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Problems named after P. Cousin [[#References|[1]]], who first solved them for certain simple domains in the complex  $  n $-
dimensional space  $  \mathbf C  ^ {n} $.

==First (additive) Cousin problem.==
Let  $  {\mathcal U} = \{ U _  \alpha  \} $
be a covering of a complex manifold  $  M $
by open subsets  $  U _  \alpha  $, 
in each of which is defined a meromorphic function  $  f _  \alpha  $; 
assume that the functions  $  f _ {\alpha \beta }  = f _  \alpha  - f _  \beta  $
are holomorphic in  $  U _ {\alpha \beta }  = U _  \alpha  \cap U _  \beta  $
for all  $  \alpha , \beta $(
compatibility condition). It is required to construct a function  $  f $
which is meromorphic on the entire manifold  $  M $
and is such that the functions  $  f - f _  \alpha  $
are holomorphic in  $  U _  \alpha  $
for all  $  \alpha $. 
In other words, the problem is to construct a global meromorphic function with locally specified polar singularities.

The functions  $  f _ {\alpha \beta }  $, 
defined in the pairwise intersections  $  U _ {\alpha \beta }  $
of elements of  $  {\mathcal U} $, 
define a holomorphic  $  1 $-
cocycle for  $  {\mathcal U} $, 
i.e. they satisfy the conditions

$$ \tag{1 }
f _ {\alpha \beta }  +
f _ {\beta \alpha }   =  0 \ \ 
 \mathop{\rm in}  U _ {\alpha \beta }  ,
$$

$$ 
f _ {\alpha \beta }  + f _ {\beta \gamma }  + f _ {\gamma \alpha }   =  0 \   \mathop{\rm in}  U _  \alpha  \cap U _  \beta  \cap U _  \gamma  ,
$$

for all  $  \alpha , \beta , \gamma $. 
A more general problem (known as the first Cousin problem in cohomological formulation) is the following. Given holomorphic functions  $  f _ {\alpha \beta }  $
in the intersections  $  U _ {\alpha \beta }  $, 
satisfying the cocycle conditions (1), it is required to find functions  $  h _  \alpha  $, 
holomorphic in  $  U _  \alpha  $, 
such that

$$ \tag{2 }
f _ {\alpha \beta }   = \ 
h _  \beta  - h _  \alpha   $$

for all  $  \alpha , \beta $. 
If the functions  $  f _ {\alpha \beta }  $
correspond to the data of the first Cousin problem and the above functions  $  h _  \alpha  $
exist, then the function

$$ 
f  = \ 
\{ f _  \alpha  + h _  \alpha  \ \ 
 \mathop{\rm in}  U _  \alpha  \}
$$

is defined and meromorphic throughout  $  M $
and is a solution of the first Cousin problem. Conversely, if  $  f $
is a solution of the first Cousin problem with data  $  \{ f _  \alpha  \} $, 
then the holomorphic functions  $  h _  \alpha  = f - f _  \alpha  $
satisfy (2). Thus, a specific first Cousin problem is solvable if and only if the corresponding cocycle is a holomorphic coboundary (i.e. satisfies condition (2)).

The first Cousin problem may also be formulated in a local version. To each set of data  $  \{ U _  \alpha  , f _  \alpha  \} $
satisfying the compatibility condition there corresponds a uniquely defined global section of the sheaf  $  {\mathcal M} / {\mathcal O} $, 
where  $  {\mathcal M} $
and  $  {\mathcal O} $
are the sheaves of germs of meromorphic and holomorphic functions, respectively; the correspondence is such that any global section of  $  {\mathcal M} / {\mathcal O} $
corresponds to some first Cousin problem (the value of the section  $  \kappa $
corresponding to data  $  \{ f _  \alpha  \} $
at a point  $  z \in U _  \alpha  $
is the element of  $  {\mathcal M} _ {z} / {\mathcal O} _ {z} $
with representative  $  f _  \alpha  $). 
The mapping of global sections  $  \phi :  \Gamma ( {\mathcal M} ) \rightarrow \Gamma ( {\mathcal M} / {\mathcal O} ) $
maps each meromorphic function  $  f $
on  $  {\mathcal M} $
to a section  $  \kappa _ {f} $
of  $  {\mathcal M} / {\mathcal O} $, 
where  $  \kappa _ {f} ( z) $
is the class in  $  {\mathcal M} _ {z} / {\mathcal O} _ {z} $
of the germ of  $  f $
at the point  $  z $, 
$  z \in M $. 
The localized first Cousin problem is then: Given a global section  $  \kappa $
of the sheaf  $  {\mathcal M} / {\mathcal O} $, 
to find a meromorphic function  $  f $
on  $  M $(
i.e. a section of  $  {\mathcal M} $) 
such that  $  \phi ( f) = \kappa $.

