Riemann Function
المؤلف:
Conway, J. H. and Guy, R. K
المصدر:
The Book of Numbers. New York: Springer-Verlag
الجزء والصفحة:
...
27-8-2020
1102
Riemann Function
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
![1/2a_0+sum_(n=1)^infty[a_ncos(nx)+b_nsin(nx)]](https://mathworld.wolfram.com/images/equations/RiemannFunction/NumberedEquation1.gif) |
(1)
|
is obtained by integrating twice term by term to obtain
![F(x)=1/4a_0x^2-sum_(n=1)^infty1/(n^2)[a_ncos(nx)+b_nsin(nx)]+Cx+D,](https://mathworld.wolfram.com/images/equations/RiemannFunction/NumberedEquation2.gif) |
(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
Here, 
is defined as the solution of the equation
 |
(7)
|
which satisfies the conditions
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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