A Picard-Fuchs equation is a linear ordinary differential equation satisfied by the periods of an algebraic differential
k-form in a one-parameter algebraic family.
For elliptic curves of the form 
, such periods arise by integrating 
along closed cycles transported as the parameter
varies (Doran 2000, Harnad 2000).
For the Legendre family 
with 
, the periods 
of 
satisfy
 |
This is the hypergeometric differential equation with 
and 
. A solution analytic at 
is proportional to the complete
elliptic integral of the first kind 
, where 
is the elliptic
modulus (Harnad 2000).
Another example uses the family in Weierstrass form with 
and 
, where 
is Klein's
absolute invariant. The periods 
of 
satisfy
 |
(Doran 2000). The equation has regular singular points at 0, 1, and 
.
