Difference between revisions of "Two-term equation"
From Encyclopedia of Mathematics
(TeX) |
m (better) |
||
| Line 1: | Line 1: | ||
{{TEX|done}} | {{TEX|done}} | ||
| − | An algebraic equation of the form $ax^n+b=0$, where $a$ and $b$ are complex | + | An algebraic equation of the form $ax^n+b=0$, where $a$ and $b$ are [[complex number]]s, with $ab\neq0$. Two-term equations have $n$ distinct complex roots |
$$x_k=\left|\frac ba\right|^{1/n}\exp\left(\frac{2\pi k+\phi}{n}i\right),$$ | $$x_k=\left|\frac ba\right|^{1/n}\exp\left(\frac{2\pi k+\phi}{n}i\right),$$ | ||
| Line 6: | Line 6: | ||
$$k=0,\ldots,n-1,\quad\phi=\arg\left(-\frac ba\right).$$ | $$k=0,\ldots,n-1,\quad\phi=\arg\left(-\frac ba\right).$$ | ||
| − | The roots of a two-term equation in the complex plane are located on the circle with radius $|b/a|^{1/n}$ and centre at the coordinate origin, at the vertices of the inscribed regular $n$-gon (cf. [[ | + | The roots of a two-term equation in the complex plane are located on the circle with radius $|b/a|^{1/n}$ and centre at the coordinate origin, at the vertices of the inscribed regular $n$-gon (cf. [[Regular polygons]]). |
Latest revision as of 17:30, 23 December 2014
An algebraic equation of the form $ax^n+b=0$, where $a$ and $b$ are complex numbers, with $ab\neq0$. Two-term equations have $n$ distinct complex roots
$$x_k=\left|\frac ba\right|^{1/n}\exp\left(\frac{2\pi k+\phi}{n}i\right),$$
$$k=0,\ldots,n-1,\quad\phi=\arg\left(-\frac ba\right).$$
The roots of a two-term equation in the complex plane are located on the circle with radius $|b/a|^{1/n}$ and centre at the coordinate origin, at the vertices of the inscribed regular $n$-gon (cf. Regular polygons).
How to Cite This Entry:
Two-term equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-term_equation&oldid=32746
Two-term equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-term_equation&oldid=32746
This article was adapted from an original article by A.I. Galochkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article