Theorems concerning the solvability of the first Cousin problem may be regarded as a multi-dimensional generalization of the [[Mittag-Leffler theorem|Mittag-Leffler theorem]] on the construction of a meromorphic function with prescribed poles. The problem in cohomological formulation, with a fixed covering  $  {\mathcal U} $, 
is solvable (for arbitrary compatible  $  \{ f _  \alpha  \} $) 
if and only if  $  H  ^ {1} ( {\mathcal U} , {\mathcal O} ) = 0 $(
the Čech cohomology for  $  {\mathcal U} $
with holomorphic coefficients is trivial).

A specific first Cousin problem on  $  M $
is solvable if and only if the corresponding section of  $  {\mathcal M} / {\mathcal O} $
belongs to the image of the mapping  $  \phi $. 
An arbitrary first Cousin problem on  $  M $
is solvable if and only if  $  \phi $
is surjective. On any complex manifold  $  M $
one has an exact sequence

$$ 
\Gamma ( {\mathcal M} )   \mathop \rightarrow \limits ^  \phi   \ 
\Gamma ( {\mathcal M} / {\mathcal O} )  \rightarrow \ 
H  ^ {1} ( M, {\mathcal O} ).
$$

If the Čech cohomology for  $  M $
with coefficients in  $  {\mathcal O} $
is trivial (i.e.  $  H  ^ {1} ( M , {\mathcal O} ) = 0 $), 
then  $  \phi $
is surjective and  $  H  ^ {1} ( {\mathcal U} , {\mathcal O} ) = 0 $
for any covering  $  {\mathcal U} $
of  $  M $. 
Thus, if  $  H  ^ {1} ( M, {\mathcal O} ) = 0 $, 
any first Cousin problem is solvable on  $  M $(
in the classical, cohomological and local version). In particular, the problem is solvable in all domains of holomorphy and on Stein manifolds (cf. [[Stein manifold|Stein manifold]]). If  $  D \subset  \mathbf C  ^ {2} $, 
then the first Cousin problem in  $  D $
is solvable if and only if  $  D $
is a domain of holomorphy. An example of an unsolvable first Cousin problem is:  $  M = \mathbf C  ^ {2} \setminus  \{ 0 \} $, 
$  U _  \alpha  = \{ z _  \alpha  \neq 0 \} $, 
$  \alpha = 1, 2 $, 
$  f _ {1} = ( z _ {1} z _ {2} )  ^ {-} 1 $, 
$  f _ {2} = 0 $.

==Second (multiplicative) Cousin problem.==
Given an open covering  $  {\mathcal U} = \{ U _  \alpha  \} $
of a complex manifold  $  M $
and, in each  $  U _  \alpha  $, 
a meromorphic function  $  f _  \alpha  $, 
$  f _  \alpha  \not\equiv 0 $
on each component of  $  U _  \alpha  $, 
with the assumption that the functions  $  f _ {\alpha \beta }  = f _  \alpha  f _  \beta   ^ {-} 1 $
are holomorphic and nowhere vanishing in  $  U _ {\alpha \beta }  $
for all  $  \alpha , \beta $(
compatibility condition). It is required to construct a meromorphic function  $  f $
on  $  M $
such that the functions  $  ff _  \alpha   ^ {-} 1 $
are holomorphic and nowhere vanishing in  $  U _  \alpha  $
for all  $  \alpha $.

The cohomological formulation of the second Cousin problem is as follows. Given the covering  $  {\mathcal U} $
and functions  $  f _ {\alpha \beta }  $, 
holomorphic and nowhere vanishing in the intersections  $  U _ {\alpha \beta }  $, 
and forming a multiplicative  $  1 $-
cocycle, i.e.

