Neumann series

A series of the form

$$ \sum _ {n=0}^ \infty a _ {n} J _ {\nu + n } ( z) , $$

where $ J _ {\nu + n } $ is the Bessel function (cylinder function of the first kind, cf. Bessel functions) and $ \nu $ is a (real or complex) number. C.G. Neumann

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

$$ f ( z) = \ \sum _ { n=0} ^ \infty a _ {n} J _ {n} ( z) , $$

where

$$ a _ {0} = f ( 0) ,\ \ a _ {n} = \frac{1}{\pi i } \int\limits _ { C } O _ {n} ( t) f ( t) dt $$

and $ O _ {n} $ is a polynomial of degree $ n+ 1 $ in $ 1 / t $:

$$ O _ {0} ( t) = \frac{1}{t} , $$

$$ O _ {n} ( t) = \frac{1}{2 t ^ {n+1} } \times $$

$$ \times \int\limits _ { 0 } ^ \infty e ^ {-x} [ ( x + \sqrt {x ^ {2} + t ^ {2} } ) ^ {n} + ( x - \sqrt {x ^ {2} + t ^ {2} } ) ^ {n} ] dx ,\ n \geq 1 ; $$

it is usually called the Neumann polynomial of order $ n $. (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:

$$ \cos ( z \sin \phi ) = \ J _ {0} ( z) + 2 \sum _ { n=1}^ \infty J _ {2n} ( z) \cos 2 n \phi , $$

$$ \sin ( z \sin \phi ) = 2 \sum _ { n=1}^ \infty J _ {2n-1} ( z) \sin ( 2n - 1 ) \phi , $$

$$ \left ( { \frac{z}{2} } \right ) ^ \mu = \sum _ { n=0} ^ \infty \frac{( \mu + 2n ) \Gamma ( \mu + n ) }{n ! } J _ {\mu + 2n } ( z) , $$

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

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

$$ \tag{1 } \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) ds = \ f ( x) ,\ \ x \in [ a , b ] , $$

a Neumann series is defined as the expansion of the resolvent $ R ( x , s ; \lambda ) $ of the kernel $ K $:

$$ \tag{2 } R ( x , s ; \lambda ) = \ \sum _ { n=1} ^ \infty \lambda ^ {n} K _ {n} ( x , s ) , $$

where the $ K _ {n} $ are the iterated kernels (of $ K $), which are defined by the recurrence formulas

$$ K _ {1} ( x , s ) = K ( x , s ) , $$

$$ K _ {n} ( x , s ) = \int\limits _ { a } ^ { b } K _ {n-1} ( x , t ) K ( t , s ) dt ,\ n \geq 2 . $$

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

$$ \tag{3 } \phi ( x) = f ( x) + \sum _ { k=1} ^ \infty \lambda ^ {n} \int\limits _ { a } ^ { b } K _ {n} ( x , s ) f ( s) ds . $$

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 $ A $ be a bounded linear operator mapping a Banach space $ X $ into itself, with norm $ \| A \| < 1 $. Then the operator $ I - A $, where $ I $ is the identity operator, has a unique bounded inverse $ ( I - A ) ^ {-1} $, which admits the expansion

$$ \tag{4 } ( I - A ) ^ {-1} = \ \sum _ { n=0} ^ \infty A ^ {n} . $$

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 $ f $, i.e.

$$ \tag{a1 } \sum _ { n=0}^ \infty A ^ {n} f $$

may converge also if $ \| A \| \geq 1 $. 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)