Mathematical Methods in the Physical Sciences · Problem 11.32
FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas
Mathematics — Methods · roots-of-equations-quadrants
Problem
As in Section 7, find out how many roots the equation has in each quadrant.Intuitive explanation
This problem asks you to locate the roots of a cubic polynomial by quadrant—a classic application of the argument principle from complex analysis. The key insight is that as you traverse a closed contour (like the boundary of a quadrant), the number of times the image curve winds around the origin tells you exactly how many roots lie inside. By carefully tracking how the polynomial's argument changes along the positive real axis, imaginary axis, and a large circular arc in each quadrant, you can count roots without finding them explicitly.
Intuitive explanation
This problem asks you to locate the roots of a cubic polynomial by quadrant—a classic application of the argument principle from complex analysis. The key insight is that as you traverse a closed contour (like the boundary of a quadrant), the number of times the image curve winds around the origin tells you exactly how many roots lie inside. By carefully tracking how the polynomial's argument changes along the positive real axis, imaginary axis, and a large circular arc in each quadrant, you can count roots without finding them explicitly.