= Solution
Starting from $f_{-1}=\sin\theta\,e^{-i\phi}$, part (i) gives $L^2f_{-1}=2\hbar^2f_{-1}$ and $L_3f_{-1}=-\hbar f_{-1}$. Apply the <angular momentum raising operator> explicitly:
$$
L_+f_{-1}=\hbar e^{i\phi}\left(\cos\theta e^{-i\phi}+i\cot\theta(-i\sin\theta e^{-i\phi})\right)=2\hbar\cos\theta.
$$
A second application gives
$$
L_+\cos\theta=-\hbar\sin\theta e^{i\phi}.
$$
Since $L_+$ commutes with $L^2$, both raised functions retain the <eigenvalue> $2\hbar^2$. Their $L_3$ <eigenvalues> follow either by differentiating their $\phi$ dependence or from $[L_3,L_+]=\hbar L_+$. Thus three valid unnormalized functions, in the requested order, are
$$
\boxed{f_{-1}=\sin\theta e^{-i\phi},\qquad f_0=\cos\theta,\qquad f_{+1}=\sin\theta e^{i\phi}.}
$$
For later <Born rule> calculations, normalize with the <solid angle> measure $d\Omega=\sin\theta\,d\theta\,d\phi$. Their squared <norms> are $8\pi/3$, $4\pi/3$, $8\pi/3$, respectively. For example $\int_0^\pi\sin^3\theta\,d\theta=4/3$ and $\int_0^\pi\cos^2\theta\sin\theta\,d\theta=2/3$. A convenient <orthonormal basis> is therefore
$$
\chi_{-1}=\sqrt{\frac3{8\pi}}\sin\theta e^{-i\phi},\qquad \chi_0=\sqrt{\frac3{4\pi}}\cos\theta,\qquad \chi_{+1}=\sqrt{\frac3{8\pi}}\sin\theta e^{i\phi}.
$$
<Orthogonality> follows from the distinct Fourier factors in $\phi$. An overall phase of each basis function is arbitrary; these choices need not impose the usual phase convention for the <spherical harmonics>.
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