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Integral Operator: Simple Definition, Examples

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Generally speaking, an integral operator is an operator that results in integration or finding the area under a curve. It is defined by the integral symbol: ∫.

It’s counterpart in calculus is the differential operator (d/dx), which results in differentiation.


The integral operator is sometimes called a standard integral operator [1] to separate it from special cases used in complex analysis, operator theory and other areas of mathematical analysis.

The term integral operator is also used as a synonym for an integral transform, which is defined via an integral and maps one function to another.

Special Cases of Integral Operator

The first operators appeared at the beginning of the 20th century, at the beginning of the theory of complex-variable functions. Many operators have been developed over the years and are defined very narrowly for special circumstances. They include:

  • Alexander integral operator: Defined for a class of analytic functions on the unit disk D [2]:
    alexander integral operator
    .
  • Fredholm operator: Arises in the Fredholm equation, an integral equation where the term containing the kernel function has constants as limits of integration.
    fredholm equation
  • The Volterra integral equation is similar to the Fredholm equation, except that it has variable integral limits.
  • A variety of pseudo-differential operators are used to study elliptic differential equations. These operators, as well as Fourier integral operators, make it possible to handle differential operators with variable coefficients in about the same way as differential operators with constant coefficients using Fourier transforms [3].

References

[1] Anderson, A. SOME CLOSED RANGE INTEGRAL OPERATORS ON SPACES
OF ANALYTIC FUNCTIONS. Retrieved April 23, 2021 from: http://www2.hawaii.edu/~austina/documents/research/aatgpaper2.5.1.pdf
[2] Gao, C. (1992). On the Starlikeness of the Alexander Integral Operator. Proc. Japan Acad. 68. Ser. A.
[3] Hormander, L. Fourier Integral Operators, I. Retrieved April 23, 2021 from: https://projecteuclid.org/journalArticle/Download?urlid=10.1007%2FBF02392052


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Stephanie Glen. "Integral Operator: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/integral-operator/
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