About the Project
4 Elementary FunctionsHyperbolic Functions

§4.38 Inverse Hyperbolic Functions: Further Properties

  1. §4.38(i) Power Series
  2. §4.38(ii) Derivatives
  3. §4.38(iii) Addition Formulas

§4.38(i) Power Series

4.38.1 arcsinhz=z12z33+1324z55135246z77+,
4.38.2 arcsinhz=ln(2z)+1212z2132414z4+13524616z6,
z>0, |z|>1.
4.38.3 arccoshz=ln(2z)1212z2132414z413524616z6,
4.38.4 arccoshz=(2(z1))1/2(1+n=1(1)n135(2n1)22nn!(2n+1)(z1)n),
z>0, |z1|2.
4.38.5 arctanhz=z+z33+z55+z77+,
|z|1, z±1.
4.38.6 arctanhz=±iπ2+1z+13z3+15z5+,
z0, |z|1.
4.38.7 arctanhz=z1z2(1+23z2z21+2435(z2z21)2+),

which requires z (=x+iy) to lie between the two rectangular hyperbolas given by

4.38.8 x2y2=12.

§4.38(ii) Derivatives

In the following equations square roots have their principal values.

4.38.9 ddzarcsinhz =(1+z2)1/2.
4.38.10 ddzarccoshz =±(z21)1/2,
4.38.11 ddzarctanhz =11z2.
4.38.12 ddzarccschz =1z(1+z2)1/2,
4.38.13 ddzarcsechz =1z(1z2)1/2.
4.38.14 ddzarccothz =11z2.

§4.38(iii) Addition Formulas

4.38.15 Arcsinhu±Arcsinhv=Arcsinh(u(1+v2)1/2±v(1+u2)1/2),
4.38.16 Arccoshu±Arccoshv=Arccosh(uv±((u21)(v21))1/2),
4.38.17 Arctanhu±Arctanhv=Arctanh(u±v1±uv),
4.38.18 Arcsinhu±Arccoshv=Arcsinh(uv±((1+u2)(v21))1/2)=Arccosh(v(1+u2)1/2±u(v21)1/2),
4.38.19 Arctanhu±Arccothv=Arctanh(uv±1v±u)=Arccoth(v±uuv±1).

The above equations are interpreted in the sense that every value of the left-hand side is a value of the right-hand side and vice-versa. All square roots have either possible value.