Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions defined using exponential functions. They have applications in physics, engineering, and calculus.

Definitions

Hyperbolic functions arise from the even/odd decomposition of the exponential function $e^x$:

$$e^x = \underbrace{\frac{e^x + e^{-x}}{2}}{\text{even}} + \underbrace{\frac{e^x - e^{-x}}{2}}{\text{odd}}$$

Hyperbolic Sine

$$\sinh x = \frac{e^x - e^{-x}}{2} = \frac{1}{2}e^x - \frac{1}{2}e^{-x}$$

  • Domain: $(-\infty, \infty)$
  • Range: $(-\infty, \infty)$
  • Parity: Odd ($\sinh(-x) = -\sinh x$)

Hyperbolic Cosine

$$\cosh x = \frac{e^x + e^{-x}}{2} = \frac{1}{2}e^x + \frac{1}{2}e^{-x}$$

  • Domain: $(-\infty, \infty)$
  • Range: $[1, \infty)$
  • Parity: Even ($\cosh(-x) = \cosh x$)

Hyperbolic Tangent

$$\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$$

  • Domain: $(-\infty, \infty)$
  • Range: $(-1, 1)$
  • Parity: Odd
  • Asymptotes: Horizontal asymptotes $y = -1$ and $y = 1$

Other Hyperbolic Functions

  • $\coth x = \frac{\cosh x}{\sinh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$
  • $\operatorname{sech} x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}}$
  • $\operatorname{cosech} x = \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}}$

Note: $\operatorname{cosech} x$ is also commonly written as $\operatorname{csch} x$.

Hyperbolic vs Trigonometric Comparison

flowchart LR
    subgraph Trig["Trigonometric"]
        direction TB
        T1["sin x"]
        T2["cos x"]
        T3["tan x"]
        T4["csc x"]
        T5["sec x"]
        T6["cot x"]
        TI["Identity: sin²x + cos²x = 1"]
    end
    subgraph Hyper["Hyperbolic"]
        direction TB
        H1["sinh x"]
        H2["cosh x"]
        H3["tanh x"]
        H4["csch x"]
        H5["sech x"]
        H6["coth x"]
        HI["Identity: cosh²x - sinh²x = 1"]
    end
    T1 -.->|Analog| H1
    T2 -.->|Analog| H2
    T3 -.->|Analog| H3
    T4 -.->|Analog| H4
    T5 -.->|Analog| H5
    T6 -.->|Analog| H6
    TI -.->|Sign Change| HI

Fundamental Identity

$$\cosh^2 x - \sinh^2 x = 1$$

Compare to: $\cos^2 x + \sin^2 x = 1$

Key Identities

  • $\sinh(-x) = -\sinh x$ (odd function)
  • $\cosh(-x) = \cosh x$ (even function)
  • $\cosh x + \sinh x = e^x$
  • $\cosh x - \sinh x = e^{-x}$
  • $1 - \tanh^2 x = \text{sech}^2 x$
  • $\coth^2 x - 1 = \text{csch}^2 x$

Hyperbolic Identities Flowchart

flowchart TD
    A["Fundamental: cosh²x - sinh²x = 1"] --> B["Divide by cosh²x"]
    A --> C["Divide by sinh²x"]
    B --> D["1 - tanh²x = sech²x"]
    C --> E["coth²x - 1 = csch²x"]
    A --> F["Addition Formulas"]
    F --> G["sinh(x±y) = sinh x cosh y ± cosh x sinh y"]
    F --> H["cosh(x±y) = cosh x cosh y ± sinh x sinh y"]
    F --> I["tanh(x±y) = (tanh x ± tanh y)/(1 ± tanh x tanh y)"]
    A --> J["Double Angle"]
    J --> K["sinh 2x = 2 sinh x cosh x"]
    J --> L["cosh 2x = cosh²x + sinh²x = 2cosh²x - 1 = 2sinh²x + 1"]
    A --> M["Half-Angle"]
    M --> N["cosh²x = (cosh 2x + 1)/2"]
    M --> O["sinh²x = (cosh 2x - 1)/2"]

