# Tagged Questions

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 $[0,2\pi]$: $$I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi.$$
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 ...
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 ...
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, ...
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 ...