Hyperbolic Functions (Real & Complex): A Step-by-Step Guide

As long as algebra is taught in school, there will be prayer in school, Cokie Roberts

Introduction

Hyperbolic functions are a family of mathematical functions that resemble or mirror many features of the trigonometric functions but arise from the geometry of hyperbolas rather than circles. Just as the circular functions $\cos\theta$ and $\sin\theta$ parametrize the unit circle $x^2 + y^2 = 1$ via $(\cos\theta, \sin\theta)$, the hyperbolic functions $\cosh t$ and $\sinh t$ parametrize the unit hyperbola $x^2 - y^2 = 1$ via $(\cosh t, \sinh t)$.

They are defined using the exponential functions $e^x$ and $e^{-x}$ and they are indeed essential in various fields, including calculus, differential equations (specifically heat transfer and wave equations), special relativity, complex analysis, and more.

Hyperbolic functions are defined via the exponential function. The six primary hyperbolic functions are:
Hyperbolic Sine: $\sinh(x) = \frac{e^x - e^{-x}}{2}$.
Hyperbolic Cosine: $\cosh(x) = \frac{e^x + e^{-x}}{2}$.
Tangent Hyperbolic: $\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$.
Reciprocal functions.
Hyperbolic Secant: $ \text{sech}(x) = \frac{1}{\cosh(x)}$.
Hyperbolic Cosecant: $\text{csch}(x) = \frac{1}{\sinh(x)}$/
Hyperbolic Cotangent: $\coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$

Properties

• Geometric interpretation. Just as $(\cos t, \sin t)$ represents a point on the unit circle $x^2+y^2=1$, the point $(\cosh t, \sinh t)$ traces the right branch of the unit hyperbola $x^2 - y^2 = 1$.
• Fundamental Identity: $\boxed{\cosh^2(x) - \sinh^2(x) = 1 \qquad \text{for all } x \in \mathbb{R}.}$
  $\cosh^2 x - \sinh^2 x = \left(\frac{e^x + e^{-x}}{2}\right)^2 - \left(\frac{e^x - e^{-x}}{2}\right)^2 = \frac{e^{2x} + 2 + e^{-2x}}{4} - \frac{e^{2x} - 2 + e^{-2x}}{4} = \frac{4}{4} = 1.$
  This is the hyperbolic analogue of $\cos^2\theta + \sin^2\theta = 1$. Note the crucial sign difference: subtraction rather than addition, reflecting the signature of the hyperbola $x^2 - y^2 = 1$ versus the circle $x^2 + y^2 = 1$.
• Pythagorean-Like Identities. Dividing the fundamental identity by $\cosh^2 x$ or $\sinh^2 x$: $1 - \tanh^2(x) = \text{sech}^2(x), \coth^2(x) - 1 = \text{csch}^2(x)$.
• Parity (Even/Odd Symmetry): $\sinh(-x) = -\sinh(x) \quad \text{(odd)}, \qquad \cosh(-x) = \cosh(x) \quad \text{(even)}.$ Consequently, $\tanh$ is odd, $\operatorname{sech}$ is even, $\operatorname{csch}$ and $\coth$ are odd.
  Proof ($\sinh$): $\sinh(-x) = \frac{e^{-x} - e^{x}}{2} = -\frac{e^x - e^{-x}}{2} = -\sinh(x)$.
• Addition formulas:: $\sinh(a \pm b) = \sinh a\cosh b \pm \cosh a\sinh b, \cosh(a \pm b) = \cosh a\cosh b \pm \sinh a\sinh b, \tanh(a \pm b) = \frac{\tanh a \pm \tanh b}{1 \pm \tanh a\tanh b}$.
• Double-angle formulas (setting $a = b = x$): $\sinh(2x) = 2\sinh x\cosh x, \cosh(2x) = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 1 + 2\sinh^2 x$.
  Compare these with the trigonometric versions: the only difference is occasional sign changes (e.g., $\cos(a+b) = \cos a\cos b - \sin a\sin b$ versus $\cosh(a+b) = \cosh a\cosh b + \sinh a\sinh b$).
• Series expansions (Maclaurin Series). Since $e^x = \sum \frac{x^n}{n!}$, we can derive: $\boxed{\sinh x=\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}}, \boxed{\cosh x=\sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}}$.
  Both series converge for all $x \in \mathbb{R}$ (and indeed for all $z \in \mathbb{C}$).
  $\sinh$ contains only odd powers (consistent with being an odd function).
  $\cosh$ contains only even powers (consistent with being an even function).
  These are identical to the series for $\sin$ and $\cos$ except that all signs are positive — no alternation.
• Derivatives. The derivatives cycle similarly to trigonometric functions but without the negative signs for sine and cosine.
  $ \frac{d}{dx} \sinh(x) = \cosh(x), \frac{d}{dx} \cosh(x) = \sinh(x), \frac{d}{dx} \tanh(x) = sech^2(x) = 1 - \tanh^2(x)$.
• Integrals: $\int \sinh(x) dx = \cosh(x) + C, \int \cosh(x) dx = \sinh(x) + C, \int \text{sech}^2(x) dx = \tanh(x) + C$.
• Differential equations. The functions $\cosh(x)$ and $\sinh(x)$ are intrinsically linked by their first derivatives: $\frac{d}{dx}\sinh(x) = \cosh(x)$ and $\frac{d}{dx}\cosh(x) = \sinh(x)$.

