Evaluating $\frac{1}{2i\pi} \oint_C \frac{z^2}{z^2+4} dz$ where $C$ is the square with vertices at $\pm 2$ and $\pm 2 + 4i$ So, I rewrote the contour integral:$\frac{1}{2i\pi} \oint_C \frac{z^2}{(z+2i)(z-2i)} dz$ then used the Cauchy Integral Theorem: $\frac{1}{2i\pi} \oint_C \frac{z^2}{(z+2i)(z-...

Let $1 < x < 4$ , $a_1 = x$ and $b_1 = 0$. Now consider the (conditional) iterations if $a_n > b_n$ then $a_{n+1} = 4(a_n - b_n - 1)$ $b_{n+1} = 2(b_n + 2)$ else ( $a_n = b_n$ or $a_n < b_n$ ) $a_{n+1} = 4 a_n$ $b_{n+1} = 2 b_n$ Now consider $c_n = \frac{b_n}{2^n}$ Now define $f(x) = \lim_{n \to ...