If the coefficients p1, p2, ... are all or any one or more of them imaginary, then the equation [f](x) = 0, separating the real and imaginary parts thereof, may be written F(x) + i[Phi](x) = 0, where F(x), [Phi](x) are each of them a function with real coefficients; and it thus appears that the equation [f](x) = 0, with imaginary coefficients, has not in general any real root; supposing it to have a real root a, this must be at once a root of each of the equations F(x) = 0 and [Phi](x) = 0. But an equation with real coefficients may have as well imaginary as real roots, and we have further the _theorem_ that for any such equation the imaginary roots enter in pairs, viz. [alpha] + [beta]i being a root, then [alpha] - [beta]i will be also a root. It follows that if the order be odd, there is always an odd number of real roots, and therefore at least one real root. 9. In the case of an equation with real coefficients, the question of the existence of real roots, and of their separation, has been already considered. In the general case of an equation with imaginary (it may be real) coefficients, the like question arises as to the situation of the (real or imaginary) roots; thus, if for facility of conception we regard the constituents [alpha], [beta] of a root [alpha] + [beta]i as the co-ordinates of a point _in plano_, and accordingly represent the root by such point, then drawing in the plane any closed curve or "contour," the question is how many roots lie within such contour. This is solved theoretically by means of a theorem of A.L. Cauchy (1837), viz. writing in the original equation x + iy in place of x, the function [f](x + iy) becomes = P + iQ, where P and Q are each of them a rational and integral function (with real coefficients) of (x, y). Imagining the point (x, y) to travel along the contour, and considering the number of changes of sign from - to + and from + to - of the fraction corresponding to passages of the fraction through zero (that is, to values for which P becomes = 0, disregarding those for which Q becomes = 0), the difference of these numbers gives the number of roots within the contour.