The behavior of the Mathieu functions is illustrated by using a variety of plots with representative examples taken from mechanics. We show how Mathieu functions can be applied to describe standing, traveling, and rotating waves in physical systems. Some background is provided on notation and analogies with other mathematical functions. Our goal is to increase the familiarity with Mathieu functions in the scientific and academic community using visualization. For this purpose we adopt a strategy based on visual recognition rather than only looking at equations and formulas.

The structural and luminescence properties of erbium doped zirconium oxide prepared by the sol-gel processes were analyzed. The annealed powders presented a concentration dependent crystallite sizes and crystalline phase, ranging from 28 to 46 nm and from 40 to 96% for the monoclinic phase, respectively. Green (545 nm) and red (680 nm) emissions bands were observed with 489 and 962 nm excitation. Experimental results showed that the emission bands can be tuned by controlling the Er3+ concentration and that the red band is almost quenched with 489 nm whereas it is enhanced with 962 nm excitation. The nature of this behavior is discussed taking into account the nonradiative energy transfer and cross-relaxation process.

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