Because differentiating either function twice returns the original function ($y'' = y$) both are valid solutions to the linear second-order ordinary differential equation (ODE) $y'' - y = 0 \iff y'' = y$. Together, they form a fundamental set of solutions because they are linearly independent and satisfy two distinct, complementary initial conditions at $x = 0$:

1. For $y(x) = \cosh(x)$: The initial conditions are $y(0) = 1$ and $y'(0) = 0$. (It starts at 1 with a flat slope).
2. For $y(x) = \sinh(x)$: The initial conditions are $y(0) = 0$ and $y'(0) = 1$. (It starts at 0 with a slope of 1).

Because the ODE $y'' = y$ is a linear, homogeneous differential equation, any valid solution can be written as a linear combination of our fundamental set. This gives us the first form of the general solution: $y(x) = A \cosh(x) + B \sinh(x)$ where $A$ and $B$ are any real constants.

However, we also know that the exponential functions $e^x$ and $e^{-x}$ both satisfy $y'' = y$. Therefore, an equally valid way to write the exact same general solution is: $y(x) = C e^x + D e^{-x}$

These two general solutions are mathematically identical. We can bridge them using the explicit exponential definitions of the hyperbolic functions: $\cosh(x) = \frac{e^x + e^{-x}}{2}$ and $\sinh(x) = \frac{e^x - e^{-x}}{2}$.

Substituting these into the first general solution gives: $y(x) = A \left( \frac{e^x + e^{-x}}{2} \right) + B \left( \frac{e^x - e^{-x}}{2} \right) = \left( \frac{A + B}{2} \right) e^x + \left( \frac{A - B}{2} \right) e^{-x}$. This proves that the two forms are perfectly equivalent, where the constants are simply related by $C = \frac{A + B}{2}$ and $D = \frac{A - B}{2}$.

• Graphs, monotonicity, asymptotics. The graphs of hyperbolic functions are similar to those of trigonometric functions, but they have some distinct differences.

$y = \sinh(x)$:
It passes through the origin: $\sinh(0) = 0$.
It is smooth and monotonically (strictly) increasing on $\mathbb{R}$ (since $\sinh'(x) = \cosh(x) > 0$ everywhere). No local maxima or minima.
Unbounded range $(−\infty, \infty)$: $\sinh(x) \to +\infty$ as $x \to +\infty$ and $\sinh(x) \to -\infty$ as $x \to -\infty$.
Asymptotic behavior: $\sinh(x) \sim \frac{1}{2}e^x$ as $x \to +\infty$ and $\sinh(x) \sim -\frac{1}{2}e^{-x}$ as $x \to -\infty$.
Odd symmetry: $\sinh(-x) = -\sinh x$ (the graph is symmetric about the origin).

$y = \cosh(x)$:
Unique global minimum at $(0, 1)$ (y-intercept): $\cosh(x) \geq 1$ for all $x$. Derivative: $\frac{d}{dx}\cosh x = \sinh x$.
Strictly decreasing on $(-\infty, 0)$ ($\sinh x < 0$ for $x < 0$) and strictly increasing on $(0, \infty)$ ($\sinh x > 0$ for $x > 0$).
Even function: symmetric about the $y$-axis, $\cosh(-x) = \cosh x$.
Asymptotic: $\cosh(x) \sim \frac{1}{2}e^{|x|}$ as $|x| \to \infty$.
The graph of $y = \cosh(x)$ is a catenary — the U-shaped curve formed by a heavy, flexible chain or cable hanging freely under its own weight between two points. It has a single minimum at x = 0, and grows symmetrically like half of $e^{|x|}$ far from the origin.

