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Harmonic Functions
Harmonic Functions
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AUTHOR'S PREFACE
This brief sketch of the Harmonic Functions and their use in Mathematical Physics was written as a chapter of Merriman and Woodward's Higher Mathematics. It was intended to give enough in the way of introduction and illustration to serve as a useful part of the equipment of the general mathematical student, and at the same time to point out to one specially interested in the subject the way to carry on his study and reading toward a broad and detailed knowledge of its more difficult portions.
Fourier's Series, Zonal Harmonics, and Bessel's Functions of the order zero are treated at considerable length, with the intention of enabling the reader to use them in actual work in physical problems, and to this end several valuable numerical tables are included in the text.
Cambridge, Mass.
* * * * * *
An excerpt from the beginning of the first chapter:
History And Description.
What is known as the Harmonic Analysis owed its origin and development to the study of concrete problems in various branches of Mathematical Physics, which however all involved the treatment of partial differential equations of the same general form.
The use of Trigonometric Series was first suggested by Daniel Bernouilli in 1753 in his researches on the musical vibrations of stretched elastic strings, although Bessel's Functions had been already (1732) employed by him and by Euler in dealing with the vibrations of a heavy string suspended from one end; and Zonal and Spherical Harmonics were introduced by Legendre and Laplace in 1782 in dealing with the attraction of solids of revolution.
The analysis was greatly advanced by Fourier in 1812-1824 in his remarkable work on the Conduction of Heat, and important additions have been made by Lame (1839) and by a host of modern investigators.
This brief sketch of the Harmonic Functions and their use in Mathematical Physics was written as a chapter of Merriman and Woodward's Higher Mathematics. It was intended to give enough in the way of introduction and illustration to serve as a useful part of the equipment of the general mathematical student, and at the same time to point out to one specially interested in the subject the way to carry on his study and reading toward a broad and detailed knowledge of its more difficult portions.
Fourier's Series, Zonal Harmonics, and Bessel's Functions of the order zero are treated at considerable length, with the intention of enabling the reader to use them in actual work in physical problems, and to this end several valuable numerical tables are included in the text.
Cambridge, Mass.
* * * * * *
An excerpt from the beginning of the first chapter:
History And Description.
What is known as the Harmonic Analysis owed its origin and development to the study of concrete problems in various branches of Mathematical Physics, which however all involved the treatment of partial differential equations of the same general form.
The use of Trigonometric Series was first suggested by Daniel Bernouilli in 1753 in his researches on the musical vibrations of stretched elastic strings, although Bessel's Functions had been already (1732) employed by him and by Euler in dealing with the vibrations of a heavy string suspended from one end; and Zonal and Spherical Harmonics were introduced by Legendre and Laplace in 1782 in dealing with the attraction of solids of revolution.
The analysis was greatly advanced by Fourier in 1812-1824 in his remarkable work on the Conduction of Heat, and important additions have been made by Lame (1839) and by a host of modern investigators.
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