It is written as
or
or
where , which is the fourth power of the
del operator and the square of the
Laplacian operator (or ), is known as the biharmonic operator or the bilaplacian operator. In
Cartesian coordinates, it can be written in dimensions as:
Because the formula here contains a summation of indices, many mathematicians prefer the notation over because the former makes clear which of the indices of the four nabla operators are contracted over.
For example, in three dimensional
Cartesian coordinates the biharmonic equation has the form
As another example, in n-dimensional
Real coordinate space without the origin ,
where
which shows, for n=3 and n=5 only, is a solution to the biharmonic equation.
A solution to the biharmonic equation is called a biharmonic function. Any
harmonic function is biharmonic, but the converse is not always true.
In two-dimensional
polar coordinates, the biharmonic equation is
which can be solved by separation of variables. The result is the
Michell solution.
Just as
harmonic functions in 2 variables are closely related to complex
analytic functions, so are biharmonic functions in 2 variables. The general form of a biharmonic function in 2 variables can also be written as
where and are
analytic functions.