Hyperbolic Sine (sinh) for Complex Numbers

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

Sinh Calculator

Hyperbolic Sine

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

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

Sinh - Properties

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

With z = x + yi

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

Half-difference of exponential functions

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

Formulas for Hyperbolic Sine of Complex Numbers

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

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

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

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

Half-difference of exponential functions

Step-by-Step Example

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

z = 3 + 5i

x = 3 (real part)

y = 5 (imaginary part)

Step 2: Calculate real part

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

\(= (10.0179) \cdot (0.28366)\)

\(\approx 2.842\)

Step 3: Calculate imaginary part

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

\(= (10.0677) \cdot (-0.95892)\)

\(\approx -9.654\)

Step 4: Result

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

\(\approx 2.842 - 9.654i\)

Observation

The magnitude \(|\sinh(3 + 5i)| \approx 10.06\) shows the exponential growth. The hyperbolic sine grows exponentially with the real part x.

More Examples

Example 1: sinh(0)

z = 0

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

\(= \frac{1 - 1}{2} = 0\)

Example 2: sinh(1)

z = 1 (real)

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

\(\approx 1.175\)

Example 3: sinh(i)

z = i (purely imaginary)

\(\sinh(i) = i\sin(1)\)

\(\approx 0.841i\)

Example 4: sinh(πi/2)

z = πi/2

\(\sinh(\pi i/2) = i\sin(\pi/2)\)

\(= i\)

Example 5: sinh(2 + i)

z = 2 + i

\(\text{Re} = \sinh(2)\cos(1) \approx 1.978\)
\(\text{Im} = \cosh(2)\sin(1) \approx 3.166\)

\(\approx 1.978 + 3.166i\)

Example 6: sinh(-2)

z = -2 (real, negative)

\(\sinh(-2) = -\sinh(2)\) (odd!)

\(\approx -3.627\)

Hyperbolic Sine - Detailed Description

Definition

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

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

For real numbers:
\[\sinh(x) = \frac{e^x - e^{-x}}{2}\]
Range: (-∞, ∞)
Zero at x = 0: sinh(0) = 0

For Complex Numbers

Calculation with z = x + yi:

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

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

Important Properties

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

Addition Formulas

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

Relation to Trigonometric Functions

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

Behavior and Growth

Exponential Growth

For large |x|:

\[\sinh(x) \approx \frac{e^{x}}{2} \text{ sgn}(x)\]

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

Zero Point

For real numbers:

\[\sinh(0) = 0\]

The only real zero!
sinh is strictly monotonically increasing

Applications

Mathematics
  • Hyperbolic geometry
  • Differential equations
  • Integral calculus
  • Complex analysis
Physics
  • Relativity theory
  • Heat equation
  • Wave equations
  • Electromagnetism
Engineering
  • Structural mechanics
  • Signal processing
  • Control engineering
  • Vibration analysis
Comparison: sinh vs. sin
sinh(x) - Hyperbolic Sine:
  • Odd function: sinh(-x) = -sinh(x)
  • Not periodic
  • Exponential growth
  • Range: (-∞, ∞)
sin(x) - Trigonometric Sine:
  • Odd function: sin(-x) = -sin(x)
  • Periodic: Period 2π
  • Oscillating
  • Range: [-1, 1]

Connection: \(\sinh(iz) = i\sin(z)\) and \(\sin(iz) = i\sinh(z)\)


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 •