांकी (exponentiation) की परिभाषा
Differential equation definition