# Modified Euler Absolute Stability Proof

Given the modified Euler method: $u_{n+1} = u_n + hf(u_n + \frac{h}{2}f(u_n))$ applied to the test equation $y' = f(y) = \lambda y$, how do you prove that no imaginary value $h\lambda$ is contained in the region of absolute stability?

I've found that the this region is $-2 \le h\lambda \le 0$, but how is it that no imaginary values are contained in that region?

Actually, the region is given by $$|1+z+\frac{z^2}2|<1,\qquad z=λh,$$ which indeed on the real axis reduces to the interval $[-2,0]$.
For $z=iy$ one finds $$|1-\frac{y^2}2+iy|<1\iff (1-\frac{y^2}2)^2+y^2<1\iff y^4<0$$ which clearly is impossible.