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Inverse Hyperbolic Sine
The inverse hyperbolic sine
(Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic sine (Harris and Stocker 1998, p. 264) is themultivalued function that is theinverse function of thehyperbolic sine.
The variants
or
(Harris and Stocker 1998, p. 263) are sometimes used to refer to explicitprincipal values of the inverse hyperbolic sine, although this distinction is not always made. Worse yet, the notation
is sometimes used for the principal value, with
being used for the multivalued function (Abramowitz and Stegun 1972, p. 87). The notations
(Jeffrey 2000, p. 124) and
(Gradshteyn and Ryzhik 2000, p. xxx) are sometimes also used. Note that in the notation
,
is thehyperbolic sine and the superscript
denotes aninverse function,not the multiplicative inverse.
Itsprincipal value of
is implemented in theWolfram Language asArcSinh[z] and in the GNU C library asasinh(double x).
The inverse hyperbolic sine is amultivalued function and hence requires abranch cut in thecomplex plane, which theWolfram Language's convention places at the line segments
and
. This follows from the definition of
as
 | (1) |
The inverse hyperbolic sine is given in terms of theinversesine by
 | (2) |
(Gradshteyn and Ryzhik 2000, p. xxx).
Thederivative of the inverse hyperbolic sine is
 | (3) |
and theindefinite integral is
 | (4) |
It has aMaclaurin series
 |  |  | (5) |
 |  |  | (6) |
 |  |  | (7) |
(OEISA055786 andA002595), where
is aLegendre polynomial. It has a Taylor series about infinity of
 |  |  | (8) |
 |  |  | (9) |
See also
Hyperbolic Sine,InverseHyperbolic FunctionsRelated Wolfram sites
http://functions.wolfram.com/ElementaryFunctions/ArcSinh/Explore with Wolfram|Alpha
More things to try:
References
Abramowitz, M. and Stegun, I. A. (Eds.). "Inverse Circular Functions." §4.4 inHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79-83, 1972.Beyer, W. H.CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142-143, 1987.GNU C Library. "Mathematics: Inverse Trigonometric Functions."http://www.gnu.org/manual/glibc-2.2.3/html_chapter/libc_19.html#SEC391.Gradshteyn, I. S. and Ryzhik, I. M.Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. xxx, 2000.Harris, J. W. and Stocker, H.Handbook of Mathematics and Computational Science. New York: Springer-Verlag, 1998.Jeffrey, A. "Inverse Trigonometric and Hyperbolic Functions." §2.7 inHandbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, pp. 124-128, 2000.Sloane, N. J. A. SequencesA002595/M4233,A052468,A052469, andA055786 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "Inverse Trigonometric Functions." Ch. 35 inAn Atlas of Functions. Washington, DC: Hemisphere, pp. 331-341, 1987.Zwillinger, D. (Ed.). "Inverse Hyperbolic Functions." §6.8 inCRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 481-483, 1995.Referenced on Wolfram|Alpha
Inverse Hyperbolic SineCite this as:
Weisstein, Eric W. "Inverse Hyperbolic Sine."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/InverseHyperbolicSine.html