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The characteristic roots (roots of the characteristic equation) also provide qualitative information about the behavior of the variable whose evolution is described by the dynamic equation. For a differential equation parameterized on time, the variable's evolution is stable if and only if the real part of each root is negative.
A quadratic with two real roots, for example, will have exactly two angles that satisfy the above conditions. For complex roots, one also needs to find a series of similar triangles, but with the vertices of the root path displaced from the polynomial path by a distance equal to the imaginary part of the root. In this case the root path will ...
Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting). For finding one root, Newton's method and other general iterative methods work generally well.
More generally, if an equation P(x) = 0 of prime degree p with rational coefficients is solvable in radicals, then one can define an auxiliary equation Q(y) = 0 of degree p – 1, also with rational coefficients, such that each root of P is the sum of p-th roots of the roots of Q.
It follows from the present theorem and the fundamental theorem of algebra that if the degree of a real polynomial is odd, it must have at least one real root. [2] This can be proved as follows. Since non-real complex roots come in conjugate pairs, there are an even number of them; But a polynomial of odd degree has an odd number of roots;
For simple roots, this results immediately from the implicit function theorem. This is true also for multiple roots, but some care is needed for the proof. A small change of coefficients may induce a dramatic change of the roots, including the change of a real root into a complex root with a rather large imaginary part (see Wilkinson's polynomial).
In mathematics, a Hurwitz polynomial (named after German mathematician Adolf Hurwitz) is a polynomial whose roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the real part of every root is zero or negative. [1] Such a polynomial must have coefficients that are positive real numbers.
Some solutions of a differential equation having a regular singular point with indicial roots = and .. In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form ″ + ′ + = with ′ and ″.
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