mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Liouville–Neumann series is a function series that results from applying the

resolvent formalism In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the ...

to solve

Fredholm integral equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to ...

s in

Fredholm theory In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given ...

Definition

The Liouville–Neumann series is defined as :

\phi\left(x\right) = \sum^\infty_ \lambda^n \phi_n \left(x\right)

which, provided that

\lambda

is small enough so that the series converges, is the unique

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

solution of the

of the second kind, If the ''n''th iterated

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...

is defined as ''n''−1 nested integrals of ''n'' operator kernels , :

K_n\left(x,z\right) = \int\int\cdots\int K\left(x,y_1\right)K\left(y_1,y_2\right) \cdots K\left(y_, z\right) dy_1 dy_2 \cdots dy_

then :

\phi_n\left(x\right) = \int K_n\left(x,z\right)f\left(z\right)dz

with :

\phi_0\left(x\right) = f\left(x\right)~,

so ''K''₀ may be taken to be , the kernel of the

identity operator Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

. The resolvent, also called the "solution kernel" for the integral operator, is then given by a generalization of the

geometric series In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...

, :

R\left(x, z;\lambda\right) = \sum^\infty_ \lambda^n K_ \left(x, z\right),

where ''K''₀ is again . The solution of the integral equation thus becomes simply :

\phi\left(x\right) = \int R\left( x, z;\lambda\right) f\left(z\right)dz.

Similar methods may be used to solve the

Volterra integral equation In mathematics, the Volterra integral equations are a special type of integral equations. They are divided into two groups referred to as the first and the second kind. A linear Volterra equation of the first kind is : f(t) = \int_a^t K(t,s)\,x( ...

References

* Mathews, Jon; Walker, Robert L. (1970), ''Mathematical methods of physics'' (2nd ed.), New York: W. A. Benjamin, * Fredholm theory Series (mathematics) Mathematical physics {{math-physics-stub

Definition

See also

References