where
h
i
(
X
,
F
)
{\displaystyle h^{i}(X,{\mathcal {F}})}
is the dimension of the * i* -th sheaf cohomology group of
F
{\displaystyle {\mathcal {F}}}
. In this case, the dimensions are all finite by Grothendieck's finiteness theorem . This is an instance of the Euler characteristic of a chain complex, where the chain complex is a finite resolution of
F
{\displaystyle {\mathcal {F}}}
by acyclic sheaves.

If the characteristic equation has complex roots of the form r 1 = a + b i {\displaystyle r_{1}=a+bi} and r 2 = a − b i {\displaystyle r_{2}=a-bi} , then the general solution is accordingly y ( x ) = c 1 e ( a + b i ) x + c 2 e ( a − b i ) x {\displaystyle y(x)=c_{1}e^{(a+bi)x}+c_{2}e^{(a-bi)x}\,} . However, by Euler's formula , which states that e i θ = cos θ + i sin θ {\displaystyle e^{i\theta }=\cos \theta +i\sin \theta \,} , this solution can be rewritten as follows: