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By Eric W. Weisstein

Upon booklet, the 1st version of the CRC Concise Encyclopedia of arithmetic received overwhelming accolades for its exceptional scope, clarity, and application. It quickly took its position one of the best promoting books within the heritage of Chapman & Hall/CRC, and its recognition maintains unabated.

Yet additionally unabated has been the commitment of writer Eric Weisstein to amassing, cataloging, and referencing mathematical proof, formulation, and definitions. He has now up-to-date many of the unique entries and multiplied the Encyclopedia to incorporate a thousand extra pages of illustrated entries.

The accessibility of the Encyclopedia in addition to its extensive insurance and within your means cost make it appealing to the widest attainable diversity of readers and definitely a needs to for libraries, from the secondary to the pro and examine degrees. For mathematical definitions, formulation, figures, tabulations, and references, this can be easily the main outstanding compendium available.

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Spanier, J. and Oldham, K. B. ” Chs. 61 and 62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 609-633, 1987. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed, Cambridge, England: Cambridge University Press, 1990. ) is the complementary complete ELLIPTIC INTEGRAL OF THE FIRST KIND, then the MODULUS k is the ROOT of an algebraic equation with INTEGER COEFFICIENTS. A MODULUS Elliptic Integral of the Third Kind Let 0 < k2 < 1. The incomplete elliptic third kind is then defined as n(n;4,k) = 4 I0 of the (l- n sin2 e)&%ZZ # (1 - nt2)&1 0 where n is a constant (1) dt -- known (2) is called de sin I integral k, such that - t2)(1 - k2t2j (2) as the CHARACTERISTIC.

Appl. Math. 45, 350-372, 1913-1914. Ellipse Caustic Curve For an ELLIPSE given by X = rcost (1) y = sint with light source - TV) cos(2t) + 2r4 - x2 - T2X2) cos(3t) + 6r(l to (2) (3) (1) and (3) gives the set of equations (4) (5) 2r2 + 4x2) + 3x(1 - 5r2) cost NY = 8~(--l + = 2r(-1 respect + 6r4 - 3x2 + 9T2x2) cost + (-2r2 + (6~ + 6r3) cos(2t) D, Combining with (4) + 6rx(l D, = 2r(l+ (1) Y2 X2 F-(1-==- (3) Y N, = 2rx(3 - 5r2) + (-6~~ for c f [0, 11. The PARTIAL DERIVATIVE c is 2Y2 -- 2x2 +(l-== C3 is x=- N;z: D, NY y=D’ Y2 X2 2+(1-l-o (2) at (x, 0), the CAUSTIC of ELLIPSES + x(1 - TV) cos(3t) ~~ - x2) sin3 t - T2 - 4x2) + 3(-x - T2) cos(2t) + x(-1 (6) (7) + 5r2) cost + r2) cos(3t).

Ex, ac . uk/pub/ cremona/dat a/. Du Val, P. Elliptic Functions and Elliptic Curves. Cambridge: Cambridge University Press, 1973. ; and Zimmer, H. G. ” Acta Arith. 68, 171-192, 1994. Ireland, K. and Rosen, M. ” Ch. 18 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 297-318, 1990. N. M. and Masur, B. Arithmetic Moduli of Elliptic Curves. Princeton, NJ: Princeton University Press, 1985. Knapp, A. W. Elliptic Curves. Princeton, NJ: Princeton University Press, 1992.

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