## A DERIVE Plot of the Level Curves: \( |N'(a + ib)| = c \)
I wanted to answer the following question: if a point \( x \) is in a circle of radius \( 1/2 \) of one of the three roots, is \( N(x) \) also in that circle? Looking at the level curve \[ |N'(a + i b)| = 1, \] we see that it gets closer than 1/2 from the roots, suggesting that the answer to the problem is no. Using the applet above we can verify this by clicking at the point 1/2 or just calculating \( N(1/2) = 5/3 \). We can find exactly where the level curve intersects the positive real axis by solving \( N'(x) = -1 \), which gives \( 5^{2/3}/5 \). So the largest circle around the roots such that \( N(x) \) stays inside the circle has radius \[ 1 - \frac{5^{2/3}}{5}. \] |

Last updated: Mon Aug 26

Ralph Freese