There are a number of functions in various branches of mathematics known as Riemann functions. Examples include the Riemann P-series, Riemann-Siegel functions, Riemann theta function, Riemann zeta function, xi-function, the function obtained by Riemann in studying Fourier series, the function appearing in the application of the Riemann method for solving the Goursat problem, the Riemann prime counting function , and the related the function obtained by replacing with in the Möbius inversion formula.
The Riemann function for a Fourier series
(1)
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is obtained by integrating twice term by term to obtain
(2)
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where and are constants (Riemann 1957; Hazewinkel 1988, vol. 8, p. 118).
The Riemann function arises in the solution of the linear case of the Goursat problem of solving the hyperbolic partial differential equation
(3)
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with boundary conditions
(4)
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(5)
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(6)
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Here, is defined as the solution of the equation
(7)
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which satisfies the conditions
(8)
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(9)
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on the characteristics and , where is a point on the domain on which (8) is defined (Hazewinkel 1988). The solution is then given by the Riemann formula
(10)
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This method of solution is called the Riemann method.