For large $|x|$, one of the exponentials dominates: (i) $x \to +\infty$: $\cosh x \sim \frac{1}{2}e^{x}$. (ii) $x \to -\infty$: $e^{-x}$ dominates, so $\cosh x \sim \frac{1}{2}e^{-x} = \frac{1}{2}e^{|x|}$. In both directions, $\cosh x$ grows like $\frac{1}{2}e^{|x|}$ and tends to $+\infty$. Range: $[1, \infty)$.

$y = \tanh(x)$:
Passes through the origin: $\tanh(0) = 0$.
Derivative: $\frac{d}{dx}\tanh x = \operatorname{sech}^2 x = \frac{1}{\cosh^2 x} > 0$ for all $x$.
Strictly increasing on $\mathbb{R}$.
Bounded: $-1 < \tanh(x) < 1$ for all $x$.
Horizontal asymptotes: $y = 1$, $\lim_{x \to +\infty}\tanh(x) = 1$ from below; and $y = -1$ $\lim_{x \to -\infty}\tanh(x) = -1$ from above.
Odd symmetry: $\tanh(-x) = -\tanh x$.
Sigmoid-like (“S”-shaped) curve: it transitions smoothly from $-1$ to $+1$; it is a bounded, increasing sigmoid function widely used in machine learning (as an activation function) and in physics.

• Inverse hyperbolic functions. For real $x$, the inverses can be expressed using natural logarithms.
  $\operatorname{arsinh}(x) = \ln\!\left(x + \sqrt{x^2 + 1}\right) \qquad \text{for all } x \in \mathbb{R},$
  $\operatorname{arcosh}(x) = \ln\!\left(x + \sqrt{x^2 - 1}\right) \qquad \text{for } x \geq 1,$
  $\operatorname{artanh}(x) = \frac{1}{2}\ln\!\left(\frac{1+x}{1-x}\right) \qquad \text{for } |x| < 1.$

Derivation of $\operatorname{arsinh}$. Set $y = \sinh(x) = \frac{e^x - e^{-x}}{2}$. Then $2y = e^x - e^{-x}$. Multiplying by $e^x$: $2ye^x = e^{2x} - 1$, or $e^{2x} - 2ye^x - 1 = 0$. This is a quadratic in $e^x$, and the solution is $e^x = \frac{2y \pm \sqrt{4y^2 + 4}}{2} = y \pm \sqrt{y^2 + 1}.$

Since $e^x > 0$ and $\sqrt{y^2+1} > |y|$, only the $+$ sign gives a positive result: $e^x = y + \sqrt{y^2+1}$, so $\boxed{x = \ln(y + \sqrt{y^2+1})}$.

Derivatives: $\frac{d}{dx}\mathbb{arsinh}(x) = \frac{1}{\sqrt{x^2+1}}, \frac{d}{dx}\mathbb{arcosh}(x) = \frac{1}{\sqrt{x-1}\sqrt{x+1}}, \frac{d}{dx}\mathbb{artanh}(x) = \frac{1}{1-x^2}.$

Examples

1. These derivatives appear frequently in integration, e.g., $\int \frac{dx}{\sqrt{x^2+1}} = \operatorname{arsinh}(x) + C = \ln(x + \sqrt{x^2+1}) + C$.
2. $\sinh(2) = \frac{e^2 - e^{-2}}{2} = \frac{7.389 - 0.135}{2} \approx 3.627.$
3. Solve the equation $\cosh(x) = 2$.
   Using the inverse: $x = \pm\operatorname{arcosh}(2) = \pm\ln(2 + \sqrt{3}) \approx \pm 1.317$.
   Verification: $\cosh(1.317) = \frac{e^{1.317} + e^{-1.317}}{2} \approx \frac{3.732 + 0.268}{2} = 2.000$ ✓.
   The $\pm$ arises because $\cosh$ is an even function, if $\cosh(x_0) = 2$, then $\cosh(-x_0) = 2$ as well.
4. Solve the equation $\tanh(x) = 0.5$.
   $x = \operatorname{artanh}(0.5) = \frac{1}{2}\ln\!\left(\frac{1.5}{0.5}\right) = \frac{1}{2}\ln 3 \approx 0.549.$

Complex Hyperbolic functions

Hyperbolic functions are analogs of trigonometric functions (sine, cosine, etc.), but they are defined using the exponential function and relate to the geometry of a hyperbola rather than a circle. When extended to complex analysis, these functions exhibit fascinating properties, periodicity, and applications in physics, engineering, and pure mathematics.

