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We obtain the asymptotic expansion of a generalised Mathieu series and its alternating variant for large complex values of the variable by means of a Mellin transform approach. Numerical examples are presented to demonstrate the accuracy of these expansions. Mathematics Subject Classification: 30E15, 30E20, 34E05doi:10.12988/ams.2013.39492 fatcat:rmsx747wyrd7ddfoolmeesyoum