Transformation of Third Order Mock Theta Functions and New $q$-Series Identities

Ramanujan introduced mock theta functions in his last letter to G.H.Hardy. He provided examples and various relations between them. G.N.Watson found transformations for the third order mock theta functions $f(q)$ and $\omega$(q). Zwegers in 2000 built on Watson's techniques to complete these mock theta functions and connected them to real analytic modular forms. We show how to derive these transformations using Lerch sums. To show the equivalence of the results involves some new $q$-series identities thus resulting in a new proof of Zwegers' theorem.