The hyperbolic sine (sinh), hyperbolic cosine (cosh), and other related functions are defined using the exponential function $e^z$, where z is a complex number. For a complex variable z = x + iy, the definitions are:

$cosh(z) = \frac{e^z + e^{-z}}{2}, sinh(z) = \frac{e^z - e^{-z}}{2}, \tanh(z) = \frac{\sinh(z)}{\cosh(z)} = \frac{e^z - e^{-z}}{e^z + e^{-z}}$

Reciprocal functions. Hyperbolic Cotangent: $\coth(z) = \frac{\cosh(z)}{\sinh(z)}$, Hyperbolic Secant: $sech(z) = \frac{1}{\cosh(z)}$, Hyperbolic Cosecant: $csch(z) = \frac{1}{\sinh(z)}$

Properties of Complex Hyperbolic Functions

• sinh(z) and cosh(z) are entire functions, meaning they are analytic (differentiable) everywhere in the complex plane. This follows because $e^z$ is entire, and sums and scalar multiples of entire functions are entire.
• $\tanh(z)$, $\coth(z)$, $\operatorname{sech}(z)$, and $\operatorname{csch}(z)$ are meromorphic, meaning that they are holomorphic on all of their domains except for a set of isolated points (or poles, the zeros of their denominators).
• Zeros and Poles: The zeros of $\sinh$ and $\cosh$ are purely imaginary, inherited from the zeros of $\sin$ and $\cos$.
  $\sinh(z) = 0$ iff $e^z = e^{-z}$ iff $e^{2z} = 1$ iff $2z = 2\pi ik$ for $k \in \mathbb{Z}$:
  $\sinh(z) = 0 \quad\Longleftrightarrow\quad \boxed{z = i\pi k}, \qquad k \in \mathbb{Z}.$
  $\cosh(z) = 0$ iff $e^z = -e^{-z}$ iff $e^{2z} = -1$ iff $2z = i(\pi + 2\pi k)$:
  $\cosh(z) = 0 \quad\Longleftrightarrow\quad \boxed{z = i\!\left(\frac{\pi}{2} + \pi k\right)}, \qquad k \in \mathbb{Z}.$
  All zeros are simple (the derivatives are nonzero there).
  Poles of $\tanh(z)$. Simple poles at the zeros of $\cosh(z)$: $z = i(\frac{\pi}{2} + \pi k)$, $k \in \mathbb{Z}$.
  Poles of $\coth(z)$. Simple poles at the zeros of $\sinh(z)$: $z = i\pi k$, $k \in \mathbb{Z}$.
• Fundamental Identity: $\boxed{\cosh^2(z) - \sinh^2(z) = 1} \qquad \text{for all } z \in \mathbb{C}.$
• Periodicity. Hyperbolic functions like sinh (x), cosh (x), and tanh (x) do indeed exhibit periodic behavior in the imaginary direction, and the period is $2\pi i$.
  $\sinh(z + 2\pi i) = \sinh(z), \qquad \cosh(z + 2\pi i) = \cosh(z), \qquad \tanh(z + 2\pi i) = \tanh(z).$
  Proof for $\cosh$: $\cosh(z + 2\pi i) = \frac{e^{z + 2\pi i} + e^{-z - 2\pi i}}{2} = \frac{e^z e^{2\pi i} + e^{-z} e^{-2\pi i}}{2} =[\text{Since } e^{2\pi i} = e^{-2\pi i} = 1] = \frac{e^z + e^{-z}}{2} = \cosh(z)$.
  Note that $\tanh$ actually has the shorter period $\pi i$ (since $\sinh(z + \pi i) = -\sinh z$ and $\cosh(z + \pi i) = -\cosh z$, so their ratio is unchanged).
• Parity. $\sinh(-z) = -\sinh(z) \text{( odd function)}, \cosh(-z) = \cosh(z) \text{( even function)}$. These mirror the parity of sine and cosine in trigonometry.
• Bridge Identities. The hyperbolic and trigonometric functions are connected by evaluating at imaginary arguments. These “bridge identities” are among the most important relationships in complex analysis.

