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This book is designed as an advanced guide to Fourier transforms or Fourier integrals. This covers the following topics: Fourier transforms, Fourier sine and cosine transforms, convolution, Fourier transforms of special functions, applications to boundary value problems, Fourier series and separation of variables. This is including the deduction of many formulas to clarify the concepts and meet all the needs of the students to reach the fundamentals of the Fourier Integrals. Finally, another objective of the book is to reach the interest and motivation of the students by the real nature of the Fourier Integrals.