A caveat about applications of the unilateral Fourier transform

Integral transforms are used in a variety of applications, for example, to evaluate certain definite integrals, to transform a partial differential equation into an ordinary differential equation, to transform an ordinary differential equation into a simpler differential equation or into an algebraic equation, and they also can play a more theoretical role in applied problems. L. Debnath and D. Bhatta wrote in the Preface to the Second Edition of their book [1]: “When the first edition of this book was published in 1995 under the sole authorship of Lokenath Debnath, it was well received, and has been used as a senior undergraduate or graduate level text and research reference in the United States and abroad for the last ten years.” It is really a wellknown book, now in its third edition [2]. Unfortunately, it is a book marked by great carelessness concerning the unilateral Fourier transform and its applications which has apparently triggered the emergence of a number of papers [3][11] emulating the misuses found there. Truthfully, one of the cited papers not only emulates but expands the misguided applications [10]. The purpose of the present work it to call attention to the catastrophic embroilments mentioned above in order to keep students on alert. It is also a warning to reduce their own risk making future misleading research efforts. Even undergraduate students can follow the argument easily. The Fourier sine and cosine transforms of f (x) are denoted by Fs {f (x)} = Fs (k) and Fc {f (x)} = Fc (k), respectively, and are defined by the integrals (see, e.g. [12][13])