Hyperbolic Tangent (tanh) for Complex Numbers
Calculation of tanh(z) - ratio of sinh to cosh
Tanh Calculator
Hyperbolic Tangent
The hyperbolic tangent tanh(z) is the ratio of sinh to cosh: \(\tanh(z) = \frac{\sinh(z)}{\cosh(z)}\). It is bounded to the interval (-1, 1) for real numbers and has no poles.
Tanh - Properties
Formula for Complex Numbers
With z = x + yi
Quotient Representation
Important Properties
- Odd function: tanh(-z) = -tanh(z)
- \(1 - \tanh^2(z) = \frac{1}{\cosh^2(z)}\)
- No poles (smooth)
- Limits: \(\lim_{x \to \pm\infty} \tanh(x) = \pm 1\)
Relations
- \(\tanh(z) = \frac{1}{\coth(z)}\)
- \(\tanh(2z) = \frac{2\tanh(z)}{1+\tanh^2(z)}\)
- \(\tanh(z \pm w) = \frac{\tanh z \pm \tanh w}{1 \pm \tanh z \tanh w}\)
- \(\tanh(iz) = i\tan(z)\)
Formulas for Hyperbolic Tangent of Complex Numbers
The hyperbolic tangent tanh(z) of a complex number z = x + yi is the ratio of sinh to cosh and combines hyperbolic with trigonometric functions.
Cartesian Form
Real part: \(\frac{\sinh(2x)}{\cosh(2x)+\cos(2y)}\)
Imaginary part: \(\frac{\sin(2y)}{\cosh(2x)+\cos(2y)}\)
Quotient Representation
Ratio of sinh to cosh
Step-by-Step Example
Calculation: tanh(1 + i)
Step 1: Apply formula
z = 1 + i
x = 1, y = 1
2x = 2, 2y = 2
Step 2: Calculate denominator
\(\cosh(2) + \cos(2)\)
\(= 3.762 + (-0.416)\)
\(\approx 3.346\)
Step 3: Calculate real part
\(\text{Re} = \frac{\sinh(2)}{3.346}\)
\(= \frac{3.627}{3.346}\)
\(\approx 1.084\)
Step 4: Calculate imaginary part
\(\text{Im} = \frac{\sin(2)}{3.346}\)
\(= \frac{0.909}{3.346}\)
\(\approx 0.272\)
Step 5: Result
\(\tanh(1 + i) = \text{Re} + i\text{Im}\)
\(\approx 1.084 + 0.272i\)
Observation
Note: \(|\tanh(1 + i)| \approx 1.118 > 1\)! For complex numbers, the magnitude of tanh can be greater than 1 (only for real x is |tanh(x)| < 1).
More Examples
Example 1: tanh(0)
z = 0
\(\tanh(0) = \frac{\sinh(0)}{\cosh(0)}\)
\(= \frac{0}{1} = 0\)
Example 2: tanh(1)
z = 1 (real)
\(\tanh(1) = \frac{e - e^{-1}}{e + e^{-1}}\)
\(\approx 0.762\)
Example 3: tanh(∞)
Limit for x → ∞
\(\lim_{x \to \infty} \tanh(x)\)
\(= 1\)
Example 4: tanh(i)
z = i (purely imaginary)
\(\tanh(i) = i\tan(1)\)
\(\approx 1.557i\)
Example 5: tanh(2)
z = 2 (real)
\(\tanh(2) = \frac{\sinh(2)}{\cosh(2)}\)
\(\approx 0.964\)
Example 6: tanh(-1)
z = -1 (real, negative)
\(\tanh(-1) = -\tanh(1)\) (odd!)
\(\approx -0.762\)
Hyperbolic Tangent - Detailed Description
Definition
The hyperbolic tangent is the ratio of sinh to cosh.
\[\tanh(z) = \frac{\sinh(z)}{\cosh(z)}\]
Exponential form:
\[\tanh(z) = \frac{e^z - e^{-z}}{e^z + e^{-z}}\]
For real numbers:
Range: (-1, 1)
No poles
For Complex Numbers
Calculation with z = x + yi:
• For complex z, |\tanh(z)| can be > 1
• No poles (smooth function)
Important Properties
- Odd function: \(\tanh(-z) = -\tanh(z)\)
- Identity: \(1 - \tanh^2(z) = \frac{1}{\cosh^2(z)}\)
- Limits: \(\lim_{x \to \pm\infty} \tanh(x) = \pm 1\)
- Derivative: \(\frac{d}{dz}\tanh(z) = \frac{1}{\cosh^2(z)}\)
Addition Formulas
\[\tanh(z \pm w) = \frac{\tanh z \pm \tanh w}{1 \pm \tanh z \tanh w}\]
Double argument:
\[\tanh(2z) = \frac{2\tanh(z)}{1+\tanh^2(z)}\]
Relations
• \(\tanh(z) = \frac{1}{\coth(z)}\) (reciprocal)
• \(\tanh(iz) = i\tan(z)\) (hyperbolic ↔ trigonometric)
• \(\tan(iz) = i\tanh(z)\) (inverse)
• \(\tanh(z) = \frac{e^{2z} - 1}{e^{2z} + 1}\)
Sigmoid Function
Activation Function
The hyperbolic tangent is a sigmoid function (S-shaped):
- Smooth and differentiable
- Monotonically increasing
- Zero point at (0,0)
- Asymptotes at y = ±1
- Maximum slope at x = 0
- \(\tanh'(0) = 1\)
- Symmetric about origin
Applications
Neural Networks
- Activation function
- Machine learning
- Deep learning
- AI algorithms
Physics
- Relativity theory
- Solitons
- Wave equations
- Nonlinear dynamics
Signal Processing
- Normalization
- Limiting
- Soft clipping
- Compression
Comparison: tanh vs. tan
- Odd function
- Range (reall