Thread: Urgent, residue and winding numbers

1. Urgent, residue and winding numbers

1. Evaluate the integrals

$\int ^{2\pi} _0 \frac{dt}{2+sint}$

and

$\int ^{\infty} _0 \frac{x^2}{x^4 +x}dx$

2. Let $z_1,z_2,z_3,z_4 \epsilon \mathbb{C}$ be the vertices of a closed quadrilateral $\Box$ such that $[z_1,z_2] \oplus [z_2,z_3]\oplus [z_3,z_4]\oplus [z_4,z_1]$ traverses $\partial \Box$ in counterclockwise direction (abusing notation, we shall write $\partial \Box$ for that curve) show that $v(\partial \Box,z)$ is 0 for $z\epsilon \mathbb{C}  \Box$ (C-minus Box) and 1 if $z$ is an interior point of $\Box$ (Hint: do not attempt to parametrize $\partial \Box$..)

2. Originally Posted by jbpellerin
1. Evaluate the integrals

$\int ^{2\pi} _0 \frac{dt}{2+sint}$

and

$\int ^{\infty} _0 \frac{x^2}{x^4 +x}dx$

2. Let $z_1,z_2,z_3,z_4 \epsilon \mathbb{C}$ be the vertices of a closed quadrilateral $\Box$ such that $[z_1,z_2] \oplus [z_2,z_3]\oplus [z_3,z_4]\oplus [z_4,z_1]$ traverses $\partial \Box$ in counterclockwise direction (abusing notation, we shall write $\partial \Box$ for that curve) show that $v(\partial \Box,z)$ is 0 for $z\epsilon \mathbb{C}  \Box$ (C-minus Box) and 1 if $z$ is an interior point of $\Box$ (Hint: do not attempt to parametrize $\partial \Box$..)
For the first ones do you need to do it by CA?

3. CA? I'm pretty sure the prof wants us to use the residue theorem
theres an example in the book with cos instead of sin and I'm working on getting it to work for this question

4. Originally Posted by jbpellerin
1. Evaluate the integrals

$\int ^{2\pi} _0 \frac{dt}{2+sint}$

Notice that,
$\int_0^{2\pi} \frac{dt}{2+\sin t} = \oint \limits_{|z|=1} \frac{1}{2+\tfrac{1}{2i}(z - z^{-1})} \cdot \frac{1}{iz} dz$

Thus, we get,
$\oint_{|z|=1} \frac{2dz}{z^2+4iz - 1}$

Now apply the residue theorem.

5. Originally Posted by jbpellerin
$\int ^{\infty} _0 \frac{x^2}{x^4 +x}dx$
By integrating over partial circular contour of angle $2\pi/n$ we get $\int_0^{\infty} \frac{dx}{x^n+1} = \frac{(\pi/n)}{\sin (\pi/n)}$ for $n\geq 2$.

$\int_0^{\infty} \frac{x^2}{x^4+x} dx=\int_0^{\infty} \frac{x}{x^3+1} = \int_0^{\infty} \frac{dx}{x^2-x+1} - \int_0^{\infty} \frac{dx}{x^3+1}$

Now,
$\int_0^{\infty} \frac{dx}{x^2-x+1} = \int_0^{\infty} \frac{dx}{\left( x - \frac{1}{2} \right)^2 + \frac{3}{4} } = \int_{-1/2}^{\infty} \frac{dt}{t^2 + \frac{3}{4}}$
That computes to,
$\frac{2}{\sqrt{3}}\tan^{-1} \left( \frac{2t}{\sqrt{3}} \right) \bigg|_{-1/2}^{\infty} = \frac{\pi}{\sqrt{3}} + \frac{\pi}{3\sqrt{3}} = \frac{4\pi}{3\sqrt{3}}$

And by above result,
$\int_0^{\infty}\frac{dx}{x^3+1} = \frac{(\pi/3)}{\sin (\pi/3)} = \frac{2\pi}{3\sqrt{3}}$

Thus, the final answer is, $\frac{4\pi}{3\sqrt{3}}-\frac{2\pi}{3\sqrt{3}} = \frac{2\pi}{3\sqrt{3}}$

6. Originally Posted by jbpellerin
2. Let $z_1,z_2,z_3,z_4 \epsilon \mathbb{C}$ be the vertices of a closed quadrilateral $\Box$ such that $[z_1,z_2] \oplus [z_2,z_3]\oplus [z_3,z_4]\oplus [z_4,z_1]$ traverses $\partial \Box$ in counterclockwise direction (abusing notation, we shall write $\partial \Box$ for that curve) show that $v(\partial \Box,z)$ is 0 for $z\epsilon \mathbb{C} ` \Box$ (C-minus Box) and 1 if $z$ is an interior point of $\Box$ (Hint: do not attempt to parametrize $\partial \Box$..)
If $z$ is outside the rectange then computing the winding number is to evaluate $\oint_{\partial \Box}\frac{dw}{w-z}=0$ by Cauchy's theorem. If $z$ is inside then it can be shown, here (post #8 with long proof), that if $C$ is a tiny circle around $z$ still inside $\partial \Box$ then $\oint_{\partial \Box} \frac{dw}{w-z} = \oint_{C} \frac{dw}{w-z} = 2\pi i$. And so the winding number is $1$.

7. Whenever I make a thread on here I always hope you're online because you always have excellent answers
thank you again for the help

8. Originally Posted by jbpellerin
Whenever I make a thread on here I always hope you're online because you always have excellent answers
You should not say that because I have a terrible ego problem.