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Mathieu ordinary differential equation is of Fuchsian types with the tworegular and one irregular singularities. In contrast, Heun equation of Fuchsiantypes has the four regular singularities. Heun equation has the four kind ofconfluent forms: (1) Confluent Heun (two regular and one irregularsingularities), (2) Doubly confluent Heun (two irregular singularities), (3)Biconfluent Heun (one regular and one irregular singularities), (4)Triconfluent Heun equations (one irregular singularity). Mathieu equation inalgebraic forms is also derived from the Confluent Heun equation by changingall coefficients.  In this paper I apply three term recurrence formula [arXiv:1303.0806] to thepower series expansion in closed forms of Mathieu equation for infinite seriesand its integral forms including all higher terms of A_n's. One interestingobservation resulting from the calculations is the fact that a modified Besselfunction recurs in each of sub-integral forms: the first sub-integral formcontains zero term of A_n's, the second one contains one term of A_n's, thethird one contains two terms of A_n's, etc. Section 5 contains two additionalexamples of Mathieu function.  This paper is 5th out of 10 in series "Special functions and three termrecurrence formula (3TRF)". See section 6 for all the papers in the series.Previous paper in series deals with asymptotic behavior of Heun function andits integral formalism [arXiv:1303.0876]. The next paper in the seriesdescribes the power series expansion in closed forms of Lame equation in thealgebraic form and its integral forms [arXiv:1303.0873].

The power series expansion of Mathieu function and its integral formalism