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secz is the trigonometric function defined by

secz=1/(cosz)
(1)
=2/(e^(iz)+e^(-iz)),
(2)

where cosz is the cosine. The secant is implemented in the Wolfram Language as Sec[z].

Note that the secant does not appear to be in consistent widespread use in Europe, although it does appear explicitly in various German and Russian handbooks (e.g., Gradshteyn and Ryzhik 2000, p. 43). Interestingly, while secz is treated on a par with the other trigonometric functions in some tabulations (Gellert et al. 1989, p. 222), it is not in others (Gradshteyn and Ryzhik 2000, who do not list it in their table of "basic functional relations" on p. 28, but do give identities involving it on p. 43).

Tropfke states, "The history of the secant function begins almost contemporaneously with that of the tangent, but ended after discovery of logarithmic calculation in the first half of the 17th century" (Tropfke 1923, pp. 28) and, "The secant naturally disappeared again from trigonometry when, after the introduction of logarithms, the appearance of trigonometric functions in the denominator no longer posed any difficulty" (Tropfke 1923, pp. 30). Harris and Stocker (1998, p. 300) call secant and cosecant "rarely used functions," but then devote an entire section to them. Because these functions do seem to be in widespread use in the United States (e.g., Abramowitz and Stegun 1972, p. 72), reports of their demise seem to be a bit premature.

The derivative is

 d/(dz)secz=secztanz,
(3)

and the indefinite integral is

 intseczdz=ln[cos(1/2z)+sin(1/2z)]-ln[cos(1/2z)-sin(1/2z)]+C,
(4)

where C is a constant of integration. For -5pi/2<z<pi/2, this can be written

intseczdz=ln[tan(1/4pi+1/2z)]+C
(5)
=ln(secz+tanz)+C.
(6)

The Maclaurin series of the secant is

secx=sum_(n=0)^(infty)((-1)^nE_(2n))/((2n)!)x^(2n)
(7)
=1+1/2x^2+5/(24)x^4+(61)/(720)x^6+(277)/(8064)x^8+...
(8)

(OEIS A046976 and A046977), where E_(2n) is an Euler number. The first few reduced numerators that are prime are 5, 61, 277, 23489580527043108252017828576198947741, ... (OEIS A092838), corresponding to n=2, 3, 4, 19, 24, ... (OEIS A092837).

SecBifurcation

A bifurcation plot of sec(x+alpha) is illustrated above (Trott 2004, p. 169). Of all the trigonometric functions, secx is apparently the only one displaying interesting bifurcation structure for iterates of this form.

The positive integer values of n giving incrementally largest values of |secn| are given by 1, 2, 5, 8, 11, 344, 699, 1054, 1409, 1764, 2119, ... (OEIS A004112), corresponding to the values 1.85082, 2.403, 3.52532, 6.87285, 225.953, 227.503, ....


See also

Alternating Permutation, Cosecant, Cosine, Euler Number, Exsecant, Hyperbolic Secant, Inverse Secant, Secant Line

Related Wolfram sites

http://functions.wolfram.com/ElementaryFunctions/Sec/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Circular Functions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 71-79, 1972.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 224, 1987.Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, 1989.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Harris, J. W. and Stocker, H. "Secant and Cosecant." §5.34 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 300-307, 1998.Jeffrey, A. "Trigonometric Identities." §2.4 in Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, pp. 111-117, 2000.Sloane, N. J. A. Sequences A004112, A046976, A046977, A092837, and A092838 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Secant sec(x) and Cosecant csc(x) Functions." Ch. 33 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 311-318, 1987.Tropfke, J. Teil IB, §3. "Die Begriffe von Sekans und Kosekans eines Winkels." In Geschichte der Elementar-Mathematik in systematischer Darstellung mit besonderer Berücksichtigung der Fachwörter, fünfter Band, zweite aufl. Berlin and Leipzig, Germany: de Gruyter, pp. 28-30, 1923.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.Zwillinger, D. (Ed.). "Trigonometric or Circular Functions." §6.1 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 452-460, 1995.

Referenced on Wolfram|Alpha

Secant

Cite this as:

Weisstein, Eric W. "Secant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Secant.html

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