Cosecant -- from Wolfram MathWorld
- ️Weisstein, Eric W.
 |
The cosecant 
is the function defined by
where 
is the sine.
The cosecant is implemented in the Wolfram
Language as Csc[z].
The notation 
is sometimes also used (Gellert et al. 1989, p. 222; Gradshteyn and Ryzhik
2000, p. xxix). Note that the cosecant does not appear to be in consistent widespread
use in Europe, although it does appear explicitly in various German and Russian handbooks
(e.g., Gellert et al. 1989, p. 222; Gradshteyn and Ryzhik 2000, pp. xxix
and p. 43). Interestingly, while 
is treated on 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).
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
 |
(3) |
and the indefinite integral is
![intcsczdz=ln[sin(1/2z)]-ln[cos(1/2z)]+C,](https://mathworld.wolfram.com/images/equations/Cosecant/NumberedEquation2.svg) |
(4) |
where 
is a constant
of integration. For 
on the real axis, this simplifies to
The Laurent series of the cosecant function is
(OEIS A036280 and A036281), where 
is a Bernoulli
number.
The positive integer values of 
giving incrementally largest values of 
are given by 1, 3, 22, 333, 355, 103993, ... (OEIS A046947), which are precisely the numerators of
the convergents of 
and correspond to the values 1.1884, 7.08617, 112.978, 113.364, 33173.7, ....
See also
Flint Hills Series, Inverse Cosecant, Secant, Sine
Related Wolfram sites
http://functions.wolfram.com/ElementaryFunctions/Csc/
Explore with Wolfram|Alpha
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. 215,
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 A036280,
A036281, and A046947
in "The On-Line Encyclopedia of Integer Sequences."Spanier,
J. and Oldham, K. B. "The Secant 
and Cosecant 
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.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
Cite this as:
Weisstein, Eric W. "Cosecant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cosecant.html