Complex Exponential Function

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)) .

z_exp.pdf