Rational Fractions. Decomposition into Partial Fractions

Decompose the following fraction into partial fractions: $\dfrac {x^{2}+1}{x(x^{2}-1)}$.

Decompose the following fraction into partial fractions: $\dfrac {2}{(x-1)(x-2)(x-3)}$.

Decompose the following fraction into partial fractions: $\dfrac {x^{5}-x^{3}-x^{2}}{x^{2}-1}$.

Decompose the following fraction into partial fractions: $\dfrac {x^{2}}{(x-1)^{2}(x+1)}$.

Decompose the following fraction into partial fractions: $\dfrac {4}{(x^{2}-1)^{2}}$.

Decompose the following fraction into partial fractions: $\dfrac {x^{6}-x^{2}+1}{(x-1)^{3}}$.

Decompose the following fraction into partial fractions: $\dfrac {x^{2}+1}{(x^{2}+x+1)^{2}}$.

Decompose the following fraction into partial fractions: $\dfrac {x^{4}+1}{x^{3}(x^{2}+1)}$.

Decompose the following fraction into partial fractions: $\dfrac {2}{x^{4}+2x^{3}+2x^{2}+2x+1}$.

Decompose the following fraction into partial fractions: $\dfrac {8}{x^{4}+4}$.

Decompose the following fraction into partial fractions: $\dfrac {2x^{4}}{x^{4}+x^{2}+1}$.

Decompose the following fraction into partial fractions: $\dfrac {x^{m-1}}{x^{n}-1}$; $m \leqslant n$.

Decompose the following fraction into partial fractions: $\dfrac {1-x^{2}}{1-2x\cos \varphi +x^{2}}$.

Decompose the following fraction into partial fractions: $\dfrac {1}{x^{2n}+1}$.

Decompose the following fraction into partial fractions: $\dfrac {1}{x^{2n+1}+1}$.

Decompose the following fraction into partial fractions: $\dfrac {3x^{3}+15x+6}{x^{4}+x^{3}+3x^{2}+2x+2}$.

Decompose the following fraction into partial fractions: $\dfrac {2x^{3}+4x}{x^{4}+x^{2}+4}$.

Decompose the following fraction into partial fractions: $\dfrac {3x^{2}-3}{x^{3}-3x+1}$.

Determine the coefficients in the expansion of the function

in decreasing powers of $x$, if the roots of the denominator are simple.

Find the sum of the series $\sum_{n=0}^{\infty }u_{n}^{2}x^{n}$, where

If $x_{1}$ is greater in absolute value than the other roots of the equation $f(x)=x^{n}+a_{1}x^{n-1}+\ldots +a_{n}=0$, then

where $u_{n}$ is the coefficient of $x^{-n}$ in the expansion of the fraction $\dfrac {\varphi (x)}{f(x)}=\sum u_{n}x^{-n}$, if $f'(x_{1}) \neq 0$. The numbers $u_{n}$ can be found successively from the equality

The series expansion of a proper rational fraction has the form $\dfrac {\varphi (x)}{\psi (x)}=u_{0}+u_{1}x+u_{2}x^{2}+u_{3}x^{3}+\ldots$, where the numbers $u_{n}$ take only the values $0$ and $\pm 1$. Prove that

where the numbers $a_{\nu }$ can likewise take only the values $0$ and $\pm 1$. (Laguerre.)

Prove the equality

where $x_{\nu }$ are the roots of the equation $x^{n}=1$ other than unity.

All the roots of the equation $f(x)=0$ of degree $n$ are real. Prove that in the interval $\left(-\dfrac {x_{\nu }-x_{\nu -1}}{n}+x_{\nu }, \dfrac {x_{\nu +1}-x_{\nu }}{n}+x_{\nu }\right)$, where $x_{\nu -1}, x_{\nu }, x_{\nu +1}$ are three consecutive roots of $f(x)$, there are no roots of $f'(x)$.

All the roots of the polynomial $f(x)$ of degree $n$ are real, as is the number $m$. Prove that the modulus of the imaginary part of each root of the equation $f(x)+imf'(x)=0$ is less than $\left|nm\right|$.