Let $\gamma_{R}:[0, \pi] \rightarrow \mathbb{C}$ s.t $\gamma_{R}(t)=Re^{it}$ $, \quad t\in [0, \pi]$ given that $$\lim _{R \rightarrow \infty} \max _{|z|=R}|f(z)|=0$$ show that for every $a>0$ $$ \lim _{R \rightarrow \infty} \int_{\gamma_{R}} e^{i a z} f(z) d z=0 $$ I was thinking on using the fact that $$|\int_{\gamma_{R}} e^{i a z} f(z) d z|\leq L_{\gamma R}\cdot max |e^{iaz}f(z)|$$ for every $z\in \gamma_{R}$ while $L_{\gamma R}= \frac{\pi R}{2}$
But I am not sure how I can use the given now because I have $R \cdot max |f(z)|$ this is like $$\infty \cdot 0$$
