Dirakova delta funkcija

Šematska reprezentacija Derekove delta funkcije linijom na čijem vrhu je strelica. Visina strelice se obično koristi za specificiranje vrednosti multiplikativne konstante, koja daje površinu ispod oblasti ispod funkcije. Druga konvencija je da se napiše površina pored strelice.

Dirakva delta funkcija je limit (u smislu distribucija) sekvenci na nuli centriranih normalnih distribucija $\delta _{a}(x)={\frac {1}{a{\sqrt {\pi }}}}\mathrm {e} ^{-x^{2}/a^{2}}$ as a → 0

Dirakova (delta) funkcija ili δ funkcija je funkcija u realnoj ravni, čija je vrednost u svim tačkama 0, osim u tački 0 kada iznosi beskonačno mnogo, definisana tako da je njen integral po celoj oblasti definsanosti 1.

δ funkciju je formulisao teoretski fizičar Pol Dirak. Diskretna analogija Dirakove funkcije je Kroneker delta funkcija, koja je obično definisana u konačnom domenu i uzima vrednosti između 0 i 1.

References

• Aratyn, Henrik; Rasinariu, Constantin (2006), A short course in mathematical methods with Maple, World Scientific, ISBN 981-256-461-6 .
• Arfken, G. B.; Weber, H. J. (2000), Mathematical Methods for Physicists (5th изд.), Boston, MA: Academic Press, ISBN 978-0-12-059825-0 .
• Bracewell, R. (1986), The Fourier Transform and Its Applications (2nd изд.), McGraw-Hill .
• Córdoba, A., „La formule sommatoire de Poisson”, C.R. Acad. Sci. Paris, Series I, 306: 373—376 .
• Courant, Richard; Hilbert, David (1962), Methods of Mathematical Physics, Volume II, Wiley-Interscience .
• Davis, Howard Ted; Thomson, Kendall T (2000), Linear algebra and linear operators in engineering with applications in Mathematica, Academic Press, ISBN 0-12-206349-X 
• Dieudonné, Jean (1976), Treatise on analysis. Vol. II, New York: Academic Press [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-215502-4, MR 0530406 .
• Dieudonné, Jean (1972), Treatise on analysis. Vol. III, Boston, MA: Academic Press, MR 0350769 
• Dirac, Paul (1958), Principles of quantum mechanics (4th изд.), Oxford at the Clarendon Press, ISBN 978-0-19-852011-5 .
• Driggers, Ronald G. (2003), Encyclopedia of Optical Engineering, CRC Press, ISBN 978-0-8247-0940-2 .
• Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, New York: Springer-Verlag, стр. xiv+676, ISBN 978-3-540-60656-7, MR 0257325 .
• Gel'fand, I.M.; Shilov, G.E. (1966—1968), Generalized functions, 1—5, Academic Press .
• Hartman, William M. (1997), Signals, sound, and sensation, Springer, ISBN 978-1-56396-283-7 .
• Hewitt, E; Stromberg, K (1963), Real and abstract analysis, Springer-Verlag .
• Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., 256, Springer, ISBN 3-540-12104-8, MR 0717035 .
• Isham, C. J. (1995), Lectures on quantum theory: mathematical and structural foundations, Imperial College Press, ISBN 978-81-7764-190-5 .
• John, Fritz (1955), Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, MR 0075429 .
• Lang, Serge (1997), Undergraduate analysis, Undergraduate Texts in Mathematics (2nd изд.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94841-6, MR 1476913 .
• Laugwitz, D. (1989), „Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820”, Arch. Hist. Exact Sci., 39 (3): 195–245, doi:10.1007/BF00329867 .
• Levin, Frank S. (2002), „Coordinate-space wave functions and completeness”, An introduction to quantum theory, Cambridge University Press, стр. 109ff, ISBN 0-521-59841-9 
• Li, Y. T.; Wong, R. (2008), „Integral and series representations of the Dirac delta function”, Commun. Pure Appl. Anal., 7 (2): 229—247, MR 2373214, doi:10.3934/cpaa.2008.7.229 .
• de la Madrid, R.; Bohm, A.; Gadella, M. (2002), „Rigged Hilbert Space Treatment of Continuous Spectrum”, Fortschr. Phys., 50 (2): 185–216, Bibcode:2002ForPh..50..185D, arXiv:quant-ph/0109154 , doi:10.1002/1521-3978(200203)50:2<185::AID-PROP185>3.0.CO;2-S .
• McMahon, D. (2005-11-22), „An Introduction to State Space”, Quantum Mechanics Demystified, A Self-Teaching Guide, Demystified Series, New York: McGraw-Hill, стр. 108, ISBN 0-07-145546-9, doi:10.1036/0071455469, Приступљено 2008-03-17 .
• van der Pol, Balth.; Bremmer, H. (1987), Operational calculus (3rd изд.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0327-6, MR 904873 .
• Rudin, W. (1991), Functional Analysis (2nd изд.), McGraw-Hill, ISBN 0-07-054236-8 .
• Soares, Manuel; Vallée, Olivier (2004), Airy functions and applications to physics, London: Imperial College Press .
• Saichev, A I; Woyczyński, Wojbor Andrzej (1997), „Chapter1: Basic definitions and operations”, Distributions in the Physical and Engineering Sciences: Distributional and fractal calculus, integral transforms, and wavelets, Birkhäuser, ISBN 0-8176-3924-1 
• Schwartz, L. (1950), Théorie des distributions, 1, Hermann .
• Schwartz, L. (1951), Théorie des distributions, 2, Hermann .
• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, ISBN 0-691-08078-X .
• Strichartz, R. (1994), A Guide to Distribution Theory and Fourier Transforms, CRC Press, ISBN 0-8493-8273-4 .
• Vladimirov, V. S. (1971), Equations of mathematical physics, Marcel Dekker, ISBN 0-8247-1713-9 .
• Yamashita, H. (2006), „Pointwise analysis of scalar fields: A nonstandard approach”, Journal of Mathematical Physics, 47 (9): 092301, Bibcode:2006JMP....47i2301Y, doi:10.1063/1.2339017 
• Yamashita, H. (2007), „Comment on "Pointwise analysis of scalar fields: A nonstandard approach" [J. Math. Phys. 47, 092301 (2006)]”, Journal of Mathematical Physics, 48 (8): 084101, Bibcode:2007JMP....48h4101Y, doi:10.1063/1.2771422 

Spoljašnje veze

• Hazewinkel Michiel, ур. (2001). „Delta-function”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104. 
• KhanAcademy.org video lesson
• The Dirac Delta function, a tutorial on the Dirac delta function.