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This chapter reviews two developments regarding integral equations and Bessel function series, which are so closely connected that they are best considered together. It focuses on integral equations for the four types of Mathieu function and expansion of the nucleus in Mathieu function series. In the study of functions of hypergeometric type, a very important part is played by their expressions as definite integrals. The chapter presents integral equations for the four types of Mathieu function. It highlights Bessel function series, Bessel function product series and Bessel nuclei. |