I have to compute the following integral: $$\oint_{|z|=2}\frac{2^z}{(3z+5)^7}$$
I see immediately that I need to use the Cauchy integral formula. I rewrite the integral as follows:
$$\oint_{|z|=2}\frac{2^z}{(3z+5)^7} = \oint_{|z|=2}\frac{(1/3)^72^z}{(z+5/3)^7}$$
Now I can use the Cauchy integral formula, since $\frac{5}{3}$ is in the contour. I used the Cauchy integral formula for derivatives, and found:
$$\oint_{|z|=2}\frac{(1/3)^72^z}{(z+5/3)^7} = \frac{2\pi i \cdot (1/3)^7 \cdot 2^{(5/3)} \cdot \log^6(2)}{6!} $$
Am I right, or did I make a mistake?
Almost. It's $-\dfrac53$ that matters here, not $\dfrac53$, and therefore you have\begin{align}\oint_{|z|=2}\frac{2^z}{(3z+5)^7} &= \frac{2\pi i\left(1/3\right)^72^{-5/3} \log^6(2)}{6}\\&=\frac{2\pi i\log^6(2)}{2^{5/3}3^76!}.\end{align}