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Thesis Riemann's P-Function

Charles Hiram Chapman

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Description:Excerpt from Thesis Riemann's P-Function: Dissertation Presented for the Degree of Doctor of Philosophy in the Johns Hopkins University The following paper was begun at the suggestion of Dr. Craig and has had the benefit of his advice and criticism. It is an attempt to set forth in a clear and readable form the properties of the P-function invented by Riemann and treated of in his collected works. Part I of the present paper is devoted to a restatement in systematic order of the properties of the function as given by Riemann, with full demonstrations of certain points at which he has only hinted, and original proofs of some propositions which he considered almost self-evident, but which admit, and perhaps require, elaborate demonstrations. Section 4 of Part I is an instance of this kind; the work therein and the table of equivalent functions are my own. The same may be said of Section 6, where, by using an expansion in powers of the variable which is known to be convergent in the proper regions, the ideas are fixed and a certain ease of statement is attained. In proving that the P-function satisfies a linear differential equation of the second order I have followed Riemann, except that certain points in the Theory of Functions have been touched more lightly, as being better known to readers at the present day. Part II is entirely my own. It is a study of the differential equation satisfied by the P-function from the point of view of the modern theory as originated by Fuchs and developed by his illustrious collaborators.Section 5 is devoted to obtaining the coefficients of the differential equation; and in subsequent sections the Spherical Harmonics, Toroidal functions and Bessel's functions are expressed as P-functions, while, in conclusion, the P-function itself is expressed as a Hypergeometric Series.We have made it easy for you to find a PDF Ebooks without any digging. And by having access to our ebooks online or by storing it on your computer, you have convenient answers with Thesis Riemann's P-Function. To get started finding Thesis Riemann's P-Function, you are right to find our website which has a comprehensive collection of manuals listed.
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ISBN: 1330112687

Description: Excerpt from Thesis Riemann's P-Function: Dissertation Presented for the Degree of Doctor of Philosophy in the Johns Hopkins University The following paper was begun at the suggestion of Dr. Craig and has had the benefit of his advice and criticism. It is an attempt to set forth in a clear and readable form the properties of the P-function invented by Riemann and treated of in his collected works. Part I of the present paper is devoted to a restatement in systematic order of the properties of the function as given by Riemann, with full demonstrations of certain points at which he has only hinted, and original proofs of some propositions which he considered almost self-evident, but which admit, and perhaps require, elaborate demonstrations. Section 4 of Part I is an instance of this kind; the work therein and the table of equivalent functions are my own. The same may be said of Section 6, where, by using an expansion in powers of the variable which is known to be convergent in the proper regions, the ideas are fixed and a certain ease of statement is attained. In proving that the P-function satisfies a linear differential equation of the second order I have followed Riemann, except that certain points in the Theory of Functions have been touched more lightly, as being better known to readers at the present day. Part II is entirely my own. It is a study of the differential equation satisfied by the P-function from the point of view of the modern theory as originated by Fuchs and developed by his illustrious collaborators.Section 5 is devoted to obtaining the coefficients of the differential equation; and in subsequent sections the Spherical Harmonics, Toroidal functions and Bessel's functions are expressed as P-functions, while, in conclusion, the P-function itself is expressed as a Hypergeometric Series.We have made it easy for you to find a PDF Ebooks without any digging. And by having access to our ebooks online or by storing it on your computer, you have convenient answers with Thesis Riemann's P-Function. To get started finding Thesis Riemann's P-Function, you are right to find our website which has a comprehensive collection of manuals listed.
Our library is the biggest of these that have literally hundreds of thousands of different products represented.

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Thesis Riemann's P-Function

Thesis Riemann's P-Function

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