Figure 1.

The convergence region of the virial expansion is the largest open disc centered at zero in the domain of definition of the Bose-Einstein integral gν(z)."

Figure 2.

The singularities of ${g}_{3/2}\left(z\right)$ and its Padé approximants ${\left[2/2\right]}_{3/2}\left(z\right)$ and ${\left[3/3\right]}_{3/2}\left(z\right)$."

Figure 3.

The convergence region of the virial expansion is the largest open disc centered at zero in the domain of definition of the Fermi-Dirac integral fν(z)."

Figure 4.

Comparison of the Padé approximants, ${\left[2/2\right]}_{{f}_{3/2}}\left(z\right)$ and ${\left[3/3\right]}_{{f}_{3/2}}\left(z\right)$, with the virial expansion and the exact result. Though the Padé approximant is constructed from the virial expansion, it apples to z > 1 where the virial expansion is invalid."

Figure 5.

Comparison of various Padé approximants. The Padé approximants ${\left[3/3\right]}_{{f}_{3/2}}\left(z\right)$, ${\left[4/2\right]}_{{f}_{3/2}}\left(z\right)$, ${\left[2/4\right]}_{{f}_{3/2}}\left(z\right)$, ${\left[1/5\right]}_{{f}_{3/2}}\left(z\right)$, and ${\left[5/1\right]}_{{f}_{3/2}}\left(z\right)$ are constructed from the virial expansion with 6 virial coefficients. The Padé approximants ${\left[2/2\right]}_{{f}_{3/2}}\left(z\right)$, ${\left[1/3\right]}_{{f}_{3/2}}\left(z\right)$, and ${\left[3/1\right]}_{{f}_{3/2}}\left(z\right)$ are constructed from the virial expansion with 4 virial coefficients."