狄拉克δ函数:修订间差异
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* [[Γ函数]] |
* [[Γ函数]] |
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* [[黎曼ζ函数]] |
* [[黎曼ζ函数]] |
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== 注释 == |
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{{Reflist|colwidth=30em}} |
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==参考文献 == |
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==外部链接== |
==外部链接== |
2014年10月30日 (四) 03:17的版本
狄拉克δ函数(Dirac Delta function),有时也说单位脉冲函数。通常用δ表示。在概念上,它是這麼一個「函數」:在除了零以外的點都等於零,而其在整個定義域上的積分等於1。嚴格來說狄拉克δ函数不能算是一個函數,因為滿足以上條件的函數是不存在的。但可以用分佈的概念來解釋,稱為狄拉克δ分布,或δ分布,但與費米-狄拉克分布是兩回事。在廣義函數論裡也可以找到δ函數的解釋,此時δ作為一個極簡單的廣義函數出現。
在實際應用中,δ函數或δ分布總是伴隨着積分一起出現。δ分布在偏微分方程、數學物理方法、傅立葉分析和概率論裡都和很多數學技巧有關。
定义
狄拉克δ函数可大概认为是实直线上的一个函数,它在原点以外的所有点函数值为0,只在原点为无穷:
并且满足约束条件
另外,还有
性质
缩放和对称
对于非零标量α,δ函数满足下面的缩放性: [1]
所以
特别地,δ函数是一个偶分布,这就是说
它是-1次齐次函数。
其他性质
表达式
以下表达式均可代表狄拉克δ函数:
参阅
注释
- ^ Strichartz 1994,Problem 2.6.2
参考文献
- Aratyn, Henrik; Rasinariu, Constantin, A short course in mathematical methods with Maple, World Scientific, 2006, ISBN 981-256-461-6 。
- Arfken, G. B.; Weber, H. J., Mathematical Methods for Physicists 5th, Boston, Massachusetts: Academic Press, 2000, ISBN 978-0-12-059825-0 。
- Bracewell, R., The Fourier Transform and Its Applications 2nd, McGraw-Hill, 1986 。
- Córdoba, A., La formule sommatoire de Poisson, C.R. Acad. Sci. Paris, Series I: 373–376 。
- Courant, Richard; Hilbert, David, Methods of Mathematical Physics, Volume II, Wiley-Interscience, 1962 。
- Davis, Howard Ted; Thomson, Kendall T, Linear algebra and linear operators in engineering with applications in Mathematica, Academic Press, 2000, ISBN 0-12-206349-X
- Dieudonné, Jean, Treatise on analysis. Vol. II, New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1976, ISBN 978-0-12-215502-4, MR 0530406 。
- Dieudonné, Jean, Treatise on analysis. Vol. III, Boston, Massachusetts: Academic Press, 1972, MR 0350769
- Dirac, Paul, Principles of quantum mechanics 4th, Oxford at the Clarendon Press, 1958, ISBN 978-0-19-852011-5 。
- Driggers, Ronald G., Encyclopedia of Optical Engineering, CRC Press, 2003, ISBN 978-0-8247-0940-2 。
- Federer, Herbert, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften 153, New York: Springer-Verlag: xiv+676, 1969, ISBN 978-3-540-60656-7, MR 0257325 。
- Gel'fand, I. M.; Shilov, G. E., Generalized functions 1–5, Academic Press, 1966–1968 。
- Hartman, William M., Signals, sound, and sensation, Springer, 1997, ISBN 978-1-56396-283-7 。
- Hewitt, E; Stromberg, K, Real and abstract analysis, Springer-Verlag, 1963 。
- Hörmander, L., The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft. 256, Springer, 1983, ISBN 3-540-12104-8, MR 0717035 。
- Isham, C. J., Lectures on quantum theory: mathematical and structural foundations, Imperial College Press, 1995, ISBN 978-81-7764-190-5 。
- John, Fritz, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955, MR 0075429 。
- Lang, Serge, Undergraduate analysis, Undergraduate Texts in Mathematics 2nd, Berlin, New York: Springer-Verlag, 1997, ISBN 978-0-387-94841-6, MR 1476913 。
