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).

- function:
`z → exp(a*z) = e^(a*z)`

- morph a :
`1 → 1+0.4*i`

- Preimage: rectangular grid:
`-1 ≦ Re(z) ≦ 1; -3.1 ≦ Im(z) ≦ 3.1`

If `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))`

.