# Neumann series

A series of the form

where is the Bessel function (cylinder function of the first kind, cf. Bessel functions) and is a (real or complex) number. C.G. Neumann

considered the special case when is an integer. He showed that if is an analytic function in a closed disc with centre at the coordinate origin, is an interior point and denotes the boundary of the disc, then

where

and is a polynomial of degree in :

it is usually called the Neumann polynomial of order . (Neumann himself called it a Bessel function of second order. Nowadays this term is used to denote one of the solutions of the Bessel equation.) Examples of the representation of functions by means of a Neumann series:

where is an arbitrary number not equal to a non-negative integer and is the gamma-function.

In the theory of Fredholm integral equations (cf. Fredholm equation)

(1) |

a Neumann series is defined as the expansion of the resolvent of the kernel :

(2) |

where the are the iterated kernels (of ), which are defined by the recurrence formulas

By means of (2) the solution of (1) for small can be represented by

(3) |

The last series is also called a Neumann series. In

the series (3) is considered in the case of an equation (1) to which the Dirichlet problem in potential theory reduces.

Let be a bounded linear operator mapping a Banach space into itself, with norm . Then the operator , where is the identity operator, has a unique bounded inverse , which admits the expansion

(4) |

In the theory of linear operators this series is called a Neumann series. The series

can be regarded as a special case of (4).

#### References

[1] | C.G. Neumann, "Theorie der Besselschen Funktionen" , Teubner (1867) |

[2] | C.G. Neumann, "Untersuchungen über das logarithmische und Newtonsche Potential" , Teubner (1877) |

[3] | G.N. Watson, "A treatise on the theory of Bessel functions" , 1 , Cambridge Univ. Press (1952) |

[4] | R.O. Kuz'min, "Bessel functions" , Moscow-Leningrad (1035) (In Russian) |

[5] | K. Yosida, "Functional analysis" , Springer (1965) |

[6] | F.G. Tricomi, "Integral equations" , Interscience (1957) |

#### Comments

The series (4), applied to a specific vector , i.e.

(a1) |

may converge also if . For necessary and sufficient conditions for convergence see [a2] (or [a3]).

#### References

[a1] | F. Smithies, "Integral equations" , Cambridge Univ. Press (1970) pp. Chapt. II |

[a2] | N. Suzuki, "On the convergence of Neumann series in Banach space" Math. Ann. , 220 (1976) pp. 143–146 |

[a3] | H.W. Engl, "A successive-approximation method for solving equations of the second kind with arbitrary spectral radius" J. Integral Eq. , 8 (1985) pp. 239–247 |

[a4] | I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981) |

[a5] | A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) pp. Chapt. 5 |

[a6] | P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian) |

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Neumann series.

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