| 1. | Jung developed with his teacher Schottky a general theory of theta functions.
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| 2. | In the last year of his life, Ramanujan discovered mock theta functions.
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| 3. | The invariants may be expressed in terms of Jacobi's theta functions.
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| 4. | One may construct generalizations of the Jacobi theta function.
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| 5. | His first significant work, published in 1896, was on theta functions.
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| 6. | This follows from the properties of the theta function:
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| 7. | Other highlights include his work on abelian functions and theta functions on Riemann surfaces.
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| 8. | These theta functions transform under the modular group.
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| 9. | However for many years there was no good definition of a mock theta function.
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| 10. | This allows one to extend many results about modular forms to mock theta functions.
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