Mock theta function

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I am unable to find a formal definition of the order of the mock theta function. Can you explain briefly, what is the order of the mock theta function?

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There is a formal definition. But Ramanujan did not define it clearly. His notion of order later turned out to correspond to the conductor of the Nebentypus character of the weight  $1⁄2$ harmonic Maass forms which admit Ramanujan's mock theta functions as their holomorphic projections. See wikipedia for definitions.