In mathematics, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of modular forms 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 Eisenstein series E2(τ) (the holomorphic part of the almost holomorphic modular form E2(τ) – 3/πIm(τ)), and derivatives of modular forms.
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| - Almost holomorphic modular form (en)
- Nästan analytisk modulär form (sv)
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| - Inom matematiken är en nästan analytiska modulära former en generalisering av modulära former som är polynom i 1/Im(τ) med koefficienter som är analytiska funktioner av τ. En kvasimodulär form är den analytiska delen av en nästan analytisk modulär form. En nästan analytisk modulär form karakteriseras av dess analytiska del, så operatorn av att ta den analytiska delen ger en isomorfi mellan rumment av nästan analytiska modulära former och rummet av kvasimodulära former. Bland de enklaste exemplen av kvasimodulära former är Eisensteinserien E2(τ) (den analytiska delen av den nästan analytiska modulära formen E2(τ) – 3/πIm(τ)) och derivator av modulära former. (sv)
- In mathematics, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of modular forms 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 Eisenstein series E2(τ) (the holomorphic part of the almost holomorphic modular form E2(τ) – 3/πIm(τ)), and derivatives of modular forms. (en)
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| - In mathematics, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of modular forms 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 Eisenstein series E2(τ) (the holomorphic part of the almost holomorphic modular form E2(τ) – 3/πIm(τ)), and derivatives of modular forms. 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. (en)
- Inom matematiken är en nästan analytiska modulära former en generalisering av modulära former som är polynom i 1/Im(τ) med koefficienter som är analytiska funktioner av τ. En kvasimodulär form är den analytiska delen av en nästan analytisk modulär form. En nästan analytisk modulär form karakteriseras av dess analytiska del, så operatorn av att ta den analytiska delen ger en isomorfi mellan rumment av nästan analytiska modulära former och rummet av kvasimodulära former. Bland de enklaste exemplen av kvasimodulära former är Eisensteinserien E2(τ) (den analytiska delen av den nästan analytiska modulära formen E2(τ) – 3/πIm(τ)) och derivator av modulära former. (sv)
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