$$ 
f _ {\alpha \beta }  f _ {\beta \alpha }   = \ 
1 \   \mathop{\rm in} \ 
U _ {\alpha \beta }  ,
$$

$$ 
f _ {\alpha \beta }  f _ {\beta \gamma }  f _ {\gamma \alpha } 
 =  1 \   \mathop{\rm in}  U _  \alpha  \cap U _  \beta  \cap U _  \gamma  ,
$$

it is required to find functions  $  h _  \alpha  $, 
holomorphic and nowhere vanishing in  $  U _  \alpha  $, 
such that  $  f _ {\alpha \beta }  = h _  \beta  h _  \alpha   ^ {-} 1 $
in  $  U _ {\alpha \beta }  $
for all  $  \alpha , \beta $. 
If the cocycle  $  \{ f _ {\alpha \beta }  \} $
corresponds to the data of a second Cousin problem and the required  $  h _  \alpha  $
exist, then the function  $  f = \{ f _  \alpha  h _ {\alpha }  \mathop{\rm in}   U _  \alpha  \} $
is defined and meromorphic throughout  $  M $
and is a solution to the given second Cousin problem. Conversely, if a specific second Cousin problem is solvable, then the corresponding cocycle is a holomorphic coboundary.

The localized second Cousin problem. To each set of data  $  \{ U _  \alpha  , f _  \alpha  \} $
for the second Cousin problem there corresponds a uniquely defined global section of the sheaf  $  {\mathcal M}  ^ {*} / {\mathcal O}  ^ {*} $(
in analogy to the first Cousin problem), where  $  {\mathcal M}  ^ {*} = {\mathcal M} \setminus  \{ 0 \} $(
with 0 the null section) is the multiplicative sheaf of germs of meromorphic functions and  $  {\mathcal O}  ^ {*} $
is the subsheaf of  $  {\mathcal O} $
in which each stalk  $  {\mathcal O} _ {z}  ^ {*} $
consists of germs of holomorphic functions that do not vanish at  $  z $. 
The mapping of global sections

$$ 
\Gamma ( {\mathcal M}  ^ {*} )   \mathop \rightarrow \limits ^  \psi   \ 
\Gamma ( {\mathcal M}  ^ {*} / {\mathcal O}  ^ {*} )
$$

maps a meromorphic function  $  f $
to a section  $  \kappa _ {f}  ^ {*} $
of the sheaf  $  {\mathcal M}  ^ {*} / {\mathcal O}  ^ {*} $, 
where  $  \kappa _ {f}  ^ {*} ( z) $
is the class in  $  {\mathcal M} _ {z}  ^ {*} / {\mathcal O} _ {z}  ^ {*} $
of the germ of  $  f $
at  $  z $, 
$  z \in M $. 
The localized second Cousin problem is: Given a global section  $  \kappa  ^ {*} $
of the sheaf  $  {\mathcal M}  ^ {*} / {\mathcal O}  ^ {*} $, 
to find a meromorphic function  $  f $
on  $  M $, 
$  f \not\equiv 0 $
on the components of  $  M $(
i.e. a global section of  $  {\mathcal M}  ^ {*} $), 
such that  $  \psi ( f  ) = \kappa  ^ {*} $.

The sections of  $  M  ^ {*} / Q  ^ {*} $
uniquely correspond to divisors (cf. [[Divisor|Divisor]]), therefore  $  {\mathcal M}  ^ {*} / {\mathcal O}  ^ {*} = {\mathcal D} $
is called the sheaf of germs of divisors. A divisor on a complex manifold  $  M $
is a formal locally finite sum  $  \sum k _ {j} \Delta _ {j} $, 
where  $  k _ {j} $
are integers and  $  \Delta _ {j} $
analytic subsets of  $  M $
of pure codimension 1. To each meromorphic function  $  f $
corresponds the divisor whose terms are the irreducible components of the zero and polar sets of  $  f $
with respective multiplicities  $  k _ {j} $, 
with multiplicities of zeros considered positive and those of poles negative. The mapping  $  \psi $
maps each function  $  f $
to its divisor  $  ( f  ) $; 
such divisors are called proper divisors. The second Cousin problem in terms of divisors is: Given a divisor  $  \Delta $
on the manifold  $  M $, 
to construct a meromorphic function  $  f $
on  $  M $
such that  $  \Delta = ( f  ) $.

