# Mock Theta Functions

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The last letter that S. Ramanujan sent to Hardy (January 12, 1920) defined 17 Jacobi-like functions for complex , called "mock theta functions" since then; they are -series [1]. Ramanujan did not rigorously define mock theta functions and their orders. G. E. Andrews in a visit to Trinity College discovered some notebooks of Ramanujan's, and called one of them the “lost notebook"; in it were seven more mock theta functions (of the sixth order) with a set of identities connecting them. More mock theta functions were discovered afterward, including some of the 10th order [2, 3]. A variety of applications appear in the fields of hypergeometric functions, number theory, Mordell integrals, probability theory, and mathematical physics, where they are used to determine critical dimensions in some string theories. In this Demonstration, plots are sampled at increments of ; for identities and more plots, see [4, 5].

Contributed by: Enrique Zeleny (October 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

References

[1] S. Saba, "A Study of a Generalization of Ramanujan's Third Order and Sixth Order Mock Theta Functions", *Applied Mathematics*, 2(5), 2012 pp. 157–165. doi:10.5923/j.am.20120205.02.

[2] G. E. Andrews, "The Fifth and Seventh Order Mock Theta Functions," *Transactions of the American Mathematical Society*, 293(1), 1986 pp. 113–134. www.ams.org/journals/tran/1986-293-01.

[3] B. Srivastava, "Ramanujan's Mock Theta Functions," *Mathematical Journal of Okayama University*, 47, 2005 pp. 163–174. www.math.okayama-u.ac.jp/mjou/mjou47/_15_Srivastava.pdf.

[4] O. Maresh. "Phase Portraits of Mock Theta Functions." (Oct 9, 2014) owen.maresh.info/mocktheta.html.

[5] Souichiro-Ikebe. "Graphics Library of Special Functions" (in Japanese). (Oct 9, 2014) math-functions-1.watson.jp/sub2_qspec_090.html.

## Permanent Citation

"Mock Theta Functions"

http://demonstrations.wolfram.com/MockThetaFunctions/

Wolfram Demonstrations Project

Published: October 17 2014