He carried out many important and fruitful investigations in number theory, in the theory of Bessel functions and of Fourier series, in ordinary and partial differential equations, and in analytical mechanics and potential theory .
Siya natupad maraming mahalaga at kapaki-pakinabang sa mga pagsisiyasat bilang teorya, sa teorya ng Bessel function at ng Fourier series, sa karaniwan at bahagyang kaugalian equation, at sa analytical mechanics at potensiyal na mga teorya.