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Description:
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In this work , we prove monotonicity results associated with certain quotients of theta functions . The monotonicity result for the second theta function has its application in the extremal problems on harmonic measure and steady -state heat distributions in special configurations . Along with these results , we include the generalization of theta functions in signature 3 and illustrate a beautiful identity of Ramanujan associated with the inversion of an analogue of an elliptic integral in signature 3 . |