The hypergeometric function $F(a,b;c;z)$ is defined by the Gauss series

on the disk , and by analytic continuation elsewhere. In general, $F(a,b;c;z)$ does not exist when $c=0,-1,-2,\mathrm{\dots }$. The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real $z$-axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of $F(a,b;c;z)$.

For all values of $c$

again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way.

Except where indicated otherwise principal branches of $F(a,b;c;z)$ and $𝐅(a,b;c;z)$ are assumed throughout the DLMF.

The difference between the principal branches on the two sides of the branch cut (§4.2(i)) is given by

On the circle of convergence, $|z|=1$, the Gauss series:

• (a)

  Converges absolutely when $\mathrm{\Re }\left(c-a-b\right)>0$.

• (b)

  Converges conditionally when and $z=1$ is excluded.

• (c)

  Diverges when $\mathrm{\Re }\left(c-a-b\right)\le -1$.

For the case $z=1$ see also §15.4(ii).

The principal branch of $𝐅(a,b;c;z)$ is an entire function of $a$, $b$, and $c$. The same is true of other branches, provided that $z=0$, $1$, and $\mathrm{\infty }$ are excluded. As a multivalued function of $z$, $𝐅(a,b;c;z)$ is analytic everywhere except for possible branch points at $z=0$, $1$, and $\mathrm{\infty }$. The same properties hold for $F(a,b;c;z)$, except that as a function of $c$, $F(a,b;c;z)$ in general has poles at $c=0,-1,-2,\mathrm{\dots }$.

Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. In particular

For example, when $a=-m$, $m=0,1,2,\mathrm{\dots }$, and $c\ne 0,-1,-2,\mathrm{\dots }$, $F(a,b;c;z)$ is a polynomial:

This formula is also valid when $c=-m-\mathrm{\ell }$, $\mathrm{\ell }=0,1,2,\mathrm{\dots }$, provided that we use the interpretation

and not

which sometimes needs to be used in §15.4. (Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does $𝐅(a,b;c;z)$, which is analytic at $c=0,-1,-2,\mathrm{\dots }$.)

For comparison of $F(a,b;c;z)$ and $𝐅(a,b;c;z)$, with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7.

Let $m$ be a nonnegative integer. Formula (15.4.6) reads $F(a,b;a;z)={(1-z)}^{-b}$. The right-hand side can be seen as an analytical continuation for the left-hand side when $a$ approaches $-m$. In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that $F(-m,b;-m;z)$ is equal to the first $m+1$ terms of the Maclaurin series for ${(1-z)}^{-b}$.