The exponential function is defined by its differential equation
exp'(z) = exp(z) with the initial value
exp(0) = 1.
For no real number x ≠ 0 can exp(x) be computed in finitely many steps. All the numbers which our computers give us are only approximations of the true value of exp(x).
z → exp(a*z) = e^(a*z)
1 → 1+0.4*i
-1 ≦ Re(z) ≦ 1; -3.1 ≦ Im(z) ≦ 3.1
Im(a) > 0 then the gridlines are spirals. The exponential function is periodic:
exp(z + 2 π * i) = exp(z). One can see that in the range: For
a = 1 the circular grid lines are about to close. The periodicity is also clear from the formula:
exp(x + i * y) = exp(x) * (cos(y) + i * sin(y))