Mellin transform of some trigonometric functions

<p>In this paper, we focus on calculating the Mellin transform of three types of trigonometric functions, namely, <span class="math inline">\(\displaystyle\sum_{k=0}^{n}c_{k}\sin(a_{k}x)\)</span>, <span class="math inline">\(\displaystyle\sum_{k=0}^{n}c_{k}\cos(a_{k}x)\)</span> and <span class="math inline">\(\displaystyle\sum_{k=1}^{n}c_{k}(1-\cos(a_{k}x))\)</span>, where <span class="math inline">\(n\)</span> is an integer, <span class="math inline">\(c_{k}\in\mathbb{R}^{*}\)</span> and <span class="math inline">\(0<a_{0}<\cdots<a_{n}\)</span>. Our approach is based on the application of techniques from linear algebra, calculus, Laplace transform, and special functions. In particular, we give an evaluation of the integral <span class="math inline">\(\displaystyle\int_{0}^{\infty}\frac{\sin^{n}x}{x^{\alpha}}dx\)</span>, <span class="math inline">\(n\in\mathbb{N}^{*}\)</span>, <span class="math inline">\(0<\alpha<n+1\)</span>.</p>