In [1]:
assume(z<1)$assume(-1<z)$
hypergeometric([1,1],[2],z);
%=hypergeometric_simp(%);
Out[1]:
\[\tag{${\it \%o}_{3}$}F\left( \left. \begin{array}{c}1,\;1\\2\end{array} \right |,z\right)\] 
SB-KERNEL:REDEFINITION-WITH-DEFUN: redefining MAXIMA::SIMP-HYPERGEOMETRIC in DEFUN
Out[1]:
\[\tag{${\it \%o}_{4}$}F\left( \left. \begin{array}{c}1,\;1\\2\end{array} \right |,z\right)=-\frac{\log \left(1-z\right)}{z}\]
In [2]:
assume(0<z)$
hypergeometric([1/2,1],[3/2],z^2);
%=hypergeometric_simp(%);
Out[2]:
\[\tag{${\it \%o}_{6}$}F\left( \left. \begin{array}{c}\frac{1}{2},\;1\\\frac{3}{2}\end{array} \right |,z^2\right)\]
Out[2]:
\[\tag{${\it \%o}_{7}$}F\left( \left. \begin{array}{c}\frac{1}{2},\;1\\\frac{3}{2}\end{array} \right |,z^2\right)=\frac{\log \left(\frac{z+1}{1-z}\right)}{2\,z}\]
In [3]:
hypergeometric([1/2,1],[3/2],-z^2);
%=hypergeometric_simp(%);
Out[3]:
\[\tag{${\it \%o}_{8}$}F\left( \left. \begin{array}{c}\frac{1}{2},\;1\\\frac{3}{2}\end{array} \right |,-z^2\right)\]
Out[3]:
\[\tag{${\it \%o}_{9}$}F\left( \left. \begin{array}{c}\frac{1}{2},\;1\\\frac{3}{2}\end{array} \right |,-z^2\right)=\frac{\arctan z}{z}\]
In [4]:
hypergeometric([1/2,1/2],[3/2],z^2);
%=hypergeometric_simp(%);
Out[4]:
\[\tag{${\it \%o}_{10}$}F\left( \left. \begin{array}{c}\frac{1}{2},\;\frac{1}{2}\\\frac{3}{2}\end{array} \right |,z^2\right)\]
Out[4]:
\[\tag{${\it \%o}_{11}$}F\left( \left. \begin{array}{c}\frac{1}{2},\;\frac{1}{2}\\\frac{3}{2}\end{array} \right |,z^2\right)=\frac{\arcsin z}{z}\]