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  2. Characteristic equation (calculus) - Wikipedia

    en.wikipedia.org/wiki/Characteristic_equation...

    By solving for the roots, r, in this characteristic equation, one can find the general solution to the differential equation. [1] [6] For example, if r has roots equal to 3, 11, and 40, then the general solution will be () = + +, where , , and are arbitrary constants which need to be determined by the boundary and/or initial conditions.

  3. Complex conjugate root theorem - Wikipedia

    en.wikipedia.org/wiki/Complex_conjugate_root_theorem

    In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P. [1]

  4. Zero of a function - Wikipedia

    en.wikipedia.org/wiki/Zero_of_a_function

    In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function, is a member of the domain of such that () vanishes at ; that is, the function attains the value of 0 at , or equivalently, is a solution to the equation () =. [1]

  5. Complex conjugate - Wikipedia

    en.wikipedia.org/wiki/Complex_conjugate

    In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}

  6. Quintic function - Wikipedia

    en.wikipedia.org/wiki/Quintic_function

    Finding the roots (zeros) of a given polynomial has been a prominent mathematical problem.. Solving linear, quadratic, cubic and quartic equations in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions.

  7. Auxiliary function - Wikipedia

    en.wikipedia.org/wiki/Auxiliary_function

    In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high order at some point.

  8. Geometrical properties of polynomial roots - Wikipedia

    en.wikipedia.org/wiki/Geometrical_properties_of...

    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).

  9. Cubic equation - Wikipedia

    en.wikipedia.org/wiki/Cubic_equation

    With one real and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's derivative. There is an interesting geometrical relationship among all these roots. The points in the complex plane representing the three roots serve as the vertices of an isosceles triangle.