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4 Elementary FunctionsHyperbolic Functions

§4.35 Identities

Contents

§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(u-v2),
4.35.6 sinhu-sinhv =2cosh(u+v2)sinh(u-v2),
4.35.7 coshu+coshv =2cosh(u+v2)cosh(u-v2),
4.35.8 coshu-coshv =2sinh(u+v2)sinh(u-v2),
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 cosh2z-sinh2z=1,
4.35.12 sech2z=1-tanh2z,
4.35.13 csch2z=coth2z-1.
4.35.14 2sinhusinhv =cosh(u+v)-cosh(u-v),
4.35.15 2coshucoshv =cosh(u+v)+cosh(u-v),
4.35.16 2sinhucoshv =sinh(u+v)+sinh(u-v).
4.35.17 sinh2u-sinh2v =sinh(u+v)sinh(u-v),
4.35.18 cosh2u-cosh2v =sinh(u+v)sinh(u-v),
4.35.19 sinh2u+cosh2v =cosh(u+v)cosh(u-v).

§4.35(iii) Multiples of the Argument

4.35.20 sinhz2=(coshz-12)1/2,
4.35.21 coshz2=(coshz+12)1/2,
4.35.22 tanhz2=(coshz-1coshz+1)1/2=coshz-1sinhz=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=2tanhz1-tanh2z,
4.35.27 cosh(2z)=2cosh2z-1=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.