Almost holomorphic modular form

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of <a href="/facts/Modular_form/PcnbYR1f">modular forms</a> that are polynomials in 1/Im(τ) with coefficients that are holomorphic functions of τ. A quasimodular form is the holomorphic part of an almost holomorphic modular form. An almost holomorphic modular form is determined by its holomorphic part, so the operation of taking the holomorphic part gives an isomorphism between the spaces of almost holomorphic modular forms and quasimodular forms. The archetypal examples of quasimodular forms are the <a href="/facts/Eisenstein_series/23vgJLFl">Eisenstein series</a> E2(τ) (the holomorphic part of the almost holomorphic modular form E2(τ) – 3/πIm(τ)),  and derivatives of modular forms.
</p><p>In terms of representation theory, modular forms correspond roughly to highest weight vectors of certain discrete series representations of SL2(R), while almost holomorphic or quasimodular forms correspond roughly to other (not necessarily highest weight) vectors of these representations.
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Almost holomorphic modular form open-in-new

Almost holomorphic modular form