# Riemann summation method

From Encyclopedia of Mathematics

A method for summing series of numbers. A series can be summed by Riemann's method to a number if

This method was first introduced and its regularity was first proved by B. Riemann in 1854 (see [1]). The Riemann summation method has been applied in the theory of trigonometric series, where it is usually stated as follows: A trigonometric series

with bounded coefficients can be summed by Riemann's method at a point to a number if the function

has, at , Riemann derivative equal to .

#### References

[1] | B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" , Gesammelte Math. Abhandlungen , Dover, reprint (1957) pp. 227–264 |

[2] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |

[3] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |

[4] | G.H. Hardy, "Divergent series" , Clarendon Press (1949) |

#### Comments

Regularity (cf. Regular summation methods) is expressed by Riemann's first theorem; the theorem stated above is called Riemann's second theorem. The function is also called the Riemann function.

#### References

[a1] | W. Beekman, "Theorie der Limitierungsverfahren" , Springer (1970) |

**How to Cite This Entry:**

Riemann summation method.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Riemann_summation_method&oldid=17922

This article was adapted from an original article by T.P. Lykashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article