  This paper is part of a general program in complex dynamics to understand
parameter spaces of transcendental maps with finitely many singular values.
  The simplest families of such functions have two asymptotic values and no
critical values. These families, up to affine conjugation, depend on two
complex parameters. Understanding their parameter spaces is key to
understanding families with more asymptotic values, just as understanding
quadratic polynomials was for rational maps more generally.
  The first such families studied were the one-dimensional slices of the
exponential family, $\exp(z) + a$, and the tangent family $\lambda \tan z$. The
exponential case exhibited phenomena not seen for rational maps: Cantor
bouquets in both the dynamic and parameter spaces, and no bounded hyperbolic
components. The tangent case, with its two finite asymptotic values $\pm
\lambda i$, is closer to the rational case, a kind of infinite degree version
of the latter.
  In this paper, we consider a general family that interpolates between
$\exp(z) + a$ and $\lambda \tan z$. Our new family has two asymptotic values
and a one-dimensional slice for which one of the asymptotic values is
constrained to be pole, the "polar asymptotic value" of the title. We show how
the dynamic and parameter planes for this slice exhibit behavior that is a
surprisingly delicate interplay between that of the $\exp(z) + a$ and $\lambda
\tan z$ families.
