For real "x", the function "f(x)=e" is well-known to be the unique real function satisfying the differential equation:
for all "x". This motivates a definition of "f(z)=e" for complex "z" as the function that satisfies the differential equation:
for all complex "z", where the derivative in "f"'("z") is defined in the sense of a complex derivative. This can be proven to yield a unique function which is well-defined everywhere on the complex plane.

Using Taylor series
टेलर श्रेणी
Differential equation definition
Multiplicative property definition