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)

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]).