Hyperbolic Cotangent

Online calculator for calculating the hyperbolic cotangent of an angle

Coth Calculator

Zero Exclusion

The Coth(x) or hyperbolic cotangent shows reciprocal behavior with singularity at x = 0.

Enter any angle except 0 (undefined at zero)
Results
Coth:

Coth Function Curve

Curve of the Coth function

The Coth function has a vertical asymptote at x = 0 and horizontal asymptotes at y = ±1.
Domain: ℝ \ {0}, Range: (-∞, -1) ∪ (1, ∞)

Reciprocal Behavior of Coth

The hyperbolic cotangent function exhibits characteristic reciprocal properties:

  • Domain: ℝ \ {0} (all real numbers except 0)
  • Range: (-∞, -1) ∪ (1, ∞)
  • Singularity: Vertical asymptote at x = 0
  • Reciprocal: Coth(x) = 1/Tanh(x)
  • Symmetry: Odd function Coth(-x) = -Coth(x)
  • Asymptotes: Horizontal at y = ±1

Exponential Representation of Coth Function

The hyperbolic cotangent function is expressed through exponential functions:

Basic Formula
\[\text{Coth}(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}\]

Ratio of hyperbolic functions for x ≠ 0

Reciprocal Relation
\[\text{Coth}(x) = \frac{1}{\tanh(x)}\]

Reciprocal of hyperbolic tangent

Formulas for the Coth Function

Definition
\[\text{Coth}(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}}\]

Ratio of hyperbolic cosine to hyperbolic sine

Alternative Forms
\[\text{Coth}(x) = \frac{1}{\tanh(x)} = \frac{e^{2x} + 1}{e^{2x} - 1} = 1 + \frac{2}{e^{2x} - 1}\]

Various equivalent expressions

Derivative
\[\frac{d}{dx} \text{Coth}(x) = -\text{Csch}^2(x) = -\frac{1}{\sinh^2(x)}\]

Derivative involves hyperbolic cosecant

Symmetry Property
\[\text{Coth}(-x) = -\text{Coth}(x)\]

Odd function (antisymmetric)

Asymptotic Behavior
\[\lim_{x \to 0^+} \text{Coth}(x) = +\infty\] \[\lim_{x \to 0^-} \text{Coth}(x) = -\infty\] \[\lim_{x \to \pm\infty} \text{Coth}(x) = \pm 1\]

Vertical asymptote at origin, horizontal at ±1

Special Values

Important Values
Coth(1) ≈ 1.313 Coth(2) ≈ 1.037 Coth(ln(2)) = 1.4
Singularity
\[x = 0: \pm\infty\]

Function has vertical asymptote at x = 0

Range Restriction
\[\text{Coth}(x) \in (-\infty, -1) \cup (1, \infty)\]

Values never between -1 and 1

Properties
  • Reciprocal function
  • Odd function symmetry
  • Vertical asymptote at x = 0
  • Horizontal asymptotes at y = ±1
Reciprocal Relation
\[\text{Coth}(x) \cdot \text{Tanh}(x) = 1\]

Perfect reciprocal of hyperbolic tangent

Applications

Statistical mechanics, plasma physics, signal processing, mathematical analysis.

Detailed Description of the Coth Function

Definition and Input

The hyperbolic cotangent function Coth(x) is the ratio of hyperbolic cosine to hyperbolic sine. It exhibits characteristic reciprocal behavior as the reciprocal of the hyperbolic tangent function.

Critical Property: The Coth function has a singularity at x = 0!
Input

The angle is given in degrees (full circle = 360°) or radians (full circle = 2π). The unit of measurement used is set using the Degrees or Radians menu. Note: The function is undefined at x = 0.

Result

The result is always greater than 1 for positive arguments or less than -1 for negative arguments. The function never takes values between -1 and 1.

Using the Calculator

Enter any angle value except 0. The Coth function calculates the hyperbolic cotangent, which is the reciprocal of the hyperbolic tangent.

Mathematical Properties

Function Properties
  • Domain: ℝ \ {0} (all real numbers except 0)
  • Range: (-∞, -1) ∪ (1, ∞)
  • Singularity: Vertical asymptote at x = 0
  • Symmetry: Odd function Coth(-x) = -Coth(x)
Reciprocal Properties
  • Reciprocal of hyperbolic tangent function
  • Vertical asymptote at x = 0
  • Horizontal asymptotes at y = ±1
  • Two separate branches on either side of x = 0
Applications
  • Statistical Mechanics: Partition functions
  • Plasma Physics: Debye-Hückel theory
  • Signal Processing: Filter design
  • Mathematical Analysis: Special functions
Practical Notes
  • Singularity at x = 0: Function diverges to ±∞
  • Range restriction: |Coth(x)| > 1 for all x ≠ 0
  • Odd function: Coth(-x) = -Coth(x)
  • Reciprocal relation: Coth(x) = 1/Tanh(x)

Calculation Examples

Small Positive Values

Coth(0.5) ≈ 2.164

Coth(1) ≈ 1.313

Coth(2) ≈ 1.037

Negative Values

Coth(-0.5) ≈ -2.164

Coth(-1) ≈ -1.313

Coth(-2) ≈ -1.037

Asymptotic Behavior

x → 0⁺: Coth(x) → +∞

x → 0⁻: Coth(x) → -∞

x → ±∞: Coth(x) → ±1

Physics and Engineering Applications

Statistical Mechanics

Partition Functions:

Z = 2·sinh(βħω/2) → Coth(βħω/2)

Quantum harmonic oscillator

Application: Thermal properties of quantum systems.

Plasma Physics

Debye-Hückel Theory:

Screening in ionic solutions

Electrostatic potential calculations

Example: Ion interaction in plasma environments.

Important Mathematical Relationships

Reciprocal Function Properties
\[\text{Coth}(x) = \frac{1}{\tanh(x)}\] \[\text{Coth}(x) \cdot \tanh(x) = 1\]

Reciprocal Relationship: Perfect inverse of hyperbolic tangent.

Calculus Properties
\[\frac{d}{dx}\text{Coth}(x) = -\text{Csch}^2(x)\] \[\int \text{Coth}(x) dx = \ln|\sinh(x)| + C\]

Derivative: Negative cosecant squared.


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Hyperbolic functions

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