This chapter starts with defining Euler’s constant \(\gamma \) and the Gamma function \(\Gamma (z)\) , which is a prerequisite for studying the Bessel functions. After this, the Bessel coefficients, \(J_n(z)\) and \(I_n(z)\) , are defined, and some of their properties are studied. Then, their generalizations, the Bessel functions, \(J_v(z)\) and \(I_v(z)\) , are presented, and some of their properties are studied. The problems given in this chapter study some of the properties of the Bessel functions. Integral representations of the Bessel coefficients are derived in the later chapters.