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More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation (+) = has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions + and .
exists and is a nonzero complex number. In this case, the point at infinity is a pole of order n if n > 0, and a zero of order | | if n < 0. For example, a polynomial of degree n has a pole of degree n at infinity. The complex plane extended by a point at infinity is called the Riemann sphere.
In fact, the same proof shows that Euler's formula is even valid for all complex numbers x. A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used ...
Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion problem. [7] Naturally also same function of one variable that depends on some complex parameter is a candidate.
He is really interested in problems 3 and 4, but the answers to the easier problems 1 and 2 are needed for proving the answers to problems 3 and 4. 1st problem [ edit ]
By the fundamental theorem of algebra, if the monic polynomial equation x 2 + bx + c = 0 has complex coefficients, it must have two (not necessarily distinct) complex roots. Unfortunately, the discriminant b 2 − 4c is not as useful in this situation, because it may be a complex number. Still, a modified version of the general theorem can be ...
In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality. A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set.
A complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the number is real. In other words, real numbers are the only fixed points of conjugation. Conjugation does not change the modulus of a complex number: | ¯ | = | |. Conjugation is an involution, that is, the conjugate of the conjugate of a complex ...
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