Answer
$Normalized \space wave \space function(\psi_{(\phi)}) = \big(\frac{1}{2\pi}\big)^{1/2} e^{i\phi}$
Work Step by Step
To Normalize a wave function $\psi_{(\phi)}$ we need to find N such that
$$N^2 \int_0^{2\pi} \psi_{(\phi)}^* \psi_{(\phi)} \space d\phi = 1$$ and the normalized wave function will be $N\psi$
Here, $\psi_{(\phi)} = e^{i\phi}$ so, $\psi_{(\phi)}^* = e^{-i\phi}$
$$N^2\int_0^{2\pi} e^{-i\phi} e^{i\phi} d\phi = 1$$
$$N^2\int_0^{2\pi} 1 d\phi = 1$$
$$N^2 \times 2\pi = 1$$
$$N = \big(\frac{1}{2\pi}\big)^{1/2}$$
And hence the normalized wave function is $$\big(\frac{1}{2\pi}\big)^{1/2} e^{i\phi}$$
