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This chapter covers two major topics: the fundamentals of the mathematics and the fundamentals of the physics involved in x-ray physics. The objective is to provide a general review of the important mathematics tools and background knowledge of x-ray physics that are used throughout the book. The chapter serves mainly as a refresher. Readers who have not been previously exposed to these topics will find a list of recommended reference materials at the end of the chapter so that detailed studies can be performed. Although these topics can be integrated into other chapters when they are encountered, many of the topics appear multiple times and at different locations. The goal is to provide a convenient and quick reference point to the readers by consolidating these subjects in a separate chapter. 2.1 Mathematics Fundamentals 2.1.1 Fourier transform and convolution The one-dimensional (1D) Fourier transform of a function f(x) is defined as F(u)=∫ ∞ −∞ f(x)e −j2πux dx, where j = √−1. In this equation, e−j2πux = cos2πux − j sin 2πux. Substituting this expression into Eq. (2.1), the 1D Fourier transform can also be expressed as F(u)=∫ ∞ −∞ f(x)cos⁡2πuxdx−j∫ ∞ −∞ f(x)sin⁡2πuxdx. When f(x) is real, the real part of Eq. (2.2) is an even function of frequency u, and the imaginary part is an odd function of frequency u.
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