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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) |
is obtained by integrating twice term by term to obtain
(2) |
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) |
with boundary conditions
(4) |
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(5) |
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(6) |
Here, is defined as the solution of the equation
(7) |
which satisfies the conditions
(8) |
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(9) |
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) |
This method of solution is called the Riemann method.
REFERENCES:
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 144-145, 1996.
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel, Vol. 4, p. 289 and Vol. 8, p. 125, 1988.
Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998.
Riemann, B. "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe." Reprinted in Gesammelte math. Abhandlungen. New York: Dover, pp. 227-264, 1957.
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