Solving linear, const.-coeff. ODEs
... yp(x) = xs eα x [Qn(x) cos(β x) + Rn(x) sin(β x)] , where Qn(x) and Rn(x) are polynomials of degree n (same degree as Pn(x)) with coefficients to be determined from (1), i.e. Qn(x) = An xn + An−1 xn−1 + · · · + A1 x + A0 , Rn(x) = Bn xn + Bn−1 xn−1 + · · · + B1 x + B0 , where An, . . . , A0 and Bn, ...
... yp(x) = xs eα x [Qn(x) cos(β x) + Rn(x) sin(β x)] , where Qn(x) and Rn(x) are polynomials of degree n (same degree as Pn(x)) with coefficients to be determined from (1), i.e. Qn(x) = An xn + An−1 xn−1 + · · · + A1 x + A0 , Rn(x) = Bn xn + Bn−1 xn−1 + · · · + B1 x + B0 , where An, . . . , A0 and Bn, ...
Module 3 Notes Prime numbers: -Prime numbers have exactly two
... -If you are an integer you are also a rational number but you may not be a whole number nor a natural number. Ordering Real Numbers -Real numbers can be plotted as points on a number line. A number line is a visual representation of real numbers. The numbers on the number line increase from left to ...
... -If you are an integer you are also a rational number but you may not be a whole number nor a natural number. Ordering Real Numbers -Real numbers can be plotted as points on a number line. A number line is a visual representation of real numbers. The numbers on the number line increase from left to ...
CHAPTER 9: COMPLEX NUMBERS 1. Introduction Although R is a
... sometimes the discriminant b2 − 4ac is negative, and in those cases we need to use complex numbers to find roots. The complex numbers allow us to use the quadratic equation successfully in all circumstances. However, the complex numbers did not arise first from quadratic equations. When a quadratic ...
... sometimes the discriminant b2 − 4ac is negative, and in those cases we need to use complex numbers to find roots. The complex numbers allow us to use the quadratic equation successfully in all circumstances. However, the complex numbers did not arise first from quadratic equations. When a quadratic ...
Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.In field theory and ring theory the notion of root of unity also applies to any ring with a multiplicative identity element. Any algebraically closed field has exactly n nth roots of unity, if n is not divisible by the characteristic of the field.