0
votes
0answers
41 views

Polynomial Expansion in a Complex Improper Integral

I want to evaluate the integral below by using the Residue theorem. $$\int_{-\infty}^{\infty}\frac{\exp(-iwt)w}{(w-ic)\sqrt{w^2-aw+b}}dw$$ There are branch cut points due to the square rooted term ...
3
votes
1answer
123 views

Roots of $z^{2n} + \alpha z^{2n -1} + \beta ^2$

I've been looking at a problem available here. The problem is: Let $n$ be a natural number, and $\alpha$, $\beta$ nonzero reals. Show that the number of roots of $p(z) = z^{2n} + \alpha z^{2n -1} + ...
1
vote
1answer
167 views

Generating Laguerre polynomials using gamma functions

An exercise given by my complex analysis assistant goes as follows: For $n \in \mathbb{N}$ and $x>0$ we define $$P_n(x) = \frac{1}{2\pi i} \oint_\Sigma ...
0
votes
1answer
111 views

Sum of all the residues of the function $a(z)/b(z)$

Let $a(z)$ and $b(z)$ be polynomials such that $ \deg(b) \ge \deg(a)+2$. Find the sum of all the residues of the function $a(z)/b(z)$. In class, I learned that $$ - \text{ sum of all residues of ...
8
votes
2answers
191 views

Why is the (-1)-th coefficient of $f^n f'$ equal to 0, without dividing by $n+1$?

Let $R$ be a commutative ring, and $n$ be a nonnegative integer. Let $f\in R\left[t,t^{-1}\right]$ be a Laurent polynomial in one variable $t$ over $R$ (this means a formal $R$-linear combination of ...
0
votes
1answer
532 views

polynomial long division - coefficients in modulo 2 coset

What's important when dividing the following two polynomials $x^4 + x + 1 \qquad \;\;\,\in \mathbb{Z}/2\mathbb{Z}[x]$ $x^3 - x^2 + 1 \qquad \in \mathbb{Z}/2\mathbb{Z}[x]$ How two calculate the ...