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The Fourier transform is a fundamental tool in the physical sciences, with applications in communications theory, electronics, engineering, biophysics and quantum mechanics. In this brief book, the essential mathematics required to understand and apply Fourier analysis is explained. The tutorial style of writing, combined with over 60 diagrams, offers a visually intuitive and rigorous account of Fourier methods. Hands-on experience is provided in the form of simple examples, written in Python and Matlab computer code. Supported by a comprehensive Glossary and an annotated list of Further Readings, this represents an ideal introduction to the Fourier transform.