Problem 1599 (difficulty: 8/10)

Let \(\displaystyle f\) be a holomorphic function on the disc \(\displaystyle |z|<1+\varepsilon\) and let \(\displaystyle |a|<1\). Find a function \(\displaystyle \varphi_a:[0,2\pi]\to\RR\) such that

\(\displaystyle f(a)=\frac1{2\pi}\int_0^{2\pi}f(e^{it})\varphi_a(t)dt.\)

Give me another random problem!