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A superposition of theta-functions (cf. [[Theta-function|Theta-function]]) of the first order  $  \Theta _ {H} ( u) $, 
$  u = ( u _ {1} \dots u _ {p} ) $, 
with half-integral characteristics  $  H $, 
and of Abelian integrals (cf. [[Abelian integral|Abelian integral]]) of the first order, used by B. Riemann in 1857 to solve the [[Jacobi inversion problem|Jacobi inversion problem]].

Let  $  F( u, w) = 0 $
be an algebraic equation which defines a compact [[Riemann surface|Riemann surface]]  $  F $
of genus  $  p $; 
let  $  \phi _ {1} \dots \phi _ {p} $
be a basis of the Abelian differentials (cf. [[Abelian differential|Abelian differential]]) of the first kind on  $  F $
with  $  ( p \times 2p) $-
dimensional period matrix

$$ 
W  =  \| \pi i E, A \|  =  \left \|

\begin{array}{cccccc}
\pi i  &\dots  & 0  &a _ {11}  &\dots  &a _ {1p}  \\
 0  &\dots  & 0  &a _ {21}  &\dots  &a _ {2p}  \\
\cdot  &\dots  &\cdot  &\cdot  &\dots  &\cdot  \\
 0  &\dots  &\pi i  &a _ {p1}  &\dots  &a _ {pp}  \\
\end{array}

\right \| .
$$

Let

$$ 
u( w)  =  \left ( u _ {1} ( w _ {1} ) =
\int\limits _ { c _ {1} } ^ { {w _ 1 } }
\phi _ {1} \dots u _ {p} ( w _ {p} ) =
\int\limits _ { c _ {p} } ^ { {w _ p } } \phi _ {p} \right )
$$

be the vector of basis Abelian integrals of the first kind, where  $  ( c _ {1} \dots c _ {p} ) $
is a fixed system of points in  $  F $
and  $  w = ( w _ {1} \dots w _ {p} ) $
is a varying system of points in  $  F $. 
For any theta-characteristic

$$ 
H  =  \left \|

\begin{array}{c}
 h  \\
h  ^  \prime   \\
\end{array}
 \right \|  = \ 
\left \| 
\begin{array}{ccc}
h _ {1}  &\dots  &h _ {p}  \\
h _ {1}  ^  \prime   &\dots  &h _ {p}  ^  \prime   \\
\end{array}
 \right \| ,
$$

where the integers  $  h _ {i} , h _ {i}  ^  \prime  $
take the values 0 or 1 only, it is possible to construct a theta-function  $  \Theta _ {H} ( u) $
with period matrix  $  W $
such that  $  \Theta _ {H} ( u) $
satisfies the fundamental relations

$$ \tag{1 }
\left . 
\begin{array}{c}
\Theta _ {H} ( u + \pi ie _  \mu  )  =  (- 1) ^ {h _  \mu  } \Theta _ {H} ( u),  \\
\Theta _ {H} ( u + e _  \mu  A)  =  (- 1) ^ {h _  \mu   ^  \prime  }
 \mathop{\rm exp} (- a _ {\mu \mu }  - 2u _  \mu  ) \cdot \Theta _ {H} ( u).  \\
\end{array}
 \right \}
$$

Here  $  e _  \mu  $
is the  $  \mu $-
th row vector of the identity matrix  $  E $, 
$  \mu = 1 \dots p $. 
If  $  z = ( z _ {1} \dots z _ {p} ) $
is a fixed vector in the complex space  $  \mathbf C  ^ {p} $, 
then the Riemann theta-function  $  \Phi _ {H} ( w) $
can be represented as the superposition

$$ \tag{2 }
\Phi _ {H} ( w)  =  \Theta _ {H} ( u( w) - z).
$$

In the domain  $  F ^ { \star } $
that is obtained from  $  F $
after removal of sections along the cycles  $  a _ {1} , b _ {1} \dots a _ {p} , b _ {p} $
of a homology basis of  $  F $, 
the Riemann theta-functions (2) are everywhere defined and analytic. When crossing through sections the Riemann theta-functions, as a rule, are multiplied by factors whose values are determined from the fundamental relations (1). In this case, a special role is played by the theta-function of the first order  $  \Phi ( u) = \Theta _ {0} ( u) $
with zero characteristic  $  H = 0 $. 
In particular, the zeros  $  \eta _ {1} \dots \eta _ {p} $
of the corresponding Riemann theta-function  $  \Phi ( w) = \Phi _ {0} ( w) $
determine the solution to the Jacobi inversion problem.

Quotients of Riemann theta-functions of the type  $  \Psi _ {H} ( w) = \Theta _ {H} ( u( w)) $
with a common denominator  $  \Psi ( w) = \Theta ( u( w)) = \Theta _ {0} ( u( w)) $
are used to construct analytic expressions solving the inversion problem. It can be seen from (1) that such quotients  $  \Psi _ {H} ( w)/ \Psi ( w) $
can have as non-trivial factors only  $  - 1 $, 
and the squares of these quotients are single-valued meromorphic functions on  $  F $, 
i.e. rational point functions on the surface  $  F $. 
The squares and other rational functions in quotients of theta-functions used in this case are special Abelian functions (cf. [[Abelian function|Abelian function]]) with  $  2p $
periods. The specialization is expressed by the fact that  $  p( p+ 1)/2 $
different elements  $  a _ {\mu \nu }  $
of the symmetric matrix  $  A $, 
when  $  p > 3 $, 
are connected by definite relations imposed by the conformal structure of  $  F $, 
so that  $  3( p- 1) $
remain independent among them.

Riemann theta-functions constructed for a hyper-elliptic surface  $  F $, 
when  $  F( u, w) = w  ^ {2} - P( u) $
where  $  P( u) $
is a polynomial of degree  $  n \geq  5 $
without multiple roots, are sometimes referred to as hyper-elliptic theta-functions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) pp. Chapt. 9 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Markushevich, "Introduction to the classical theory of Abelian functions" , Moscow (1979) (In Russian) {{MR|0544988}} {{ZBL|0493.14023}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Krazer, "Lehrbuch der Thetafunktionen" , Chelsea, reprint (1970) {{MR|}} {{ZBL|0212.42901}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Conforto, "Abelsche Funktionen und algebraische Geometrie" , Springer (1956) {{MR|0079316}} {{ZBL|0074.36601}} </TD></TR></table>

====Comments====
Nowadays a Riemann theta-function is defined as a theta-function of the first order with half-integral characteristic corresponding to the [[Jacobi variety|Jacobi variety]] of an algebraic curve (or a compact Riemann surface). A general theta-function corresponds to an arbitrary [[Abelian variety|Abelian variety]]. The problem of distinguishing the Riemann theta-functions among the general theta-functions is called the Schottky problem. It has been solved (see [[Schottky problem|Schottky problem]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , '''1–2''' , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Arbarello, "Periods of Abelian integrals, theta functions, and differential equations of KdV type" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , '''I''' , Amer. Math. Soc. (1987) pp. 623–627 {{MR|0934264}} {{ZBL|0696.14019}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Mumford, "Tata lectures on Theta" , '''1–2''' , Birkhäuser (1983–1984) {{MR|2352717}} {{MR|2307769}} {{MR|2307768}} {{MR|1116553}} {{MR|0742776}} {{MR|0688651}} {{ZBL|1124.14043}} {{ZBL|1112.14003}} {{ZBL|1112.14002}} {{ZBL|0744.14033}} {{ZBL|0549.14014}} {{ZBL|0509.14049}} </TD></TR></table>