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For a general discussion of these equations see, e.g., Ref. 14, Chap. 1, Sec. 7. p. 72 (Sec. 1.7, p. 75 of the English translation), Ref. 15, Sec. 10.1, p. 475, Refs. 16, 17; in the context of purely fermionic theories these equations can be found displayed, e.g., in Ref. 18, Sec. 4.C, Exercise 4.C.2, p. 54, Refs. 19, 20.
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0[{χ̄},{χ}]. For a discussion of the Fourier-Laplace transformation in a Grassmann algebra see Ref. 66, Ref. 14, Chap. 1, Sec. 6, Subsec. 2, pp. 62/63 (Subsec. 1.6.2, p. 63 of the English translation), Ref. 67, Appendix A, p. 355, Ref. 68, Appendix A, p. 3709, Ref. 69, Subsec. 10.5.4., p. 339, Ref. 9, Subsec. 2.3. p. 72 (p. 74 of the English translation).
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Only by accident, in preparing the final draft of the present paper we became aware of the fact that the concept proposed in Ref. 73 (preprint version, University of Leipzig Preprint NTZ 16/1993, Physics E-Print arXiv:hep-th/9312137) has also been proposed earlier by Prokhorov (Ref. 28).
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M. Marcus, Finite Dimensional Multilinear Algebra, 2 Parts. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 23 (Dekker, New York, Part 1. 1973; Part 2, 1975).
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note
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(2n) is a 1 × 1 matrix with a length n row and column index.
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85
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0345658206
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note
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(4)* [cf. Sec. III A, Eq. (139)].
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87
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0345658211
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note
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(4)).
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88
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0345226327
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(2)= - 1
-
(2)= - 1.
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89
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0345226318
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note
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4.
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90
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0344795395
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Compare, e.g., Ref. 86, Ref. 88, Chap. IX. p. 245. For an early account of the history of self-reciprocal functions see Ref. 89. Eigenfunctions to the eigenvalue - 1 are often called skew-reciprocal. In optics, following a paper by Caola (who seems to not have been aware of the history of the subject in mathematics, Ref. 90) in recent years self-reciprocal functions are often referred to as "self-Fourier functions" (in the context of the Fourier transformation). Incidentally, also note the comment in Ref. 91 on the earlier optics literature on the subject.
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91
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0344795390
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Reprinted in Ref. 87, Chap. 1 (a), pp. 167-202 (including some corrections, p. 202)
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G. H. Hardy and E. C. Titchmarsh, Q. J. Math. 1, 196 (1930). Reprinted in Ref. 87, Chap. 1 (a), pp. 167-202 (including some corrections, p. 202).
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Hardy, G.H.1
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0344795394
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note
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Incidentally, in considering the bosonic (real/complex analysis) analogue [Eq. (201)] of the Grassmann integral equation (133) Eq. (191) can also be disregarded as it does not lead to any convergent integral.
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99
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Functional Equations in a Single Variable
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PWN - Polish Scientific Publishers, Warsaw
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Reprinted in Ref. 110, pp. 93-123
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C. Babbage, "Examples of the solutions of functional equations," in A Collection of Examples of the Applications of the Differential and Integral Calculus, edited by C. Babbage, J. F. W. Herschel, G. Peacock (J. Deighton and Sons, Cambridge, 1820), Part III. Reprinted in Ref. 110, pp. 287-326. An extract of this work written by J. D. Gergonne (the editor of the Annales de Mathematiques Pures et Appliquees) has been published in French: C. Babbage (Extrait par M. [Extract by Mr.] Gergonne): Des équations fonctionnelles [Functional equations], Ann. Math. Pures Appl. 12, 73 (1821/22).
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125
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0344795382
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note
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Also see in this respect, e.g., Ref. 120, Chap. IV, Part I, Art. 4.11, p. 120, Ref. 121, Sec. 7, p. 14, Ref. 122, Part III.B, p. 227. For further references see, e.g., Ref. 123.
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130
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0345658203
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-
note
-
For a general discussion of eigenfunctions and eigenvalues of integral transformations see Ref. 125, for a discussion concerning the Fourier transformation see, e.g., Ref. 88, Subsec. 3.8, p. 81. Incidentally and as an aside, we mention here that the Laplace transformation does not have any real eigenfunctions to the eigenvalue 1 (Ref. 126, Ref. 125, Sec. 4, p. 121).
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131
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0345226313
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(1940/41) [in German]
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G. Doetsch, Math. Ann. 117, 106 (1940/41) [scanned pages of this article are freely available online at the Göttingen Digitalization Center (GDZ) site: http://gdz.sub.uni-goettingen.de/en/index.html] [in German].
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Doetsch, G.1
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132
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