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The Art of Mathematics

Help 6.17

arsinh or arsh — arc-hyperbolic sine function

1. Definition

Arc-hyperbolic sine is inverse of hyperbolic sine function.

arsinhx ≡ sinh ^invx

With the help of natural logarithm it can be represented as:

arsinhx ≡ ln[x + √(x² + 1)]

2. Graph

Arc-hyperbolic sine is antisymmetric function defined everywhere on real axis. Its graph is depicted below — fig. 1.

[画像:Fig. 1. Graph y = arsinh x.] Fig. 1. Graph of the arc-hyperbolic sine function y = arsinhx.

Function codomain is entire real axis.

3. Identities

Property of antisymmetry:

arsinh−x = −arsinhx

Reciprocal argument:

arsinh(1/x) = arcschx

Sum and difference:

arsinhx + arsinhy = arsinh[x√(y² + 1) + y√(x² + 1)]
arsinhx − arsinhy = arsinh[x√(y² + 1) − y√(x² + 1)]
arsinhx + arcoshy = arsinh{xy + √[(x² + 1)(y² − 1)]} = arcosh[y√(x² + 1) + x√(y² − 1)]

4. Derivative and indefinite integral

Arc-hyperbolic sine derivative:

arsinh′x = 1 /√(1 + x²)

Indefinite integral of the arc-hyperbolic sine:

∫ arsinhx dx = x arsinhx − √(1 + x²) + C

where C is an arbitrary constant.

5. How to use

To calculate arc-hyperbolic sine of the number:

arsinh(−1);

To get arc-hyperbolic sine of the complex number:

arsinh(−1+i);

To get arc-hyperbolic sine of the current result:

arsinh(rslt);

To get arc-hyperbolic sine of the number z in calculator memory:

arsinh(mem[z]);

6. Support

Arc-hyperbolic sine of the real argument is supported in free version of the Librow calculator.

Arc-hyperbolic sine of the complex argument is supported in professional version of the Librow calculator.

Function 1
√ or sqrt

Function 2
Γ or Gamma

Function 3
∏ or Prod

Function 4
∑ or Sum

Function 5
abs

Function 6
arccos

Function 7
arccot or arcctg

Function 8
arccsc or arccosec

Function 9
arcosh or arch

Function 10
arcoth or arcth

Function 11
arcsch

Function 12
arcsec

Function 13
arcsin

Function 14
arctan or arctg

Function 15
arg

Function 16
arsech or arsch

Function 17
arsinh or arsh

Function 18
artanh or arth

Function 19
ceil

Function 20
conj

Function 21
cos

Function 22
cosh or ch

Function 23
cot or ctg

Function 24
coth or cth

Function 25
csc or cosec

Function 26
csch

Function 27
exp

Function 28
fact

Function 29
floor

Function 30
I

Function 31
im

Function 32
J

Function 33
K

Function 34
ln

Function 35
log or lg

Function 36
^

Function 37
rand

Function 38
re

Function 39
sec

Function 40
sech

Function 41
sign

Function 42
sin

Function 43
sinh or sh

Function 44
tan or tg

Function 45
tanh or th

Function 46
Y

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