- Lange, Rutger-Jan, Potential theory, path integrals and the Laplacian of the indicator, Journal of High Energy Physics (Springer), 2012, 2012 (11): 29–30, Bibcode:2012JHEP...11..032L, arXiv:1302.0864 , doi:10.1007/JHEP11(2012)032 。
- Laugwitz, D., Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820, Arch. Hist. Exact Sci., 1989, 39 (3): 195–245, doi:10.1007/BF00329867 。
- Levin, Frank S., Coordinate-space wave functions and completeness, An introduction to quantum theory, Cambridge University Press: 109ff, 2002, ISBN 0-521-59841-9
- Li, Y. T.; Wong, R., Integral and series representations of the Dirac delta function, Commun. Pure Appl. Anal., 2008, 7 (2): 229–247, MR 2373214, doi:10.3934/cpaa.2008.7.229 。
- de la Madrid, R.; Bohm, A.; Gadella, M., Rigged Hilbert Space Treatment of Continuous Spectrum, Fortschr. Phys., 2002, 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., An Introduction to State Space, Quantum Mechanics Demystified, A Self-Teaching Guide, Demystified Series, New York: McGraw-Hill: 108, 2005-11-22 [2008-03-17], ISBN 0-07-145546-9, doi:10.1036/0071455469 。
- van der Pol, Balth.; Bremmer, H., Operational calculus 3rd, New York: Chelsea Publishing Co., 1987, ISBN 978-0-8284-0327-6, MR 0904873 。
- Rudin, W., Functional Analysis 2nd, McGraw-Hill, 1991, ISBN 0-07-054236-8 。
- Soares, Manuel; Vallée, Olivier, Airy functions and applications to physics, London: Imperial College Press, 2004 。
- Saichev, A I; Woyczyński, Wojbor Andrzej, Chapter1: Basic definitions and operations, Distributions in the Physical and Engineering Sciences: Distributional and fractal calculus, integral transforms, and wavelets, Birkhäuser, 1997, ISBN 0-8176-3924-1
- Schwartz, L., Théorie des distributions 1, Hermann, 1950 。
- Schwartz, L., Théorie des distributions 2, Hermann, 1951 。
- Stein, Elias; Weiss, Guido, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971, ISBN 0-691-08078-X 。
- Strichartz, R., A Guide to Distribution Theory and Fourier Transforms, CRC Press, 1994, ISBN 0-8493-8273-4 。
- Vladimirov, V. S., Equations of mathematical physics, Marcel Dekker, 1971, ISBN 0-8247-1713-9 。
- 埃里克·韦斯坦因. Delta Function. MathWorld.
- Yamashita, H., Pointwise analysis of scalar fields: A nonstandard approach, Journal of Mathematical Physics, 2006, 47 (9): 092301, Bibcode:2006JMP....47i2301Y, doi:10.1063/1.2339017
- Yamashita, H., Comment on "Pointwise analysis of scalar fields: A nonstandard approach" [J. Math. Phys. 47, 092301 (2006)], Journal of Mathematical Physics, 2007, 48 (8): 084101, Bibcode:2007JMP....48h4101Y, doi:10.1063/1.2771422
外部链接
- Hazewinkel, Michiel (编), Delta-function, 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4 (英文)
- KhanAcademy.org录像课 (英文)
- 关于狄拉克δ函数 (英文),对狄拉克δ函数的教程。
- 视频讲座-讲座23 (英文),讲座者Arthur Mattuck。
- 狄拉克δ函数 (英文),在PlanetMath上
- 狄拉克δ测度是一个超函数 (英文)
- We show the existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure (英文)
- Non-Lebesgue measures on R. Lebesgue-Stieltjes measure, Dirac delta measure. (英文)