Addition Formulas

  • $\sinh(x \pm y) = \sinh x \cosh y \pm \cosh x \sinh y$
  • $\cosh(x \pm y) = \cosh x \cosh y \pm \sinh x \sinh y$
  • $\tanh(x \pm y) = \frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y}$

Double Angle Formulas

  • $\sinh(2x) = 2\sinh x \cosh x$
  • $\cosh(2x) = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 2\sinh^2 x + 1$

Half-Angle (Power-Reduction) Formulas

  • $\cosh^2 x = \dfrac{\cosh 2x + 1}{2}$
  • $\sinh^2 x = \dfrac{\cosh 2x - 1}{2}$

Derivatives

Basic Derivatives

  • $\frac{d}{dx}\sinh x = \cosh x$
  • $\frac{d}{dx}\cosh x = \sinh x$
  • $\frac{d}{dx}\tanh x = \text{sech}^2 x$
  • $\frac{d}{dx}\coth x = -\text{csch}^2 x$
  • $\frac{d}{dx}\text{sech } x = -\text{sech } x \tanh x$
  • $\frac{d}{dx}\text{csch } x = -\text{csch } x \coth x$

Chain Rule Forms

If $u$ is a differentiable function of $x$:

  • $\frac{d}{dx}\sinh u = \cosh u \cdot \frac{du}{dx}$
  • $\frac{d}{dx}\cosh u = \sinh u \cdot \frac{du}{dx}$
  • $\frac{d}{dx}\tanh u = \text{sech}^2 u \cdot \frac{du}{dx}$
  • $\frac{d}{dx}\coth u = -\text{csch}^2 u \cdot \frac{du}{dx}$
  • $\frac{d}{dx}\text{sech } u = -\text{sech } u \tanh u \cdot \frac{du}{dx}$
  • $\frac{d}{dx}\text{csch } u = -\text{csch } u \coth u \cdot \frac{du}{dx}$

Integrals

Basic Integrals

  • $\int \sinh x , dx = \cosh x + C$
  • $\int \cosh x , dx = \sinh x + C$
  • $\int \text{sech}^2 x , dx = \tanh x + C$
  • $\int \text{csch}^2 x , dx = -\coth x + C$
  • $\int \text{sech } x \tanh x , dx = -\text{sech } x + C$
  • $\int \text{csch } x \coth x , dx = -\text{csch } x + C$

General Forms

  • $\int \sinh u , du = \cosh u + C$
  • $\int \cosh u , du = \sinh u + C$
  • $\int \text{sech}^2 u , du = \tanh u + C$
  • $\int \text{csch}^2 u , du = -\coth u + C$
  • $\int \text{sech } u \tanh u , du = -\text{sech } u + C$
  • $\int \text{csch } u \coth u , du = -\text{csch } u + C$

Common Hyperbolic Integrals by Substitution

  • $\int \tanh x , dx = \ln(\cosh x) + C$
    (Let $u = \cosh x$, $du = \sinh x , dx$)

Inverse Hyperbolic Functions

Definitions and Domain/Range

Inverse hyperbolic functions are established by reflecting the graphs of hyperbolic functions (with appropriate restrictions) about the line $y = x$.

Function Definition Domain Range
$\sinh^{-1} x$ Inverse of $\sinh x$ $(-\infty, \infty)$ $(-\infty, \infty)$
$\cosh^{-1} x$ Inverse of $\cosh x$ (principal branch) $(1, \infty)$ $[0, \infty)$
$\tanh^{-1} x$ Inverse of $\tanh x$ $(-1, 1)$ $(-\infty, \infty)$
$\coth^{-1} x$ Inverse of $\coth x$ $(-\infty, -1) \cup (1, \infty)$ $(-\infty, 0) \cup (0, \infty)$
$\text{sech}^{-1} x$ Inverse of $\text{sech } x$ (principal branch) $(0, 1)$ $[0, \infty)$
$\text{csch}^{-1} x$ Inverse of $\text{csch } x$ $(-\infty, 0) \cup (0, \infty)$ $(-\infty, 0) \cup (0, \infty)$

Note: For $\cosh^{-1} x$ and $\text{sech}^{-1} x$, the restriction $x \geq 0$ on the original function selects the principal branch, making the function invertible.

