Formulas: Hyperbolic Functions #
Hyperbolic Functions #
1. Hyperbolic Sine: \( \pmb{\sinh(x) = \frac{e^x – e^{-x}}{2}} \)
2. Hyperbolic Cosine: \( \pmb{\cosh(x) = \frac{e^x + e^{-x}}{2}} \)
3. Hyperbolic Tangent: \( \pmb{\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x – e^{-x}}{e^x + e^{-x}}} \)
4. Hyperbolic Cotangent: \( \pmb{\coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x – e^{-x}}} \)
5. Hyperbolic Secant: \( \pmb{\text{sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}}} \)
6. Hyperbolic Cosecant: \( \pmb{\text{csch}(x) = \frac{1}{\sinh(x)} = \frac{2}{e^x – e^{-x}}} \)
Identities of Hyperbolic Functions #
1. \( \pmb{\sinh(x \pm y) = \sinh(x) \cosh(y) \pm \cosh(x) \sinh(y)} \)
2. \( \pmb{\cosh(x \pm y) = \cosh(x) \cosh(y) \pm \sinh(x) \sinh(y)} \)
3. \( \pmb{\tanh(x \pm y) = \frac{\tanh(x) \pm \tanh(y)}{1 \pm \tanh(x) \tanh(y)}} \)
4. \( \pmb{\coth(x \pm y) = \frac{\coth(x) \coth(y) \pm 1}{\coth(y) \pm \coth(x)}} \)
5. \( \pmb{\cosh^2(x) – \sinh^2(x) = 1} \)
6. \( \pmb{\tanh^2(x) + \text{sech}^2(x) = 1} \)
7. \( \pmb{\coth^2(x) – \text{csch}^2(x) = 1} \)
8. \( \pmb{2 \sinh(x) \cosh(y) = \sinh(x + y) + \sinh(x – y)} \)
9. \( \pmb{2 \cosh(x) \sinh(y) = \sinh(x + y) – \sinh(x – y)} \)
10. \( \pmb{2 \sinh(x) \sinh(y) = \cosh(x + y) – \cosh(x – y)} \)
11. \( \pmb{2 \cosh(x) \cosh(y) = \cosh(x + y) + \cosh(x – y)} \)
12. \( \pmb{\sinh(x) – \sinh(y) = 2 \sinh \left(\frac{x – y}{2}\right) \cosh \left(\frac{x + y}{2}\right)} \)
13. \( \pmb{\sinh(x) + \sinh(y) = 2 \sinh \left(\frac{x + y}{2}\right) \cosh \left(\frac{x – y}{2}\right)} \)
14. \( \pmb{\cosh(x) + \cosh(y) = 2 \cosh \left(\frac{x + y}{2}\right) \cosh \left(\frac{x – y}{2}\right)} \)
15. \( \pmb{\cosh(x) – \cosh(y) = 2 \sinh \left(\frac{x + y}{2}\right) \sinh \left(\frac{x – y}{2}\right)} \)
Derivatives of Hyperbolic Functions #
1. \( \pmb{\frac{d}{dx}[\sinh(x)] = \cosh(x)} \)
2. \( \pmb{\frac{d}{dx}[\cosh(x)] = \sinh(x)} \)
3. \( \pmb{\frac{d}{dx}[\tanh(x)] = \text{sech}^2(x)} \)
4. \( \pmb{\frac{d}{dx}[\coth(x)] = -\text{csch}^2(x)} \)
5. \( \pmb{\frac{d}{dx}[\text{sech}(x)] = -\text{sech}(x)\tanh(x)} \)
6. \( \pmb{\frac{d}{dx}[\text{csch}(x)] = -\text{csch}(x)\coth(x)} \)
Derivatives of Inverse Hyperbolic Functions #
1. \( \pmb{\frac{d}{dx}[\text{arsinh}(x)] = \frac{1}{\sqrt{x^2 + 1}}} \)
2. \( \pmb{\frac{d}{dx}[\text{arcosh}(x)] = \frac{1}{\sqrt{x^2 – 1}}} \)
3. \( \pmb{\frac{d}{dx}[\text{artanh}(x)] = \frac{1}{1 – x^2}} \)
4. \( \pmb{\frac{d}{dx}[\text{arcoth}(x)] = \frac{1}{1 – x^2}} \) (for \( \pmb{|x| > 1} \))
5. \( \pmb{\frac{d}{dx}[\text{arsech}(x)] = \frac{-1}{x\sqrt{1 – x^2}}} \)
6. \( \pmb{\frac{d}{dx}[\text{arcsch}(x)] = \frac{-1}{|x|\sqrt{1 + x^2}}} \)
Integrals of Hyperbolic Functions #
1. \( \pmb{\int \sinh(x) \, dx = \cosh(x) + C} \)
2. \( \pmb{\int \cosh(x) \, dx = \sinh(x) + C} \)
3. \( \pmb{\int \text{sech}^2(x) \, dx = \tanh(x) + C} \)
4. \( \pmb{\int \text{csch}^2(x) \, dx = -\coth(x) + C} \)
5. \( \pmb{\int \text{sech}(x) \, dx = 2\arctan[\tanh\left(\frac{x}{2}\right)] + C} \)
6. \( \pmb{\int \text{csch}(x) \, dx = \ln\left|\tanh\left(\frac{x}{2}\right)\right| + C} \)
Inverse Hyperbolic Functions #
\( \pmb{\text{arsinh}(x) = \ln\left(x + \sqrt{x^2 + 1}\right)} \)
\( \pmb{\text{arcosh}(x) = \ln\left(x + \sqrt{x^2 – 1}\right)} \), for \( \pmb{x \geq 1} \)
\( \pmb{\text{artanh}(x) = \frac{1}{2} \ln\left(\frac{1 + x}{1 – x}\right)} \), for \( \pmb{|x| \lt 1} \)
\( \pmb{\text{arcoth}(x) = \frac{1}{2} \ln\left|\frac{x + 1}{x – 1}\right|} \), for \( \pmb{|x| > 1} \)
\( \pmb{\text{arsech}(x) = \ln\left(x + \sqrt{x^2 – 1}\right)} \), for \( \pmb{0 \lt x \leq 1} \)
\( \pmb{\text{arcsch}(x) = \ln\left|x + \sqrt{x^2 + 1}\right|} \)
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