  Based on the Hilb type formula between Jacobi polynomials and Bessel
functions, optimal decay rates on Jacobi expansion coefficients are derived, by
applying van der Corput type lemmas, for functions of logarithmatic
singularities, which leads to the optimal convergence rates on the Jacobi,
Gegenbauer and Chebyshev orthogonal projections. It is interesting to see that
for boundary singularities, one may get faster convergence rate on the Jacobi
or Gegenbauer projection as $(\alpha,\beta)$ and $\lambda$ increases. The
larger values of parameter, the higher convergence rates can be achieved. In
particular, the Legendre projection has one half order higher than Chebyshev.
Moreover, if $\min\{\alpha,\beta\}>0$ and $\lambda>\frac{1}{2}$, the Jacobi and
Gegenbauer orthogonal projections have higher convergence orders compared with
Legendre. While for interior singularity, the convergence order is independent
of $(\alpha,\beta)$ and $\lambda$.
