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Showing content from https://dlmf.nist.gov/4.35 below:

§4.35 Identities ‣ Hyperbolic Functions ‣ Chapter 4 Elementary 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 ) = sinh ⁡ u ⁢ cosh ⁡ v ± cosh ⁡ u ⁢ sinh ⁡ v , 4.35.2 cosh ⁡ ( u ± v ) = cosh ⁡ u ⁢ cosh ⁡ v ± sinh ⁡ u ⁢ sinh ⁡ v , 4.35.3 tanh ⁡ ( u ± v ) = tanh ⁡ u ± tanh ⁡ v 1 ± tanh ⁡ u ⁢ tanh ⁡ v , 4.35.4 coth ⁡ ( u ± v ) = ± coth ⁡ u ⁢ coth ⁡ v + 1 coth ⁡ u ± coth ⁡ v . 4.35.5 sinh ⁡ u + sinh ⁡ v = 2 ⁢ sinh ⁡ ( u + v 2 ) ⁢ cosh ⁡ ( u − v 2 ) , 4.35.6 sinh ⁡ u − sinh ⁡ v = 2 ⁢ cosh ⁡ ( u + v 2 ) ⁢ sinh ⁡ ( u − v 2 ) , 4.35.7 cosh ⁡ u + cosh ⁡ v = 2 ⁢ cosh ⁡ ( u + v 2 ) ⁢ cosh ⁡ ( u − v 2 ) , 4.35.8 cosh ⁡ u − cosh ⁡ v = 2 ⁢ sinh ⁡ ( u + v 2 ) ⁢ sinh ⁡ ( u − v 2 ) , 4.35.9 tanh ⁡ u ± tanh ⁡ v = sinh ⁡ ( u ± v ) cosh ⁡ u ⁢ cosh ⁡ v , 4.35.10 coth ⁡ u ± coth ⁡ v = sinh ⁡ ( v ± u ) sinh ⁡ u ⁢ sinh ⁡ v . §4.35(ii) Squares and Products 4.35.14 2 ⁢ sinh ⁡ u ⁢ sinh ⁡ v = cosh ⁡ ( u + v ) − cosh ⁡ ( u − v ) , 4.35.15 2 ⁢ cosh ⁡ u ⁢ cosh ⁡ v = cosh ⁡ ( u + v ) + cosh ⁡ ( u − v ) , 4.35.16 2 ⁢ sinh ⁡ u ⁢ cosh ⁡ v = sinh ⁡ ( u + v ) + sinh ⁡ ( u − v ) . 4.35.17 sinh2 ⁡ u − sinh2 ⁡ v = sinh ⁡ ( u + v ) ⁢ sinh ⁡ ( u − v ) , 4.35.18 cosh2 ⁡ u − cosh2 ⁡ v = sinh ⁡ ( u + v ) ⁢ sinh ⁡ ( u − v ) , 4.35.19 sinh2 ⁡ u + cosh2 ⁡ v = cosh ⁡ ( u + v ) ⁢ cosh ⁡ ( u − v ) . §4.35(iii) Multiples of the Argument

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

4.35.23 sinh ⁡ ( − z ) = − sinh ⁡ z , 4.35.24 cosh ⁡ ( − z ) = cosh ⁡ z , 4.35.25 tanh ⁡ ( − z ) = − tanh ⁡ z . §4.35(iv) Real and Imaginary Parts; Moduli

With z = x + i ⁢ y

4.35.34 sinh ⁡ z = sinh ⁡ x ⁢ cos ⁡ y + i ⁢ cosh ⁡ x ⁢ sin ⁡ y , 4.35.35 cosh ⁡ z = cosh ⁡ x ⁢ cos ⁡ y + i ⁢ sinh ⁡ x ⁢ sin ⁡ y , 4.35.36 tanh ⁡ z = sinh ⁡ ( 2 ⁢ x ) + i ⁢ sin ⁡ ( 2 ⁢ y ) cosh ⁡ ( 2 ⁢ x ) + cos ⁡ ( 2 ⁢ y ) , 4.35.37 coth ⁡ z = sinh ⁡ ( 2 ⁢ x ) − i ⁢ sin ⁡ ( 2 ⁢ y ) cosh ⁡ ( 2 ⁢ x ) − cos ⁡ ( 2 ⁢ y ) .

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