Complex Trigonometric Functions

Definition

Complex trigonometric functions are extensions of the real sine and cosine functions to the complex plane, defined in terms of the complex exponential function.

• How to read: “Trigonometric functions evaluated for complex arguments.”
• Meaning: Generalizing circular motion to include growth and decay components in the complex plane.

Why It Matters

These functions allow for the analysis of wave phenomena with damping or gain, which are common in physics and engineering. They reveal deep connections between circular and hyperbolic functions, providing a unified framework for solving differential equations.

Core Concepts

• Exponential Definitions: $\cos z = \frac{e^{iz} + e^{-iz}}{2}, \quad \sin z = \frac{e^{iz} - e^{-iz}}{2i}$
  • How to read: “The cosine of z equals the quantity e to the i z plus e to the negative i z, all over two; the sine of z equals their difference all over two i.”
  • Meaning: Defining trig functions via complex exponentials allows them to accept any complex number $z$.
• Unboundedness: Unlike real sine and cosine, which are bounded between $-1$ and $1$, complex trigonometric functions are unbounded and grow exponentially as the imaginary part of $z$ increases.
• Hyperbolic Connection: $\cos(iy) = \cosh y, \quad \sin(iy) = i\sinh y$
  • How to read: “The cosine of i y equals the hyperbolic cosine of y, and the sine of i y equals i times the hyperbolic sine of y.”
  • Meaning: Rotating a real argument by $90^\circ$ into the imaginary axis transforms circular functions into hyperbolic functions.

Connected Concepts

• Complex Exponential
• Trigonometric Functions: Any Angle
• Hyperbolic Functions