Harmonic Xxxxx Forms Sample Clauses

Harmonic Xxxxx Forms. ‌ A harmonic Xxxxx form is a certain kind of nonholomorphic modular form that has a natural decomposition into a holomorphic and nonholomorphic part. The holomor- phic part of a harmonic Xxxxx form is called a mock modular form, and every mock modular form naturally has a cusp form associated to it called its shadow. In this sec- tion, we define level 1 harmonic Xxxxx forms (harmonic Xxxxx forms of level greater than 1 are defined as the natural generalization of the definition given here) and state some of their important properties. For more on mock modular forms and harmonic Xxxxx forms, see references such as [BF04], [Ono09], and [BFOR17]. Throughout, the variable z lies in H with z = x + iy, where x, y ∈ R. The weight k hyperbolic Laplacian operator ∆k is defined as follows: ∆k := −y ∂x2 + ∂y2