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. 2014 Jul 24;10(7):e1003616. doi: 10.1371/journal.pcbi.1003616

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© 2014 Weber et al

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.

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as a function of for fixed values of For (green circle) the real part of Q takes, depending on a value in the interval The values for x are increasing from A-D, while and remain unchanged. For relatively low values of (A-B) the real part is positive for After one or several turns, i.e by increasing the spiral can potentially cross the origin only once (empty circle). In A the spiral misses the origin, while in B the spiral crosses the origin after one turn. Crossing of the origin means that the corresponding complex number is a root of Q. In C the spiral starts at the origin. This represents the only real positive root of Q. For initially negative values of (D) the spiral can never cross the origin because the distance to the center point (gray circle) is already in the beginning for larger than the distance between the latter and the origin. By increasing y this distance will even grow further according to Eq. 33.

Inline graphic — as a function of for fixed values of For (green circle) the real part of Q takes, depending on a value in the interval The values for x are increasing from A-D, while and remain unchanged. For relatively low values of (A-B) the real part is positive for After one or several turns, i.e by increasing the spiral can potentially cross the origin only once (empty circle). In A the spiral misses the origin, while in B the spiral crosses the origin after one turn. Crossing of the origin means that the corresponding complex number is a root of Q. In C the spiral starts at the origin. This represents the only real positive root of Q. For initially negative values of (D) the spiral can never cross the origin because the distance to the center point (gray circle) is already in the beginning for larger than the distance between the latter and the origin. By increasing y this distance will even grow further according to Eq. 33.