  In this paper we present a new compact expression of the elliptic genus of
SL(2)/U(1)-supercoset theory by making use of the `spectral flow method' of the
path-integral evaluation. This new expression is written in a form like a
Poincare series with a non-holomorphic Gaussian damping factor, and manifestly
shows the modular and spectral flow properties of a real analytic Jacobi form.
As a related problem, we present similar compact formulas for the modular
completions of various mock modular forms which appear in the representation
theory of N=2,4 superconformal algebras.
  We further discuss the generalization to the cases of arbitrary
spin-structures, that is, the world-sheet fermions with twisted boundary
conditions parameterized by a continuous parameter. This parameter is naturally
identified with the `u-variable' in the Appell-Lerch sum.
