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4 Elementary FunctionsHyperbolic Functions

§4.38 Inverse Hyperbolic Functions: Further Properties


§4.38(i) Power Series

4.38.1 arcsinhz=z-12z33+1324z55-135246z77+,
4.38.2 arcsinhz=ln(2z)+1212z2-132414z4+13524616z6-,
z>0, |z|>1.
4.38.3 arccoshz=ln(2z)-1212z2-132414z4-13524616z6-,
4.38.4 arccoshz=(2(z-1))1/2(1+n=1(-1)n135(2n-1)22nn!(2n+1)(z-1)n),
z>0, |z-1|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=z1-z2(1+23z2z2-1+2435(z2z2-1)2+),

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

4.38.8 x2-y2=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 =±(z2-1)-1/2,
4.38.11 ddzarctanhz =11-z2.
4.38.12 ddzarccschz =1z(1+z2)1/2,
4.38.13 ddzarcsechz =-1z(1-z2)1/2.
4.38.14 ddzarccothz =11-z2.

§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±((u2-1)(v2-1))1/2),
4.38.17 Arctanhu±Arctanhv=Arctanh(u±v1±uv),
4.38.18 Arcsinhu±Arccoshv=Arcsinh(uv±((1+u2)(v2-1))1/2)=Arccosh(v(1+u2)1/2±u(v2-1)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.