Decomposition of an Arbitrary Function into Plane Waves

Fritz John
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Fritz John: New York University, Courant Institute of Mathematical Sciences

Chapter Chapter I in Plane Waves and Spherical Means, 1981, pp 7-13 from Springer

Abstract: Abstract In what follows the letters x, y, z, X, Y, Z, ξ, η, ζ will always stand for the vectors (x 1, ..., x n ), (y 1, ..., y n ), ..., (ζ1, ..., ζ n ) in n-dimensional space where n ≧ 2. All other letters will stand for scalar variables. The scalar product ∑ i = 1 n X i Y i $$\sum\limits_{i = 1}^n {{X_i}} {Y_i}$$ of the vectors x and y will be denoted by x · y, the length (x · x)1/2 of the vector x by | x |. The volume element dx 1, ..., dx n will be abbreviated to dx, while dS x will denote the surface element of a hyper-surface in n-dimensional space. The spherical surface of radius 1 about the origin in x-space will be denoted by Ω ∞, its surface element by dω x , its total surface measure by ω n . The volume of the unit-sphere in n-space is then (1/n)ω n . Integrations are carried out over the whole range of a variable, unless other limits are indicated.

Keywords: Plane Wave; Arbitrary Function; Spherical Surface; Surface Element; Plane Integral (search for similar items in EconPapers)
Date: 1981
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DOI: 10.1007/978-1-4613-9453-2_2

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