In mathematics, the

crenel An embrasure (or crenel or crenelle; sometimes called gunhole in the domain of gunpowder-era architecture) is the opening in a battlement between two raised solid portions (merlons). Alternatively, an embrasure can be a space hollowed out ...

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

is a periodic

discontinuous Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of a ...

function ''P''(''x'') defined as 1 for ''x'' belonging to a given interval and 0 outside of it. It can be presented as a difference between two

Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...

s of amplitude 1. It is used in crystallography to account for irregularities in the occupation of atomic sites by given atoms in solids, such as periodic domain structures, where some regions are enriched and some are depleted with certain atoms. Mathematically, :

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The coefficients of its Fourier series are :

P_k(\Delta,x) = \frac = \Delta\cdot\mathrm(\pi k\Delta)\cdot\mathrm^.

with the Sinc function.

References

{{reflist, refs= {{Cite journal , doi = 10.1107/S0108767395000365, title = On the use of crenel functions for occupationally modulated structures, journal = Acta Crystallographica Section A, volume = 51, issue = 4, pages = 529, year = 1995, last1 = Petříček , first1 = V., last2 = Van Der Lee , first2 = A., last3 = Evain , first3 = M. {{cite book, author=Malliakas, Christos D. , title=Charge Density Waves and Structural Modulations in Polytelluride Compounds, url=https://books.google.com/books?id=vSYU5Ih-W1QC&pg=PA30, year=2008, publisher=ProQuest, isbn=978-0-549-61737-2, pages=30–31 Special functions Generalized functions