False theta functions closely resemble ordinary theta functions; however, they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications for a variety of contexts where false and partial theta series appear. To exemplify the utility of this derivation, we discuss the details of its use on two cases. First, we derive a convergent Rademacher-type exact formula for the number of unimodal sequences via the circle method and extend earlier work on their asymptotic properties. Secondly, we show how quantum modular properties of the limits of false theta functions can be rederived directly from the modular completion of false theta functions proposed in this paper.

中文翻译：

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虚假theta函数的模块化属性的框架

虚假theta函数与普通theta函数非常相似；但是，它们不具有theta函数具有的模块化转换属性。在本文中，我们发现了虚假theta函数的模块化补全，这尤其提供了一种有效的方法来计算其对模块化的障碍。这对于出现假和部分theta系列的各种情况有潜在的应用。为了举例说明此推导的效用，我们在两种情况下讨论了其用法的详细信息。首先，我们通过圆法推导了单峰序列数的收敛的Rademacher型精确公式，并扩展了其渐近性质的早期工作。其次，