Quasi-Spherical harmonics, Y m ℓ (θ, φ) are derived and presented for half-odd-integer values of ℓ and m. The form of the φ factor is identical to that in the case of integer ℓ and m: exp (imφ). However, the domain of these functions in the half-odd-integer case is 0≤φ<4π rather than the domain 0≤φ<2π in the case of integer ℓ and m (the true spherical harmonics). The form of the θ factor, P |m| ℓ (θ) (an associated Legendre function) is (as in the integer case) the factor (sin θ) |m| multiplied by a polynomial in cos θ of degree (ℓ−|m|) (an associated Legendre polynomial). A substantial difference between the spherical (integer ℓ and m) and quasi-spherical (half-odd-integer ℓ and m) Legendre functions is that the latter have an irrational factor of √ sin θ whereas the factor of the truly spherical functions is an integer power of sin θ. The domain of both the true and quasi spherical associated Legendre functions is the same: 0≤θ<π. A table of the Associated Legendre Functions is presented for both integer and half-odd-integer values of ℓ and m, for |m| = 0, 1 2 , 1 . . . 11 2 , and for (ℓ−|m|) = 0, 1, 2, 3, 4, 5. The table displays the similarity between the functions for integer ℓ and m (which are well known) and those for half-odd-integer ℓ and m (which have not been recognized previously).