0
votes
1answer
46 views

Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi. $$
2
votes
2answers
86 views

Show these approximations of $\cos$, $\sin$ and $\tan$ are exact.

A while back I was looking for an approximation to $\cos(x)$ and I constructed a polynomial with zeros in the same places as the first few zeros of $cos(x)$: $$c_n(x) = \frac{\prod_{i=1}^n ...
1
vote
2answers
151 views

Use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $?

How do I use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter- clockwise ...
2
votes
1answer
294 views

Singularities of $ \ \frac{z-1}{z^2 \sin z} \ $

Find all singularities of $ \ \frac{z-1}{z^2 \sin z} \ $ Determine if they are isolated or nonisolated. This is not hard, it is z = 0 and z = k*pi. But how do I: For isolated singularities, ...
12
votes
2answers
626 views

Sum of the squares of the reciprocals of the fixed points of the tangent function

The sum of the squares of the reciprocals of the positive fixed points of the tangent function is $1/10$. I've seen this proved by means of residues, but I don't remember the details. I've also ...