It's well known that the Schrödinger equation can be solved in elementary functions for a spherically symmetric 1/r potential, but I had to dig pretty hard to find the complete formula for the resulting wave function. Here it is, correctly normalized so that its norm squared integrates to one. These are eigenstates for energy, angular momentum, and the z-projection of angular momentum. To get the real-valued wave functions that you see in chemistry books, you have to take the sum or difference of two functions with opposite m values and rescale appropriately. Also note that theta is latitude, ranging from zero at the north pole to pi at the south pole, and phi is longitude. Kudos to Wikipedia for having all of this information available, and to MathJax for making it possible to render mathematical formulas in HTML.
\[ \Psi_{nlm}(r,\theta,\phi) = \] \[ \left ( \frac{2Z}{na_0} \right ) ^ {3/2} \sqrt{\frac{(n-l-1)!}{2n(n+l)!}} \sqrt{\frac{2l+1}{4\pi}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} \frac{1}{2^ll!} \] \[ e^{im\phi} \sin^{|m|}\theta \left [ \frac{d^{l+|m|}}{dx^{l+|m|}} (x^2 - 1)^l \right ] _{x=\cos \theta} e^{\frac{-Zr}{na_0}} \left ( \frac{2Zr}{na_0} \right ) ^l \] \[ \left [ \Sigma_{k=0}^{n-l-1} \frac{(-1)^k}{k!} \binom{n+l}{n-l-1-k} \left ( \frac{2Zr}{na_0} \right ) ^k \right ] \]Here n, l, and m are the principal, angular momentum, and z-projection of angular momentum quantum numbers, respectively. Z is the atomic number of the nucleus, and a0 is the Bohr radius (technically, this should be the reduced Bohr radius of the nucleus-electron system, but that differs by only one part in 1836 for the hydrogen atom and less for heavier nuclei).