Theorems concerning the solvability of the second Cousin problem may be regarded as multi-dimensional generalizations of Weierstrass' theorem on the construction of a meromorphic function with prescribed zeros and poles. As in the case of the first Cousin problem, a necessary and sufficient condition for the solvability of any second Cousin problem in cohomological version is that  $  H  ^ {1} ( M, {\mathcal O}  ^ {*} ) = 0 $. 
Unfortunately, the sheaf  $  {\mathcal O}  ^ {*} $
is not coherent, and this condition is less effective. The attempt to reduce a given second Cousin problem to a first Cousin problem by taking logarithms encounters an obstruction in the form of an integral  $  2 $-
cocycle, and one obtains an exact sequence

$$ 
H  ^ {1} ( M, {\mathcal O} )  \rightarrow \ 
H  ^ {1} ( M, {\mathcal O}  ^ {*} )   \mathop \rightarrow \limits ^  \alpha   \ 
H  ^ {2} ( M, \mathbf Z ),
$$

where  $  \mathbf Z $
is the constant sheaf of integers. Thus, if  $  H  ^ {1} ( M, {\mathcal O} ) = H  ^ {2} ( M, \mathbf Z ) = 0 $, 
any second Cousin problem is solvable on  $  M $, 
and any divisor is proper. If  $  M $
is a Stein manifold, then  $  \alpha $
is an isomorphism; hence the topological condition  $  H  ^ {2} ( M, \mathbf Z ) = 0 $
on a Stein manifold  $  M $
is necessary and sufficient for the second Cousin problem in cohomological version to be solvable. The composite mapping  $  c = \alpha \circ \beta $,

$$ 
\Gamma ( {\mathcal D} )   \mathop \rightarrow \limits ^  \beta   \ 
H  ^ {1} ( M, {\mathcal O}  ^ {*} )   \mathop \rightarrow \limits ^  \alpha   \ 
H  ^ {2} ( M, \mathbf Z )
$$

maps each divisor  $  \Delta $
to an element  $  c( \Delta ) $
of the group  $  H  ^ {2} ( M, \mathbf Z ) $, 
which is known as the Chern class of  $  \Delta $. 
The specific second Cousin problem corresponding to  $  \Delta $
is solvable, assuming  $  H  ^ {1} ( M, {\mathcal O} ) = 0 $, 
if and only if the Chern class of  $  \Delta $
is trivial:  $  c( \Delta ) = 0 $. 
On a Stein manifold, the mapping  $  c $
is surjective; moreover, every element in  $  H  ^ {2} ( M, \mathbf Z ) $
may be expressed as  $  c( \Delta ) $
for some divisor  $  \Delta $
with positive multiplicities  $  k _ {j} $. 
Thus, the obstructions to the solution of the second Cousin problem on a Stein manifold  $  M $
are completely described by the group  $  H  ^ {2} ( M, \mathbf Z ) $.

===Examples.===

1)  $  M = \mathbf C  ^ {2} \setminus  \{ z _ {1} = z _ {2} ,  | z _ {1} | = 1 \} $; 
the first Cousin problem is unsolvable; the second Cousin problem is unsolvable, e.g., for the divisor  $  \Delta = M \cap \{ z _ {1} = z _ {2} ,  | z _ {1} | < 1 \} $
with multiplicity 1.

2)  $  M = \{ | z _ {1}  ^ {2} + z _ {2}  ^ {2} + z _ {3}  ^ {2} - 1 | < 1 \} \subset  \mathbf C  ^ {3} $, 
$  \Delta $
is one of the components of the intersection of  $  M $
and the plane  $  z _ {2} = iz _ {1} $
with multiplicity 1. The second Cousin problem is unsolvable ( $  M $
is a domain of holomorphy, the first Cousin problem is solvable).

3) The first and second Cousin problems are solvable in domains  $  D = D _ {1} \times \dots \times D _ {n} \subset  \mathbf C  ^ {n} $, 
where  $  D _ {j} $
are plane domains and all  $  D _ {j} $, 
with the possible exception of one, are simply connected.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> P. Cousin, "Sur les fonctions de $n$ variables" ''Acta Math.'' , '''19''' (1895) pp. 1–62 {{MR|1554861}} {{ZBL|26.0456.02}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR>
</table>

====Comments====
The Cousin problems are related to the [[Poincaré problem|Poincaré problem]] (is a meromorphic function given on a complex manifold  $  X $
globally the quotient of two holomorphic functions whose germs are relatively prime for all  $  x \in X $?) 
and to the, more algebraic, Theorems A and B of H. Cartan and J.-P. Serre, cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.A. Cazacu, "Theorie der Funktionen mehreren komplexer Veränderlicher" , Birkhäuser (1975) (Translated from Rumanian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German) {{MR|0580152}} {{ZBL|0433.32007}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Sect. 4.5 {{MR|0344507}} {{ZBL|0271.32001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 6 {{MR|0635928}} {{ZBL|0471.32008}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 6 {{MR|0847923}} {{ZBL|}} </TD></TR></table>