Formulas and Identities for Hyperbolic Functions
Definitions of Hyperbolic Functions
- Hyperbolic sine: \( \sinh x = \dfrac{e^x - e^{-x}}{2} \)
- Hyperbolic cosine: \( \cosh x = \dfrac{e^x + e^{-x}}{2} \)
- Hyperbolic tangent: \( \tanh x = \dfrac{\sinh x}{\cosh x} \)
- Hyperbolic cotangent: \( \coth x = \dfrac{1}{\tanh x} = \dfrac{\cosh x}{\sinh x} \)
- Hyperbolic secant: \( \operatorname{sech} x = \dfrac{1}{\cosh x} \)
- Hyperbolic cosecant: \( \operatorname{csch} x = \dfrac{1}{\sinh x} \)
Formulas for Negative Arguments
- \( \sinh (-x) = - \sinh x \) , odd function , graph symmetric with respect to the origin.
- \( \cosh (-x) = \cosh x \) , even function , graph symmetric with respect to the \( y \)-axis.
- \( \tanh (-x) = - \tanh x \) , odd function , graph symmetric with respect to the origin.
Hyperbolic Functions of Sums/Differences
- \( \cosh(A \pm B) = \cosh A \cosh B \pm \sinh A \sinh B \)
- \( \sinh(A \pm B) = \sinh A \cosh B \pm \cosh A \sinh B \)
- \( \tanh(A \pm B) = \dfrac{\tanh A \pm \tanh B}{1 \pm \tanh A \tanh B} \)
Sum of Hyperbolic Functions into Products of Hyperbolic Functions
- \( \sinh A + \sinh B = 2 \sinh \left( \dfrac {A+B}{2} \right) \cosh \left( \dfrac {A-B}{2} \right) \)
- \( \sinh A - \sinh B = 2 \sinh \left( \dfrac {A - B}{2} \right) \cosh \left( \dfrac {A + B}{2} \right) \)
- \( \cosh A + \cosh B = 2 \cosh \left( \dfrac {A+B}{2} \right) \cosh \left( \dfrac {A-B}{2} \right) \)
- \( \cosh A - \cosh B = 2 \sinh \left( \dfrac {A+B}{2} \right) \sinh \left( \dfrac {A-B}{2} \right) \)
Product of Hyperbolic Functions into Sum of Hyperbolic Functions
- \( \sinh A \, \cosh B = \dfrac{1}{2} \; [ \sinh(A+B) + \sinh(A-B) ] \)
- \( \cosh A \, \cosh B = \dfrac{1}{2} \; [ \cosh(A+B) + \cosh(A-B) ] \)
- \( \sinh A \, \sinh B = \dfrac{1}{2} \; [ \cosh(A+B) - \cosh(A-B) ] \)
Hyperbolic Functions of Multiple Arguments
- \( \sinh (2 A) = 2 \sinh A \cosh A \)
- \( \cosh (2 A) = 2 \sinh^2 A + 1 = 2 \cosh^2 A - 1 \)
- \( \sinh (3 A) = 4 \sinh^3 A + 3 \sinh A \)
- \( \cosh (3 A) = 4\cosh^3 A - 3 \cosh A \)
Relations Between Hyperbolic Functions
- \( \cosh^2 A - \sinh^2 A = 1 \)
- \( \tanh^2 A + \operatorname{sech}^2 A = 1 \)
- \( \coth^2 A - \operatorname{csch}^2 A = 1 \)
Power Reduction Identities
- \( \sinh^2 A = \dfrac{1}{2} [\cosh (2A) - 1] \)
- \( \cosh^2 A = \dfrac{1}{2} [\cosh (2A) + 1] \)
- \( \sinh^3 A = \dfrac{\sinh(3A) - 3 \sinh A}{4} \)
- \( \cosh^3 A = \dfrac{\cosh(3A) + 3 \cosh A}{4} \)
- \( \sinh^4 A = \dfrac{\cosh (4A) - 4 \cosh (2A) + 3}{8} \)
- \( \cosh^4 A = \dfrac{\cosh (4A) + 4 \cosh (2A) + 3}{8} \)
- \( \sinh^5 A = \dfrac{\sinh (5A) - 5 \sinh (3A) + 10 \sinh A}{16} \)
- \( \cosh^5 A = \dfrac{\cosh (5A) + 5 \cosh (3A) + 10 \cosh A}{16} \)
Inverse Hyperbolic Functions
- \( \sinh^{-1} x = \ln (x+\sqrt{x^2+1}) \)
- \( \cosh^{-1} x = \ln (x+\sqrt{x^2-1}) \; , \; x \ge 1 \)
- \( \tanh^{-1} x = \dfrac{1}{2}\ln \left(\dfrac{1+x}{1-x} \right) \; , \; -1 \lt x \lt 1 \)
- \( \operatorname{csch}^{-1} x = \ln \left(\dfrac{1}{x}+\sqrt{\dfrac{1}{x^2} + 1} \right) \; , \; x \ne 0 \)
- \( \operatorname{sech}^{-1} x = \ln \left(\dfrac{1}{x}+\sqrt{\dfrac{1}{x^2} - 1} \right) \; , \; 0 \lt x \le 1 \)
- \( \coth^{-1} x = \dfrac{1}{2}\ln \left(\dfrac{x+1}{x-1} \right) \; , \; |x| \gt 1 \)
Derivatives of Hyperbolic Functions and Their Inverses
\( f(x) \) | \( \dfrac{d f(x)}{dx} \) |
|---|---|
| \( \sinh x \) | \( \cosh x \) |
| \( \cosh x \) | \( \sinh x \) |
| \( \tanh x \) | \( \text{sech}^2 x \) |
| \( \coth x \) | \( - \text{csch}^2 x \) |
| \( \text{sech} \; x \) | \( -\text{sech} \; x \tanh x \) |
| \( \text{csch} \; x \) | \( - \text{csch} \; x \coth x\) |
| \( \sinh^{-1} x\) | \( \dfrac{1}{\sqrt{x^2+1}} \) |
| \( \cosh^{-1} x\) | \( \dfrac{1}{\sqrt{x^2-1}} \) |
| \( \tanh^{-1} x\) | \( \dfrac{1}{1-x^2} \) |
| \( \coth^{-1} x\) | \( \dfrac{1}{1-x^2} \) |
Indefinite Integrals of Hyperbolic Functions
The constant of integration is omitted here but should be added if necessary.
\( f(x) \) | \( \displaystyle \int f(x) dx \) |
|---|---|
| \( \sinh x \) | \( \cosh x \) |
| \( \cosh x \) | \( \sinh x \) |
| \( \tanh x \) | \( \ln( \cosh x) \) |
| \( \coth x \) | \( \ln( \sinh x) \) |
| \( \text{sech} \; x \) | \( 2 \tan^{-1}(e^x) \) |
| \( \text{csch} \; x \) | \( -\ln (\coth x + \text{csch}\; x) \) |
More References and Links
Handbook of Mathematical Functions
Mathematics for Engineers with Examples and Solutions