Large antenna arrays at the base station can facilitate power efficient single user downlink communication due to the inherent array power gain, i.e., under an average only total transmit power constraint, for a fixed desired information rate, the required total transmit power can be reduced by increasing the number of base station antennas (e. g. with i.i.d. fading, the required total transmit power can be reduced by roughly 3 dB with every doubling in the number of base station antennas, i.e., an O(N) array power gain can be achieved with N antennas). However, in practice, building power efficient large antenna arrays would require power efficient amplifiers/analog RF components. With current technology, highly linear power amplifiers generally have low power efficiency, and therefore linearity constraints on power amplifiers must be relaxed. Under such relaxed linearity constraints, the transmit signal that suffers the least distortion is a signal with constant envelope (CE). In this paper, we consider a single user Gaussian multiple-input single-output (MISO) downlink channel where the signal transmitted from each antenna is constrained to have a constant envelope (i.e., for every channel-use the amplitude of the signal transmitted from each antenna is constant, irrespective of the channel realization). We show that under such a per-antenna CE constraint, the complex noise-free received signal lies in the interior of a "doughnut" shaped region in the complex plane. The per-antenna CE constrained MISO channel is therefore equivalent to a doughnut channel, i.e., a single-input single-output (SISO) AWGN channel where the channel input is constrained to lie inside a "doughnut" shaped region. Using this equivalence, we analytically compute a closed-form expression for an achievable information rate under the per-antenna CE constraint. We then show that, for a broad class of fading channels (i.i.d. and direct-line-of-sight (DLOS)), even under the more stringent per-antenna CE constraint (compared to the average only total power constraint), an O(N) array power gain can still be achieved with N base station antennas. We also show that with N >> 1, compared to the average only total transmit power constrained channel, the extra total transmit power required under the per-antenna CE constraint, to achieve a desired information rate is small and bounded for a broad class of fading channels (i.i.d. and DLOS). We also propose novel CE precoding algorithms. The analysis and algorithms presented are general and therefore applicable to conventional systems with a small number of antennas. Analytical results are supported with numerical results for the i.i.d. Rayleigh fading channel.