= Polyharmonic Maass form
{title2=$\Delta^mF=0$}
In the integer-depth convention, a <polyharmonic Maass form> is a smooth automorphic function annihilated by a positive integer power of its weight-appropriate hyperbolic Laplacian, with specified growth at the cusps. In weight zero, using $\Delta=-y^2(\partial_x^2+\partial_y^2)$, the equation is $\Delta^mF=0$. It is the zero-shift case of a <shifted polyharmonic Maass form>. Exact depth is the least such integer $m$. Some refined conventions also use half-integer depths, so the convention must be stated.<Polyharmonic Maass forms for PSL(2,Z) by Jeffrey C. Lagarias and Robert C. Rhoades>[Introduction and harmonic-depth convention]{ref}
A global depth-two example comes from the finite part of a completed <nonholomorphic Eisenstein series>. Near $s=1$, write
$$
E^*(z,s)=\frac{c}{s-1}+F(z)+O(s-1),
$$
where $c$ is a nonzero constant and $\Delta E^*=s(1-s)E^*$. Comparing constant Laurent coefficients gives $\Delta F=-c$, hence $\Delta^2F=0$ but $\Delta F\ne0$. The finite part is automorphic and has moderate growth.<Polyharmonic Maass forms for PSL(2,Z) by Jeffrey C. Lagarias and Robert C. Rhoades>[equations 1.3–1.4 and Theorem 3.6, completed Eisenstein series and poles]{ref}
This demonstrates why higher depth is substantive: the space contains functions that are not harmonic but become harmonic after one application of the Laplacian. Merely solving a local differential equation is insufficient; automorphy and cusp growth remain part of the definition.
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