From the definitions: $\cosh(iz) = \frac{e^{iz} + e^{-iz}}{2} = \cos(z), \qquad \sinh(iz) = \frac{e^{iz} - e^{-iz}}{2} = \frac{2i\sin(z)}{2i} \cdot i = i\sin(z).$

$$\boxed{\begin{aligned} \cos(z) &= \cosh(iz), & \cosh(z) &= \cos(iz), \\ \sin(z) &= -i\sinh(iz), & \sinh(z) &= -i\sin(iz), \\ \tan(z) &= -i\tanh(iz), & \tanh(z) &= -i\tan(iz). \end{aligned}}$$

Trigonometric and hyperbolic functions are essentially the same functions, related by a rotation of $\pi/2$ in the complex plane (multiplication by $i$). The “oscillatory” behavior of $\cos$ and $\sin$ on the real axis transforms into the “exponential growth” behavior of $\cosh$ and $\sinh$ — and vice versa.

• Derivatives: $\frac{d}{dz}(cosh(z)) = sinh(z), \frac{d}{dz}(sinh(z)) = cosh(z), \frac{d}{dz}(\tanh(z)) = 1 - \tanh^2(z)$.
  These are identical to the real-variable formulas and hold for all $z \in \mathbb{C}$.

• Addition formulas: $\sinh(z_1 + z_2) = \sinh z_1\cosh z_2 + \cosh z_1\sinh z_2,$ $\cosh(z_1 + z_2) = \cosh z_1\cosh z_2 + \sinh z_1\sinh z_2.$
  Double-angle: $\sinh(2z) = 2\sinh z\cosh z, \qquad \cosh(2z) = \cosh^2 z + \sinh^2 z.$

• Real and Imaginary Parts. Using the addition formulas with $z = x + iy$: $\boxed{\cosh(x + iy) = \cosh(x)\cos(y) + i\sinh(x)\sin(y)}$, $\boxed{\sinh(x + iy) = \sinh(x)\cos(y) + i\cosh(x)\sin(y)}$.

Derivation for $\cosh$: $\cosh(x + iy) = \cosh x\cosh(iy) + \sinh x\sinh(iy) = \cosh x\cos y + i\sinh x\sin y,$ using $\cosh(iy) = \cos y$ and $\sinh(iy) = i\sin y$.

Connecting to the trigonometric decompositions. These formulas are consistent with the decompositions of $\cos(z)$ and $\sin(z)$ derived in the section on complex trigonometric functions. Indeed, the bridge identities guarantee: $\cos(x + iy) = \cosh(i(x+iy)) = \cosh(ix - y)$
Using: $\boxed{\cosh(x + iy) = \cosh(x)\cos(y) + i\sinh(x)\sin(y)}$, but now the real part is -y and the imaginary part is x, so:
$\cosh(-y)\cos(x) + i\sinh(-y)\sin(x) =[\cosh(-y) = \cosh(-y), \sinh(-y) = -\sinh(y)]$ Thus,
$\cos x\cosh y - i\sin x\sinh y.$
$Similarly, \sin (x+iy)=\sin x\cosh y+i\cos x\sinh y$.

• Taylor series expansion. Hyperbolic functions are entire functions (analytic everywhere in the complex plane), and their Taylor series around z = 0 are:
  $\sinh z = z + \frac{z^3}{3!} + \frac{z^5}{5!} + \frac{z^7}{7!} + \dots = \sum_{n=0}^{\infty}\frac{z^{2n+1}}{(2n+1)!}$.
  $\cosh z = 1 + \frac{z^2}{2!} + \frac{z^4}{4!} + \frac{z^6}{6!} + \dots = \sum_{n=0}^{\infty}\frac{z^{2n}}{(2n)!}$ valid for all $z \in \mathbb{C}$

The radius of convergence is $\infty$ (both are entire functions). Comparing with the trigonometric series: $\sin z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!}, \qquad \cos z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!}.$

The only difference is the absence of the alternating factor $(-1)^n$ in the hyperbolic series.