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Mock theta function
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Mock Theta Functions: An Overview Mock theta functions were first introduced by the brilliant Indian mathematician Srinivasa Ramanujan in his last letter to G. H. Hardy in 1920. These functions exhibit intriguing behavior, resembling certain aspects of theta functions while diverging significantly in other ways. Unlike ordinary theta functions, mock theta functions are not modular. Let’s explore their properties further.
Definition:
A mock theta function is essentially the holomorphic part of a harmonic weak Maass form.
Specifically, it is a mock modular form of weight 1/2.
Ramanujan initially defined 17 mock theta functions, which he referred to as “mock” because they mimic the behavior of theta functions but with distinct asymptotic properties.
Behavior:
Mock theta functions are q-series (power series in the variable q) that converge when |q| < 1.
They exhibit exponential singularities, terminating for some power t^N.
If f(q) is not a Jacobi theta function, it becomes a mock theta function if, for each root of unity ρ, there exists an approximation of the form: [ f(q) \approx \sum_{n=0}^\infty a_n q^n ] where the coefficients a_n are related to the roots of unity.
Strong mock theta functions have additional properties, including boundedness as the argument approaches the unit circle.
Examples:
Ramanujan’s original 17 mock theta functions include:
(f_1(q) = q^{1/5} + 4q^{6/5} + 6q^{11/5} + \ldots)
(f_2(q) = q^{1/7} + 6q^{8/7} + 8q^{15/7} + \ldots)
And others with similar patterns.
Dragonette (1952) explored properties of these functions, revealing connections to partition functions and other sequences.
Order:
While the concept of “order” for mock theta functions lacks a formal definition, it serves as a convenient label.
The order corresponds to the number of terms in the series.
For example, the order-3 mock theta function (f_1(q)) has three terms.
In summary, mock theta functions are captivating mathematical entities that blur the boundaries between ordinary theta functions and something altogether novel. Their intricate behavior continues to inspire research and exploration. 🌠
If you’d like to dive deeper into specific mock theta functions or explore related topics, feel free to ask! 🤓1234
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