Logarithmic Forms

All inverse hyperbolic functions can be expressed using natural logarithms:

$$\sinh^{-1} x = \ln(x + \sqrt{x^2 + 1})$$

$$\cosh^{-1} x = \ln(x + \sqrt{x^2 - 1})$$

$$\tanh^{-1} x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$$

$$\coth^{-1} x = \frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)$$

$$\text{sech}^{-1} x = \ln\left(\frac{1 + \sqrt{1-x^2}}{x}\right)$$

$$\text{csch}^{-1} x = \ln\left(\frac{1}{x} + \frac{\sqrt{1+x^2}}{|x|}\right)$$

Derivatives

$f(x)$ $\frac{d}{dx}f(x)$ Domain
$\sinh^{-1} x$ $\dfrac{1}{\sqrt{1+x^2}}$ all $x$
$\cosh^{-1} x$ $\dfrac{1}{\sqrt{x^2-1}}$ $x > 1$
$\tanh^{-1} x$ $\dfrac{1}{1-x^2}$ $
$\coth^{-1} x$ $\dfrac{1}{1-x^2}$ $
$\text{sech}^{-1} x$ $\dfrac{-1}{x\sqrt{1-x^2}}$ $0 < x < 1$
$\text{csch}^{-1} x$ $\dfrac{-1}{ x

Note: $\frac{d}{dx}\tanh^{-1} x$ and $\frac{d}{dx}\coth^{-1} x$ have the same formula but apply on different domains.

Integrals Leading to Inverse Hyperbolic Functions

The following standard forms produce inverse hyperbolic results. They are often reached after a preliminary $u$-substitution.

  • Inverse sine hyperbolic form: $$\int \frac{dx}{\sqrt{a^2+x^2}} = \sinh^{-1}\left(\frac{x}{a}\right) + C = \ln\left|x + \sqrt{a^2+x^2}\right| + D$$

  • Inverse cosine hyperbolic form: $$\int \frac{dx}{\sqrt{x^2-a^2}} = \cosh^{-1}\left(\frac{x}{a}\right) + C$$

  • Inverse tangent/cotangent hyperbolic form (piecewise): $$\int \frac{dx}{a^2-x^2} = \begin{cases} \displaystyle\frac{1}{a}\tanh^{-1}\left(\frac{x}{a}\right) + C, & |x| < a \ \displaystyle\frac{1}{a}\coth^{-1}\left(\frac{x}{a}\right) + C, & |x| > a \end{cases}$$

  • Inverse cosecant hyperbolic form: $$\int \frac{dx}{x\sqrt{a^2+x^2}} = -\frac{1}{a}\operatorname{csch}^{-1}\left(\frac{x}{a}\right) + C$$

  • Inverse secant hyperbolic form: $$\int \frac{dx}{x\sqrt{a^2-x^2}} = -\frac{1}{a}\operatorname{sech}^{-1}\left(\frac{x}{a}\right) + C$$

Integration Techniques

$u$-Substitution with Hyperbolic Functions

When one factor is the derivative of another hyperbolic term, substitute:

  • $u = \sinh x$ when $\cosh x , dx$ is present
  • $u = \cosh x$ when $\sinh x , dx$ is present
  • $u = \tanh x$ when $\operatorname{sech}^2 x , dx$ is present
  • $u = \coth x$ when $\operatorname{csch}^2 x , dx$ is present

Trigonometric vs. Hyperbolic Substitution

For radical forms, either substitution works, but hyperbolic substitution often yields the inverse hyperbolic form directly:

Form Trig Substitution Hyperbolic Substitution
$\sqrt{a^2+x^2}$ $x = a\tan\theta$ $x = a\sinh u$
$\sqrt{x^2-a^2}$ $x = a\sec\theta$ $x = a\cosh u$

Example — $\displaystyle\int \frac{dx}{\sqrt{a^2+x^2}}$

  • With $x = a\sinh u$: $\displaystyle\int du = u + C = \sinh^{-1}!\left(\frac{x}{a}\right) + C$
  • With $x = a\tan\theta$: reduces to $\ln\bigl|x + \sqrt{a^2+x^2}\bigr| + D$

Both answers are equivalent because $\sinh^{-1}(x/a) = \ln\bigl|x + \sqrt{a^2+x^2}\bigr| - \ln a$.

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