Hyperbolic Cosine (cosh) for Complex Numbers

Calculation of cosh(z) - hyperbolic function in the complex plane

Cosh Calculator

Hyperbolic Cosine

The hyperbolic cosine cosh(z) of a complex number z = x + yi combines hyperbolic and trigonometric functions. It grows exponentially and is closely related to the exponential function: \(\cosh(z) = \frac{e^z + e^{-z}}{2}\)

Argument z = x + yi
+
i
Calculation Result
cosh(z) =
cosh(z) grows exponentially for large |Re(z)| and can take very large values!

Cosh - Properties

Formula for Complex Numbers
\[\cosh(z) = \cosh(x)\cos(y) + i\sinh(x)\sin(y)\]

With z = x + yi

Exponential Representation
\[\cosh(z) = \frac{e^z + e^{-z}}{2}\]

Average of exponential functions

Even function cosh(-z) = cosh(z)
Minimum cosh(0) = 1
Important Properties
  • Even function: cosh(-z) = cosh(z)
  • \(\cosh^2(z) - \sinh^2(z) = 1\)
  • Minimum: cosh(0) = 1
  • Grows exponentially for |z| → ∞
Relations
  • \(\cosh(iz) = \cos(z)\)
  • \(\cosh(2z) = \cosh^2(z) + \sinh^2(z)\)
  • \(\cosh(z \pm w) = \cosh z \cosh w \pm \sinh z \sinh w\)
  • \(\frac{d}{dz}\cosh(z) = \sinh(z)\)

Formulas for Hyperbolic Cosine of Complex Numbers

The hyperbolic cosine cosh(z) of a complex number z = x + yi combines hyperbolic functions (cosh, sinh) with trigonometric functions (cos, sin).

Cartesian Form
\[\cosh(x + yi) = \cosh(x)\cos(y) + i\sinh(x)\sin(y)\]

Real part: \(\cosh(x)\cos(y)\)
Imaginary part: \(\sinh(x)\sin(y)\)

Exponential Form
\[\cosh(z) = \frac{e^z + e^{-z}}{2}\]

Average of exponential functions

Step-by-Step Example

Calculation: cosh(3 + 5i)
Step 1: Apply formula

z = 3 + 5i

x = 3 (real part)

y = 5 (imaginary part)

Step 2: Calculate real part

\(\text{Re} = \cosh(3) \cdot \cos(5)\)

\(= (10.0677) \cdot (0.28366)\)

\(\approx 2.856\)

Step 3: Calculate imaginary part

\(\text{Im} = \sinh(3) \cdot \sin(5)\)

\(= (10.0179) \cdot (-0.95892)\)

\(\approx -9.606\)

Step 4: Result

\(\cosh(3 + 5i) = \text{Re} + i\text{Im}\)

\(\approx 2.856 - 9.606i\)

Observation

The magnitude \(|\cosh(3 + 5i)| \approx 10.02\) is significantly larger than 1. The hyperbolic cosine grows exponentially with the real part x.

More Examples

Example 1: cosh(0)

z = 0

\(\cosh(0) = \frac{e^0 + e^{-0}}{2}\)

\(= \frac{1 + 1}{2} = 1\)

Example 2: cosh(1)

z = 1 (real)

\(\cosh(1) = \frac{e + e^{-1}}{2}\)

\(\approx 1.543\)

Example 3: cosh(i)

z = i (purely imaginary)

\(\cosh(i) = \cos(1)\)

\(\approx 0.540\)

Example 4: cosh(πi)

z = πi

\(\cosh(\pi i) = \cos(\pi)\)

\(= -1\)

Example 5: cosh(2 + i)

z = 2 + i

\(\text{Re} = \cosh(2)\cos(1) \approx 2.033\)
\(\text{Im} = \sinh(2)\sin(1) \approx 3.166\)

\(\approx 2.033 + 3.166i\)

Example 6: cosh(-2)

z = -2 (real, negative)

\(\cosh(-2) = \cosh(2)\) (even!)

\(\approx 3.762\)

Hyperbolic Cosine - Detailed Description

Definition

The hyperbolic cosine is one of the hyperbolic functions, analogous to the trigonometric cosine.

Exponential representation:
\[\cosh(z) = \frac{e^z + e^{-z}}{2}\]

For real numbers:
\[\cosh(x) = \frac{e^x + e^{-x}}{2}\]
Range: [1, ∞)
Minimum at x = 0: cosh(0) = 1

For Complex Numbers

Calculation with z = x + yi:

\[\cosh(z) = \cosh(x)\cos(y) + i\sinh(x)\sin(y)\]

• Real part: \(\cosh(x)\cos(y)\)
• Imaginary part: \(\sinh(x)\sin(y)\)
• Grows exponentially with |Re(z)|

Important Properties

  • Even function: \(\cosh(-z) = \cosh(z)\)
  • Hyperbolic identity: \(\cosh^2(z) - \sinh^2(z) = 1\)
  • Minimum: cosh(0) = 1
  • Derivative: \(\frac{d}{dz}\cosh(z) = \sinh(z)\)

Addition Formulas

Sum formula:
\[\cosh(z \pm w) = \cosh z \cosh w \pm \sinh z \sinh w\]
Double argument:
\[\cosh(2z) = \cosh^2(z) + \sinh^2(z)\]
\[= 2\cosh^2(z) - 1 = 2\sinh^2(z) + 1\]

Relation to Trigonometric Functions

• \(\cosh(iz) = \cos(z)\) (important connection!)
• \(\cos(iz) = \cosh(z)\) (inverse)
• \(e^z = \cosh(z) + \sinh(z)\)
• \(e^{-z} = \cosh(z) - \sinh(z)\)

Behavior and Growth

Exponential Growth

For large |x|:

\[\cosh(x) \approx \frac{e^{|x|}}{2}\]

The hyperbolic cosine grows exponentially!
e.g.: cosh(5) ≈ 74.2, cosh(10) ≈ 11013.2

Minimum Point

For real numbers, cosh has a minimum:

\[\min_{x \in \mathbb{R}} \cosh(x) = \cosh(0) = 1\]

For all x ∈ ℝ: cosh(x) ≥ 1

Applications

Mathematics
  • Hyperbolic geometry
  • Catenary (rope, chain)
  • Differential equations
  • Integral calculus
Physics
  • Relativity theory
  • Heat equation
  • Wave equations
  • Electromagnetism
Engineering
  • Bridge construction (suspension bridges)
  • Structural mechanics
  • Signal processing
  • Control engineering
The Catenary

The most famous application of the hyperbolic cosine is the catenary (also called catenoid):

\[y(x) = a \cosh\left(\frac{x}{a}\right)\]

This curve describes the shape of a freely hanging chain or rope under its own weight. It is optimal for suspension bridges and arches, as it distributes tension evenly.


More complex functions

Absolute value (abs) • Angle • Conjugate • Division • Exponent • Logarithm to base 10 • Multiplication • Natural logarithm • Polarform • Power • Root • Reciprocal • Square root •
Cosh • Sinh • Tanh •
Acos • Asin • Atan • Cos • Sin • Tan •
Airy function • Derivative Airy function •
Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye •