By Andre Mercier
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Additional resources for Analytical and canonical formalism in physics
Complex numbers are introduced later in this book (chapter 3) and, when the reader has studied them, he or she may wish to justify the procedure set out below. It can be shown to be equivalent to that already given, but the zeros of h(x) are now allowed to be complex and terms that are complex conjugates of each other are combined to leave only real terms. e. of the form Bi x + Ci . Thus, in the expansion, linear terms (ﬁrst-degree polynomials) in the denominator have constants (zerodegree polynomials) in their numerators, whilst quadratic terms (second-degree polynomials) in the denominator have linear terms (ﬁrst-degree polynomials) in their numerators.
40) with α and β both set equal to zero. Although the algebra is more complicated, the same method can be used to derive the equations for the ellipse and the hyperbola. In these cases the distance from the ﬁxed point is a deﬁnite fraction, e, known as the eccentricity, of the distance from the ﬁxed line. For an ellipse 0 < e < 1, for a circle e = 0, and for a hyperbola e > 1. The parabola corresponds to the case e = 1. 3 Construction of a parabola using the point (a, 0) as the focus and the line x = −a as the directrix.
Similar expressions for the derivatives of other polynomials are used later in this chapter. e. 2). Having brieﬂy mentioned the derivative of a function and Rolle’s theorem, we now use them to establish whether g(x) has one or three real zeros. e. g(αk ) = 0 for k = 1, 2, 3, then it follows from Rolle’s theorem that between any consecutive pair of them (say α1 and α2 ) there must be some real value of x at which g (x) = 0. Similarly, there must be a further zero of g (x) lying between α2 and α3 .