1902/05/01 by E. T. Whittaker · 2 citations
Computer Science · Physics and Astronomy · #Music Technology and Sound Studies #Chaos control and synchronization #Advanced Mathematical Theories and Applications
paper · doi:10.1112/plms/s1-35.1.417
openalex publication_date 1902/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
The following paper is a study of the functions defined by the differential equation•where Z is a quadratic function of z.. To these functions the name functions of the parabolic cylinder may be given, as it was shown by Weber (Math Ann., Vol.i.) that the harinonic functions applicable to the parabolic cylinder satisfy this equation ; some power series which represent these harmonic functions and satisfy the differential equation have been found by Baer (Diss., Gdstrin) and Haentzschel (Zeitsehrift fur Math., Vol.xxxni.)The differential equation, however, has an importance quite apart from its occurrence in harmonic analysis ; for, owing to its simplicity, jt naturally is presented to us as the first object of study in the general scheme of defining new transcendental functions by means of differential equations; and it will be evident from the results found in the present paper that many definite integrals, which have been suggested at various times by different writers as suitable definitions for new transcendental functions, are really special cases of definite integrals which satisfy the differential equation of the parabolic cylinder. Description of Paper.In the present paper the differential equation associated with the parabolic cylinder is first ( §3) solved by means of a family of definite integrals.It is then ( § 4) shown that one of the integx'als of this family possesses properties which fit it to be taken, as a standard solution, and this integral is denoted by-D,,(z).It is shown that i),,(z) possesses an asymptotic expansion which is well suited for the purpose of calculating its numerical values, and that in certain cases it degenerates into an elementai-y function.It is then ( § 5) shown that the general solution of the equation does not introduce any VOL.xxxv.-NO.816.