The cosecant 
is the function defined by
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is the sine
and 
. 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,](/images/equations/Cosecant/NumberedEquation2.svg) |
(4)
|
where 
is a constant
of integration and compatible local branches of the natural
logarithm are chosen. For 
on the real axis,
a real antiderivative is
 |  |  |
(5)
|
 |  |  |
(6)
|
The Laurent series of the cosecant function is
 |  |  |
(7)
|
 |  |  |
(8)
|
(OEIS A036280 and A036281), where 
is a Bernoulli
number and 
.
For arguments measured in radians, the positive integer values of 
giving incrementally largest values of 
are given by 1, 3, 22, 333, 355, 103993, ... (OEIS A046947), which, apart from the initial term 1,
are the numerators of the convergents
of 
and correspond to the values 1.1884,
7.08617, 112.978, 113.364, 33173.7, ....
and Cosecant 
Functions." Ch. 33 in