A Systematic Approach to Delay Functions

Christopher N. Angstmann, Stuart James M. Burney, Bruce I. Henry, Byron A. Jacobs, Zhuang Xu

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into their series representations. Using this approach, we have defined delay exponential functions, delay trigonometric functions and delay fractional Mittag-Leffler functions, among others. We obtained Laplace transforms of the delay functions and demonstrated how they can be employed in finding solutions to delay differential equations. Our results, which extend and unify previous work, offer a consistent framework for defining and using delay functions.

Original languageEnglish
Article number4526
Issue number21
Publication statusPublished - Nov 2023


  • delay differential equations
  • fractional differential equations
  • integral transforms
  • special functions

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • General Mathematics
  • Engineering (miscellaneous)


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