  We present analytic descriptions of atomic interaction at ultracold
temperatures using both single-channel and multichannel quantum-defect
theories. In the case of a single channel, addressed in this paper, the
expansion of [B. Gao, Phys. Rev. A \textbf{58}, 4222 (1998)] is generalized to
higher orders for angular momentum $l\ge 2$ to give a more complete description
of ultracold scattering, including an analytic description of ultracold shape
resonances of arbitrary $l$. We also introduce a generalized scattering length
that is well defined and useful for all partial waves, to replace the
traditional definition that fails for $l\ge 2$ due to the long-range van der
Waals interaction. The results are used in a companion paper to derive analytic
descriptions of atomic interaction around a magnetic Feshbach resonance of
arbitrary angular momentum $l$.
