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4 Elementary FunctionsHyperbolic Functions

§4.35 Identities

Contents
  1. §4.35(i) Addition Formulas
  2. §4.35(ii) Squares and Products
  3. §4.35(iii) Multiples of the Argument
  4. §4.35(iv) Real and Imaginary Parts; Moduli

§4.35(i) Addition Formulas

4.35.1 sinh(u±v) =sinhucoshv±coshusinhv,
4.35.2 cosh(u±v) =coshucoshv±sinhusinhv,
4.35.3 tanh(u±v) =tanhu±tanhv1±tanhutanhv,
4.35.4 coth(u±v) =±cothucothv+1cothu±cothv.
4.35.5 sinhu+sinhv =2sinh(u+v2)cosh(uv2),
4.35.6 sinhusinhv =2cosh(u+v2)sinh(uv2),
4.35.7 coshu+coshv =2cosh(u+v2)cosh(uv2),
4.35.8 coshucoshv =2sinh(u+v2)sinh(uv2),
4.35.9 tanhu±tanhv =sinh(u±v)coshucoshv,
4.35.10 cothu±cothv =sinh(v±u)sinhusinhv.

§4.35(ii) Squares and Products

4.35.11 cosh2zsinh2z=1,
4.35.12 sech2z=1tanh2z,
4.35.13 csch2z=coth2z1.
4.35.14 2sinhusinhv =cosh(u+v)cosh(uv),
4.35.15 2coshucoshv =cosh(u+v)+cosh(uv),
4.35.16 2sinhucoshv =sinh(u+v)+sinh(uv).
4.35.17 sinh2usinh2v =sinh(u+v)sinh(uv),
4.35.18 cosh2ucosh2v =sinh(u+v)sinh(uv),
4.35.19 sinh2u+cosh2v =cosh(u+v)cosh(uv).

§4.35(iii) Multiples of the Argument

4.35.20 sinhz2=(coshz12)1/2,
4.35.21 coshz2=(coshz+12)1/2,
4.35.22 tanhz2=(coshz1coshz+1)1/2=coshz1sinhz=sinhzcoshz+1.

The square roots assume their principal value on the positive real axis, and are determined by continuity elsewhere.

4.35.23 sinh(z) =sinhz,
4.35.24 cosh(z) =coshz,
4.35.25 tanh(z) =tanhz.
4.35.26 sinh(2z)=2sinhzcoshz=2tanhz1tanh2z,
4.35.27 cosh(2z)=2cosh2z1=2sinh2z+1=cosh2z+sinh2z,
4.35.28 tanh(2z)=2tanhz1+tanh2z,
4.35.29 sinh(3z)=3sinhz+4sinh3z,
4.35.30 cosh(3z)=3coshz+4cosh3z,
4.35.31 sinh(4z) =4sinh3zcoshz+4cosh3zsinhz,
4.35.32 cosh(4z) =cosh4z+6sinh2zcosh2z+sinh4z.
4.35.33 cosh(nz)±sinh(nz)=(coshz±sinhz)n,
n.

§4.35(iv) Real and Imaginary Parts; Moduli

With z=x+iy

4.35.34 sinhz =sinhxcosy+icoshxsiny,
4.35.35 coshz =coshxcosy+isinhxsiny,
4.35.36 tanhz =sinh(2x)+isin(2y)cosh(2x)+cos(2y),
4.35.37 cothz =sinh(2x)isin(2y)cosh(2x)cos(2y).
4.35.38 |sinhz|=(sinh2x+sin2y)1/2=(12(cosh(2x)cos(2y)))1/2,
4.35.39 |coshz|=(sinh2x+cos2y)1/2=(12(cosh(2x)+cos(2y)))1/2,
4.35.40 |tanhz|=(cosh(2x)cos(2y)cosh(2x)+cos(2y))1/2.