The Summability of the Developments in Bessel Functions: with Applications
The object of the present book is to establish sufficient conditions for the summability at the origin and the uniform summability in the neighborhood of the origin of the developments in Bessel's functions. It has also been the aim of the writer to obtain summability of as low an index as possible without placing any considerable restriction on the function to be developed.
The study of the behavior of the development in the neighborhood of the origin presents much greater difficulties than the study of the development in intervals that do not include the origin. This accounts for the fact that previous discussions of the summability of the development apply only to intervals of the latter type, and that most of the previous discussions of the convergence of the development are incomplete in the same manner.
The difficulty in studying the development in the neighborhood of the origin arises from the fact that the terms in which the asymptotic expansion of the Bessel's functions can be used to advantage begin later and later in the series as we approach the origin. Hence, in studying the series in an interval that reaches up to the origin, the point at which we start using the asymptotic expansion is continually shifting. This gives rise to many complications and accounts for the length of that portion of this paper which deals with the uniform summability of the development in the neighborhood of the origin.
It may be noted that many of the lemmas obtained, particularly those in sections 7-17, have an interest of their own and many possible applications in other investigations, such as the study of Fourier's series. There has been no attempt to point out explicitly these applications, as this would detract from the unity of the book and unnecessarily increase its length.
* * *
Friedrich Wilhelm Bessel was able to achieve the feat for which he is best remembered today: he is credited with being the first to use parallax in calculating the distance to a star.
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The study of the behavior of the development in the neighborhood of the origin presents much greater difficulties than the study of the development in intervals that do not include the origin. This accounts for the fact that previous discussions of the summability of the development apply only to intervals of the latter type, and that most of the previous discussions of the convergence of the development are incomplete in the same manner.
The difficulty in studying the development in the neighborhood of the origin arises from the fact that the terms in which the asymptotic expansion of the Bessel's functions can be used to advantage begin later and later in the series as we approach the origin. Hence, in studying the series in an interval that reaches up to the origin, the point at which we start using the asymptotic expansion is continually shifting. This gives rise to many complications and accounts for the length of that portion of this paper which deals with the uniform summability of the development in the neighborhood of the origin.
It may be noted that many of the lemmas obtained, particularly those in sections 7-17, have an interest of their own and many possible applications in other investigations, such as the study of Fourier's series. There has been no attempt to point out explicitly these applications, as this would detract from the unity of the book and unnecessarily increase its length.
* * *
Friedrich Wilhelm Bessel was able to achieve the feat for which he is best remembered today: he is credited with being the first to use parallax in calculating the distance to a star.
The Summability of the Developments in Bessel Functions: with Applications
The object of the present book is to establish sufficient conditions for the summability at the origin and the uniform summability in the neighborhood of the origin of the developments in Bessel's functions. It has also been the aim of the writer to obtain summability of as low an index as possible without placing any considerable restriction on the function to be developed.
The study of the behavior of the development in the neighborhood of the origin presents much greater difficulties than the study of the development in intervals that do not include the origin. This accounts for the fact that previous discussions of the summability of the development apply only to intervals of the latter type, and that most of the previous discussions of the convergence of the development are incomplete in the same manner.
The difficulty in studying the development in the neighborhood of the origin arises from the fact that the terms in which the asymptotic expansion of the Bessel's functions can be used to advantage begin later and later in the series as we approach the origin. Hence, in studying the series in an interval that reaches up to the origin, the point at which we start using the asymptotic expansion is continually shifting. This gives rise to many complications and accounts for the length of that portion of this paper which deals with the uniform summability of the development in the neighborhood of the origin.
It may be noted that many of the lemmas obtained, particularly those in sections 7-17, have an interest of their own and many possible applications in other investigations, such as the study of Fourier's series. There has been no attempt to point out explicitly these applications, as this would detract from the unity of the book and unnecessarily increase its length.
* * *
Friedrich Wilhelm Bessel was able to achieve the feat for which he is best remembered today: he is credited with being the first to use parallax in calculating the distance to a star.
The study of the behavior of the development in the neighborhood of the origin presents much greater difficulties than the study of the development in intervals that do not include the origin. This accounts for the fact that previous discussions of the summability of the development apply only to intervals of the latter type, and that most of the previous discussions of the convergence of the development are incomplete in the same manner.
The difficulty in studying the development in the neighborhood of the origin arises from the fact that the terms in which the asymptotic expansion of the Bessel's functions can be used to advantage begin later and later in the series as we approach the origin. Hence, in studying the series in an interval that reaches up to the origin, the point at which we start using the asymptotic expansion is continually shifting. This gives rise to many complications and accounts for the length of that portion of this paper which deals with the uniform summability of the development in the neighborhood of the origin.
It may be noted that many of the lemmas obtained, particularly those in sections 7-17, have an interest of their own and many possible applications in other investigations, such as the study of Fourier's series. There has been no attempt to point out explicitly these applications, as this would detract from the unity of the book and unnecessarily increase its length.
* * *
Friedrich Wilhelm Bessel was able to achieve the feat for which he is best remembered today: he is credited with being the first to use parallax in calculating the distance to a star.
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The Summability of the Developments in Bessel Functions: with Applications
The Summability of the Developments in Bessel Functions: with Applications
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Product Details
| BN ID: | 2940016330099 |
|---|---|
| Publisher: | OGB |
| Publication date: | 03/18/2013 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| File size: | 2 MB |
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