Calculus How To

Integral Operator: Simple Definition, Examples

Share on

Integrals >

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].


OF ANALYTIC FUNCTIONS. Retrieved April 23, 2021 from:
[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:

Stephanie Glen. "Integral Operator: Simple Definition, Examples" From Calculus for the rest of us!

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Leave a Reply

Your email address will not be published. Required fields are marked *